# Noncooperative Herding With Control Barrier Functions:
Theory and Experiments

Jaskaran Grover∗1, Nishant Mohanty∗1, Wenhao Luo2, Changliu Liu1, Katia
Sycara1 1J. Grover, N. Mohanty, C. Liu, K. Sycara are with the Robotics
Institute at Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA
15213, USA<EMAIL_ADDRESS>Luo is with the
Department of Computer Science, University of North Carolina at Charlotte.
<EMAIL_ADDRESS>Equal Contribution

###### Abstract

In this paper, we consider the problem of protecting a high-value unit from
inadvertent attack by a group of agents using defending robots. Specifically,
we develop a control strategy for the defending agents that we call “dog
robots” to prevent a flock of “sheep agents” from breaching a protected zone.
We take recourse to control barrier functions to pose this problem and exploit
the interaction dynamics between the sheep and dogs to find dogs’ velocities
that result in the sheep getting repelled from the zone. We solve a QP
reactively that incorporates the defending constraints to compute the desired
velocities for all dogs. Owing to this, our proposed framework is composable
i.e. it allows for simultaneous inclusion of multiple protected zones in the
constraints on dog robots’ velocities. We provide a theoretical proof of
feasibility of our strategy for the one dog/one sheep case. Additionally, we
provide empirical results of two dogs defending the protected zone from upto
ten sheep averaged over a hundred simulations and report high success rates.
We also demonstrate this algorithm experimentally on non-holonomic robots.
Videos of these results are available at https://tinyurl.com/4dj2kjwx.

##  I Introduction

In the last decade, multi-robot systems (MRSs) have advanced from being
researched in labs to being deployed in the real-world for solving practical
problems [1, 2], [3]. The redundancy offered by an aggregated system provides
resilience to faults and distributed acquisition of information. Several
control algorithms have been developed that make multiple robots come together
to solve team-level, global tasks using local interaction rules [4, 5]. These
algorithms are (a) local (i.e. individual robots act on information locally
available to them), (b) safe (i.e. result in collision-free motions amongst
robots) and (c) emergent (i.e. global properties result from using local
interaction rules) [6].

These characteristics can be treated as the insider’s perspective i.e.
principles borne in mind by the control engineer when programming their own
robots for a given task. Complementary to this is the outsider’s perspective
i.e. the perspective of an external agent watching a group carry out a task by
executing motions consistent with these characteristics [7]. Viewing the the
motion of a group from the vantage point of an external observer is equally
important. For example, if group is adversarial, potentially by posing a
threat to a high-value unit, then the observer must predict the group’s motion
and conscript robots (the defenders) to defend the unit [8, 9]. This requires
the observer to orchestrate motions for their robots to prevent breach of the
high-value unit. In this paper, we investigate how to develop provably correct
control inputs for a group of defenders (“dog robots”) to prevent another
group (the “sheep agents”) from breaching a protected zone. This is a
challenging problem because the dog robots cannot directly command the
actuators of the sheep agents, they must rely on their interaction dynamics
(collision-avoidance behavior) with the sheep agents to influence the sheep’s
behavior. This results in a non-collocated control problem. Additionally, this
is also challenging because usually there are not as many defending robots as
agents in the herd. Therefore, from the perspective of the dog robots, the
control problem can become highly underactuated.

(a) Preventing breaching of protected zone

(b) Preventing escape from protected zone

Figure 1: Demonstration of our results showing (a) how to prevent sheep (red)
from breaching a protect zone (green) and (b) preventing sheep (red) from
escaping the protected zone using dog robots (blue).

In this paper, we investigate how to solve this problem using ideas from
control barrier functions. Specifically, we develop a centralized control
technique that computes velocity inputs for the dog robots to ensure that the
sheep agents do not breach a protected zone. We convert these requirements to
constraints on the velocities of the dog robots. Our proposed framework is
compositional in nature i.e. we can consider more sheep as well as more
protected zones by just adding more constraints on the velocities.
Furthermore, our approach relies on using automatic differentiation and
symbolic computation tools, owing to which, we can easily change behavioral
requirements from the sheep. For example, instead of preventing them from
breaching a protected zone (Fig. 1(a)), we can prevent them from escaping a
zone (Fig. 1(b)). We provide numerical results showing the success of our
approach for multiple dogs v/s multiple sheep agents. Additionally, to test
the repeatability of our algorithm, we conduct Monte Carlo simulations with
increasing number of dogs and sheep averaged over 100 runs each and show high-
success rates. Finally, we demonstrate our algorithm on real robots and
demonstrate that we can prevent breaching of multiple zones from two sheep
using one dog robot.

The outline of this paper is as follows: in section II we briefly review the
prior work in this area. In section III, we give a mathematical formulation of
the problem statement. In section IV, we show how to use control barrier
functions to derive constraints on velocities of dog robots to pose the
requirement of defense against the sheep. We consider additional collision
avoidance constraints on dog robots’ velocities. In section V, we provide both
simulation as well as experimental results demonstrating our approach.
Finally, we summarize in section VI with directions for future work.

##  II Prior Work

Influencing group behavior has applications beyond just the adversarial
context. For example, shepherding behaviors, specifically, are one class of
flocking behaviors in which one or more external agents (called shepherds)
attempt to control the motion of another group of agents (called a flock) by
exerting repulsive forces on them [10, 11]. A successful practical
demonstration of robotic herding was achieved in the Robot Sheepdog Project
[12, 13]. Here an autonomous wheeled mobile robot (the external
agent/shepherd) was used to gather a flock of ducks and manoeuvred them to a
specified goal position.

Several prior works have considered the problem of noncooperative shepherding
using robots. Some of these include [11],[14],[15],[16],[17],[18]. They refer
to the shepherding problem as noncooperative because the flock agents are not
necessarily adversarial i.e. they do not work against the robots, but at the
same time are not cooperative because the flock agents repel from the robots.
These works exploit this repulsive interaction to develop feedback controllers
for the robots to steer the flock agents to a designated region. While
successful, one issue common among these works is that they fail to consider
the self-motivated dynamics of the flock agents i.e. their nominal dynamics
without any robots in the picture. As a result, the flock agents’ motions are
solely driven by repulsions from the robots. Additionally, these approaches
tend to be handcrafted for generating a specific behavior in the sheep, for
example: herding to a given location. Finally, many papers do not consider
scalability with respect to the number of agents.

Differently from prior work, we do not omit the self-motivated dynamics in the
sheeps’ motions. We synthesize the inputs for the dogs while considering
cohesion, inter-sheep and dog/sheep repulsions and the sheep agents’
attraction to their goal (the self-motivated part). Moreover, our proposed
approach uses control barrier functions which only requires expected behaviors
from sheep to be expressed as symbolic functions. Using automatic
differentiation, we can generate constraints on dog velocities for any given
behavioral requirement from the sheep. Lastly, in our Monte Carlo study, we
obtain high success rates even when there are many more sheep than the number
of dog robots in the system. This provides an empirical evidence of
scalability of our approach.

##  III Problem Formulation

Suppose there are $n$ “sheep” agents (the herd) and $m$ dog robots (the
defenders). We assume that the sheep are exhibiting flocking dynamics i.e.
moving towards a common goal while staying close enough to each other and
repelling from the dogs. Given this dynamics, it is possible that while en-
route to their goal, they end up breaching a high-value unit i.e. the
protected zone. From the perspective of the dogs, the sheep represent a non-
cooperative group because they are not intentionally aiming towards the
protected zone but may inadvertently end up breaching it. Therefore, the
objective of the dog robots is to steer the sheep away from the protected
zone. Let us pose this requirement mathematically.

Denote the position of the $i^{th}$ sheep as
$\boldsymbol{x}_{S_{i}}\in\mathbb{R}^{2}$ and the collective positions of the
herd as
$\boldsymbol{x}^{all}_{S}\coloneqq(\boldsymbol{x}_{S_{1}},\boldsymbol{x}_{S_{2}},...,\boldsymbol{x}_{S_{n}})$.
Likewise, we denote the position of the $k^{th}$ dog as
$\boldsymbol{x}_{D_{k}}\in\mathbb{R}^{2}$ and the collective positions of the
defending robots’ group as
$\boldsymbol{x}^{all}_{D}\coloneqq(\boldsymbol{x}_{D_{1}},\boldsymbol{x}_{D_{2}},...,\boldsymbol{x}_{D_{m}})$.
We assume both sheep and dogs have single-integrator dynamics i.e. they are
velocity controlled. For the $i^{th}$ sheep, we have:

$\displaystyle\dot{\boldsymbol{x}}_{S_{i}}$
$\displaystyle=\boldsymbol{u}_{S_{i}}$ (1)
$\displaystyle=k_{S}\sum_{j\in\mathcal{S}}\left(1-\frac{R_{S}^{3}}{\left\lVert\boldsymbol{x}_{S_{j}}-\boldsymbol{x}_{S_{i}}\right\rVert^{3}}\right)(\boldsymbol{x}_{S_{j}}-\boldsymbol{x}_{S_{i}})$
$\displaystyle+k_{G}\left(\boldsymbol{x}_{G}-\boldsymbol{x}_{S_{i}}\right)+k_{D}\sum_{k\in\mathcal{D}}\frac{\boldsymbol{x}_{S_{i}}-\boldsymbol{x}_{D_{k}}}{\left\lVert\boldsymbol{x}_{S_{i}}-\boldsymbol{x}_{D_{k}}\right\rVert^{3}}$
$\displaystyle\coloneqq\boldsymbol{f}_{i}(\boldsymbol{x}_{S_{1}},...,\boldsymbol{x}_{S_{n}},\boldsymbol{x}_{D_{1}},...,\boldsymbol{x}_{D_{n}})$
(2)

Here the first term represents cohesion of the flock, the second represents
attraction to goal and the third represents repulsion from dog robots. The
attraction to the goal represents the self-motivated part of the dynamics of
the sheep agents. This term is often neglected in prior work. $R_{S}$ is the
safety radius for sheep $i$ to avoid collisions with the other sheep, $x_{G}$
is its desired goal position (common for all sheep) and $k_{S},k_{G},k_{D}$
are proportional gains corresponding to forces in the dynamics. For each dog
we have:

$\displaystyle\dot{\boldsymbol{x}}_{D_{k}}=\boldsymbol{u}_{D_{k}}\hskip
5.69046pt\forall k\in\\{1,2,\cdots,m\\}$ (3)

We denote the protected zone as $\mathcal{P}\subset\mathbb{R}^{2}$ and for
this paper, assume that it is a disc centered at $\boldsymbol{x}_{p}$ and
radius $R_{p}$:

$\displaystyle\mathcal{P}\coloneqq\\{\boldsymbol{x}\in\mathbb{R}^{2}|\left\lVert\boldsymbol{x}-\boldsymbol{x}_{p}\right\rVert\leq
R_{p}\\}$ (4)

We denote the set excluding the protected zone as
$\mathcal{P}^{c}\coloneqq\mathbb{R}^{2}\backslash\mathcal{P}$. The sheep are
assumed to have no knowledge about the presence of $\mathcal{P}$. The dog
robots need to ensure that the sheep remain in $\mathcal{P}^{c}$ if they are
initially in $\mathcal{P}^{c}$ by finding suitable control inputs
$\\{\boldsymbol{u}_{D_{1}},\cdots,\boldsymbol{u}_{D_{m}}\\}$. We make the
following assumption on the dog’s knowledge before posing the problem:

######  Assumption 1.

The dog robots have knowledge about the sheep’s dynamics i.e. (1) and can
measure the sheep’s positions accurately.

This is not a stringent assumption because if the dynamics are unknown, the
dog robots can learn the dynamics online using multiagent system
identification algorithms, some of which we have developed in our prior work
[19, 20] and use certainty equivalence to design the controllers. We can pose
the dog robots’ problem as follows:

######  Definition 1.

Assuming that the initial positions of the sheep
$\boldsymbol{x}^{all}_{S}(0)\in\mathcal{P}^{c}$, the dog robots’ problem is to
synthesize controls
$\\{\boldsymbol{u}_{D_{1}},\cdots,\boldsymbol{u}_{D_{m}}\\}$ such that
$\boldsymbol{x}^{all}_{S}(t)\in\mathcal{P}^{c}$ $\forall t\geq 0$. If
$\boldsymbol{x}^{all}_{S}(0)\notin\mathcal{P}^{c}$, the dog robots’ problem is
to synthesize controls
$\\{\boldsymbol{u}_{D_{1}},\cdots,\boldsymbol{u}_{D_{m}}\\}$ such that
$\boldsymbol{x}^{all}_{S}(t)\leadsto\mathcal{P}^{c}$ in a finite time.

Additionally, we require that the dog robots never collide with the sheep. In
the next section, we show how to address this problem using control barrier
functions.

##  IV Controller Design

In this section, we discuss our proposed approach to solve the problem of
defending the protected zone as stated before. Given the protected zone as
defined (4), we first pose the requirement for defending against one sheep,
say sheep $i$ located at $\boldsymbol{x}_{S_{i}}$. Subsequently, we will
generalize this to the rest of the sheep in the herd. For this sheep, define a
safety index $h(\cdot):\mathbb{R}^{2}\longrightarrow\mathbb{R}$ as follows:

$\displaystyle
h=\left\lVert\boldsymbol{x}_{S_{i}}-\boldsymbol{x}_{p}\right\rVert^{2}-R_{p}^{2}$
(5)

By construction, $h\geq 0$ $\forall\boldsymbol{x}_{S_{i}}\in\mathcal{P}^{c}$
i.e. non-negative whenever $i$ is on the boundary or outside the protected
zone. Thus, assuming that at $t=0$, $h(\boldsymbol{x}_{S_{i}}(0))\geq 0$, we
require $h(\boldsymbol{x}_{S_{i}}(t))\geq 0$ $\forall t\geq 0$. Treating
$h(\cdot)$ as a control barrier function [21], this can be achieved if the
derivative of $h(\cdot)$ satisfies the following constraint:

$\displaystyle\dot{h}(\boldsymbol{x}_{S_{1}},\cdots,\boldsymbol{x}_{S_{n}},\boldsymbol{x}_{D_{1}},\cdots,\boldsymbol{x}_{D_{m}})+p_{1}h(\boldsymbol{x}_{S_{i}})\geq
0$ $\displaystyle\implies$ $\displaystyle
2(\boldsymbol{x}_{S_{i}}-\boldsymbol{x}_{p})^{T}\dot{\boldsymbol{x}}_{S_{i}}+p_{1}h(\boldsymbol{x}_{S_{i}})\geq
0$ $\displaystyle\implies$ $\displaystyle
2(\boldsymbol{x}_{S_{i}}-\boldsymbol{x}_{p})^{T}\boldsymbol{f}_{i}+p_{1}h(\boldsymbol{x}_{S_{i}})\geq
0$ (6)

Define $\boldsymbol{x}=(\boldsymbol{x}^{all}_{S},\boldsymbol{x}^{all}_{D})$,
we rewrite this as

$\displaystyle
2(\boldsymbol{x}_{S_{i}}-\boldsymbol{x}_{p})^{T}\boldsymbol{f}_{i}(\boldsymbol{x})+p_{1}h(\boldsymbol{x}_{S_{i}})\geq
0$ (7)

Here $p_{1}$ is a design parameter that we choose to ensure that

$\displaystyle p_{1}>0\quad\text{and}\quad
p_{1}>-\frac{\dot{h}(\boldsymbol{x}(0))}{h(\boldsymbol{x}(0))}$ (8)

The first condition on $p_{1}$ requires that the pole is real and negative.
The second depends on the initial positions $\boldsymbol{x}(0)$ of all the
sheep and dogs relative to the protected zone. Now while (IV) depends on the
positions of the sheep and dogs, it is the velocities of the dogs that are
directly controllable not their positions (3). Since
$\boldsymbol{u}^{all}_{D}$ does not show up in (IV), we define another
function $v(\cdot):\mathbb{R}^{2(m+n)}\longrightarrow\mathbb{R}$:

$\displaystyle v=\dot{h}+p_{1}h$ (9)

Like before, in order to ensure $v\geq 0$ is always maintained, its derivative
needs to satisfy

$\displaystyle\dot{v}(\boldsymbol{x})+p_{2}v(\boldsymbol{x})\geq 0.$ (10)

Here $p_{2}$ is another design parameter which we choose $p_{2}$ to ensure
that the following is satisfied at $t=0$

$\displaystyle p_{2}>0\quad\text{and}\quad
p_{2}>-\frac{\ddot{h}(\boldsymbol{x}(0))+p_{1}\dot{h}(\boldsymbol{x}(0))}{\dot{h}(\boldsymbol{x}(0))+p_{1}h(\boldsymbol{x}(0))}$
(11)

Using (9) in (10), we get:

$\displaystyle\ddot{h}(\boldsymbol{x})+(p_{1}+p_{2})\dot{h}(\boldsymbol{x})+p_{1}p_{2}h(\boldsymbol{x})\geq
0$
$\displaystyle\implies\ddot{h}(\boldsymbol{x})+\alpha\dot{h}(\boldsymbol{x})+\beta
h(\boldsymbol{x})\geq 0$ (12)

where we have defined $\alpha\coloneqq p_{1}+p_{2}$ and $\beta\coloneqq
p_{1}p_{2}$. The time derivatives of the control-barrier function $h(\cdot)$
required in (IV) are obtained as:

$\displaystyle\dot{h}(\boldsymbol{x})$
$\displaystyle=2(\boldsymbol{x}_{S_{i}}-\boldsymbol{x}_{P})^{T}\dot{\boldsymbol{x}}_{S_{i}}$
$\displaystyle=2(\boldsymbol{x}_{S_{i}}-\boldsymbol{x}_{P})^{T}\boldsymbol{f}_{i}(\boldsymbol{x}_{S_{1}},\cdots,\boldsymbol{x}_{S_{n}},\boldsymbol{x}_{D_{1}},\cdots,\boldsymbol{x}_{D_{m}})$
(13) $\displaystyle\ddot{h}(\boldsymbol{x})$
$\displaystyle=2\dot{\boldsymbol{x}}_{S_{i}}^{T}\dot{\boldsymbol{x}}_{S_{i}}$
$\displaystyle+2(\boldsymbol{x}_{S_{i}}-\boldsymbol{x}_{P})^{T}\bigg{(}\sum_{j=1}^{n}\mathbb{J}_{ji}^{S}\dot{\boldsymbol{x}}_{S_{i}}+\sum_{k=1}^{m}\mathbb{J}_{ki}^{D}\boldsymbol{u}_{D_{k}}\bigg{)}$
$\displaystyle=2\boldsymbol{f}^{T}_{i}\boldsymbol{f}_{i}$
$\displaystyle+2(\boldsymbol{x}_{S_{i}}-\boldsymbol{x}_{P})^{T}\bigg{(}\sum_{j=1}^{n}\mathbb{J}_{ji}^{S}\boldsymbol{f}_{i}+\sum_{k=1}^{m}\mathbb{J}_{ki}^{D}\boldsymbol{u}_{D_{k}}\bigg{)}$
(14)

where $\mathbb{J}^{S}_{ji}$ and $\mathbb{J}^{D}_{ki}$ are

$\displaystyle\mathbb{J}^{S}_{ji}$
$\displaystyle\coloneqq\nabla_{\boldsymbol{x}_{S_{j}}}\boldsymbol{f}_{i}(\boldsymbol{x}_{S_{1}},\cdots,\boldsymbol{x}_{S_{n}},\boldsymbol{x}_{D_{1}},\cdots,\boldsymbol{x}_{D_{m}})$
$\displaystyle\mathbb{J}^{D}_{ki}$
$\displaystyle\coloneqq\nabla_{\boldsymbol{x}_{D_{k}}}\boldsymbol{f}_{i}(\boldsymbol{x}_{S_{1}},\cdots,\boldsymbol{x}_{S_{n}},\boldsymbol{x}_{D_{1}},\cdots,\boldsymbol{x}_{D_{m}})$

Note here that $\ddot{h}(\boldsymbol{x})$ contains the velocities of dogs as
we wanted. Using (IV) and (IV) in (IV), we get the following linear
constraints on dog velocities to ensure that the $i^{th}$ sheep stays outside
the protected zone $\mathcal{P}$:

$\displaystyle A^{H}_{i}\boldsymbol{u}^{all}_{D}\leq b^{H}_{i},\hskip
14.22636pt\mbox{where}$ (15) $\displaystyle A^{H}_{i}$
$\displaystyle\coloneqq(\boldsymbol{x}_{P}-\boldsymbol{x}_{S_{i}})^{T}\begin{bmatrix}\mathbb{J}_{1i}^{D}&\mathbb{J}_{2i}^{D}&.....&\mathbb{J}_{mi}^{D}\end{bmatrix}$
$\displaystyle b^{H}_{i}$
$\displaystyle\coloneqq\boldsymbol{f}^{T}_{i}\boldsymbol{f}_{i}+(\boldsymbol{x}_{S_{i}}-\boldsymbol{x}_{P})^{T}\sum_{j=1}^{n}\mathbb{J}_{ji}^{S}\boldsymbol{f}_{j}$
$\displaystyle+\alpha(\boldsymbol{x}_{S_{i}}-\boldsymbol{x}_{P})^{T}\boldsymbol{f}_{i}+\beta\frac{h}{2}$

To ensure all $n$ sheep stay away from $\mathcal{P}$, we compose constraints
(15) for all the herd as follows:

$\displaystyle\left[\begin{matrix}A^{H}_{1}\\\ \vdots\\\
A^{H}_{n}\end{matrix}\right]\boldsymbol{u}^{all}_{D}\leq\left[\begin{matrix}b^{H}_{1}\\\
\vdots\\\
b^{H}_{n}\end{matrix}\right]\implies\mathcal{A}^{H}\boldsymbol{u}^{all}_{D}\leq\boldsymbol{b}^{H}$
(16)

Here $\mathcal{A}^{H}\in\mathbb{R}^{n\times 2m}$ and
$\boldsymbol{b}^{H}\in\mathbb{R}^{n}$. Given these constraints on the dogs’
velocities, we can pose the following QP that searches for the min-norm
velocities that satisfies these constraints

$\displaystyle\boldsymbol{u}^{*all}_{D}=\underset{\boldsymbol{u}^{all}_{D}}{\arg\min}\left\lVert\boldsymbol{u}^{all}_{D}\right\rVert^{2}$
$\displaystyle\text{subject
to}\quad\mathcal{A}^{H}\boldsymbol{u}^{all}_{D}\leq\boldsymbol{b}^{H}$ (17)

Here $\boldsymbol{u}^{*all}_{D}$ are the optimal velocities for all the dog
robots to ensure both defending $\mathcal{P}$ and collision avoidance
simultaneously. By construction, our approach is centralized i.e. it computes
velocities of all dog robots together. Future work will consider ways to
decentralize this approach.
Considering multiple protected zones: While in the above derivation, we
considered preventing the sheep from breaching only one protected zone, we can
just as easily consider another protected zone by formulating similar
constraints
$\mathcal{A}^{H}_{2}\boldsymbol{u}^{all}_{D}\leq\boldsymbol{b}^{H}_{2}$ on the
dogs’ velocities. By augmenting (IV) with these constraints for the other
zone, we will be able to defend both zones from all sheep simultaneously. This
compositionality is a benefit offered by our constraint based framework. An
experimental validation of this is shown in Fig. 5. In the following
discussion, we prove that for the one dog v/s one sheep case, (IV) is always
feasible:

######  Theorem 1.

If there is one dog and one sheep, then (IV) always has a solution.

###### Proof.

Let the position of the dog be $\boldsymbol{x}_{D}$ and that of the sheep be
$\boldsymbol{x}_{S}$. The sheep dynamics can be simplified to

$\displaystyle\dot{\boldsymbol{x}}_{S}=\boldsymbol{f}(\boldsymbol{x}_{S},\boldsymbol{x}_{D})=k_{G}\left(\boldsymbol{x}_{G}-\boldsymbol{x}_{S}\right)+k_{D}\frac{\boldsymbol{x}_{S}-\boldsymbol{x}_{D}}{\left\lVert\boldsymbol{x}_{S}-\boldsymbol{x}_{D}\right\rVert^{3}}$
(18)

The only case when (IV) does not have a solution is when the defending
constraint is infeasible i.e. when $A^{H}\boldsymbol{u}_{D}\leq b^{H}$ is
infeasible. This can occur when

  * •

either when $A^{H}=\boldsymbol{0}$ and $b^{H}<0$ (possibility 1)

  * •

or when $b^{H}=-\infty$ (possibility 2).

For this case $A^{H}$ is:

$\displaystyle
A^{H}=(\boldsymbol{x}_{P}-\boldsymbol{x}_{S})^{T}\mathbb{J}_{11}^{D}$ (19)

Thus, if $\mathbb{J}_{11}^{D}$ is non-singular,
$(\boldsymbol{x}_{P}-\boldsymbol{x}_{S})^{T}\mathbb{J}_{11}^{D}\neq\boldsymbol{0}$.
From our calculations, we find that the determinant of $\mathbb{J}_{11}^{D}$
is

$\displaystyle
det(\mathbb{J}_{11}^{D})=\frac{-2k_{D}^{2}}{\left\lVert\boldsymbol{x}_{D}-\boldsymbol{x}_{S}\right\rVert^{3}}$
(20)

As long as the distance between the dog and the sheep is finite,
$det(\mathbb{J}_{11}^{D})$ is always non zero. Thus, there exists no null
space for the jacobian matrix $\mathbb{J}_{11}^{D}$. This implies
$A^{H}\neq\boldsymbol{0}$
$\forall\boldsymbol{x}_{S}\in\mathbb{R}^{n},\boldsymbol{x}_{D}\in\mathbb{R}^{2}$.
This rules out possibility 1 for infeasibility. For possibility 2, we need to
examine when does $b^{H}\longrightarrow-\infty$. The expression for $b^{H}$
is:

$\displaystyle
b^{H}=\boldsymbol{f}^{T}\boldsymbol{f}+(\boldsymbol{x}_{S}-\boldsymbol{x}_{P})^{T}\mathbb{J}_{11}^{S}\boldsymbol{f}+\alpha(\boldsymbol{x}_{S}-\boldsymbol{x}_{P})^{T}\boldsymbol{f}+\beta\frac{h}{2}$

We want to find the worst case lower bound of $b^{H}$. Here
$\boldsymbol{f}^{T}\boldsymbol{f}\geq 0$ always. We assume that at the current
time step, the sheep is outside the $\mathcal{P}$, this ensures
$\beta\frac{h}{2}\geq 0$.

######  Assumption 2.

Assume that the following bounds hold
$\left\lVert\boldsymbol{x}_{S}-\boldsymbol{x}_{G}\right\rVert\leq
M_{1},\left\lVert\boldsymbol{x}_{S}-\boldsymbol{x}_{P}\right\rVert\leq M_{2}$
and $\left\lVert\boldsymbol{x}_{S}-\boldsymbol{x}_{D}\right\rVert\geq M_{3}$.

With these assumptions, we can lower bound $b^{H}$ as follows:

$\displaystyle
b^{H}\geq(\boldsymbol{x}_{S}-\boldsymbol{x}_{P})^{T}\mathbb{J}_{11}^{S}\boldsymbol{f}+\alpha(\boldsymbol{x}_{S}-\boldsymbol{x}_{P})^{T}\boldsymbol{f}$
$\displaystyle\geq-(\sigma_{max}(\mathbb{J}_{11})+\alpha)\left\lVert\boldsymbol{f}\right\rVert\left\lVert\boldsymbol{x}_{S}-\boldsymbol{x}_{P}\right\rVert$
$\displaystyle\geq-(\sigma_{F}(\mathbb{J}_{11})+\alpha)\left\lVert\boldsymbol{f}\right\rVert\left\lVert\boldsymbol{x}_{S}-\boldsymbol{x}_{P}\right\rVert$
(21)

Here $\left\lVert\boldsymbol{f}\right\rVert\leq
k_{G}\left\lVert\boldsymbol{x}_{S}-\boldsymbol{x}_{G}\right\rVert+\frac{k_{D}}{\left\lVert\boldsymbol{x}_{S}-\boldsymbol{x}_{D}\right\rVert^{2}}$
using triangle inequality on (18). This gives
$\left\lVert\boldsymbol{x}_{S}-\boldsymbol{x}_{P}\right\rVert\left\lVert\boldsymbol{f}\right\rVert\leq
k_{G}M_{1}M_{2}+\frac{k_{D}M_{2}}{M_{3}^{2}}$. We can show that
$\sigma_{F}(\mathbb{J}_{11})\leq\lambda_{M}\coloneqq\sqrt{2k_{G}^{2}+5\frac{k^{2}_{D}}{M^{6}_{3}}+\frac{2k_{G}k_{D}}{M_{3}^{2}}}$.
Thus, using this, we obtain the following lower bound for $b^{H}$

$\displaystyle
b^{H}\geq-(\lambda_{M}+\alpha)\bigg{(}k_{G}M_{1}M_{2}+\frac{k_{D}M_{2}}{M_{3}^{2}}\bigg{)}$
(22)

This shows that $b^{H}$ is lower bounded and thus does not reach $-\infty$.
Hence possibility 2 is also ruled out. Thus, (IV) is always feasible. ∎

###  IV-A Incorporating collision avoidance constraints

The defending constraints we posed above do not guarantee that the dog robots
won’t collide with the sheep. Even though the sheep dynamics have repulsions
from the dogs, the velocities computed can result in aggressive behavior.
Thus, we augment the defending constraints with additional constraints to
ensure collision free behavior. Following the approach in [22], we define a
pairwise safety index
$b^{ik}(\cdot):\mathbb{R}^{2}\times\mathbb{R}^{2m}\longrightarrow\mathbb{R}$
as:

$\displaystyle
b^{ik}(\boldsymbol{x}_{S_{i}},\boldsymbol{x}_{D_{1}},\cdots,\boldsymbol{x}_{D_{m}})$
$\displaystyle=\left\lVert\boldsymbol{x}_{S_{i}}-\boldsymbol{x}_{D_{k}}\right\rVert^{2}-R_{S}^{2}$
$\displaystyle=\left\lVert\boldsymbol{x}_{S_{i}}-C_{k}\boldsymbol{x}^{all}_{D}\right\rVert^{2}-R_{S}^{2}$

$b^{ik}(\cdot)\geq 0$ iff dog $k$ is atleast $R_{S}$ distance away from sheep
$i$. Here $C_{k}$ is a matrix defined appropriately to extract the position of
the $k^{th}$ dog from $\boldsymbol{x}^{all}_{D}$. If
$b^{ik}(\boldsymbol{x}_{S_{i}}(0),\boldsymbol{x}^{all}_{D}(0))\geq 0$ $\forall
k\in\\{1,2,\cdots,m\\}$, we would like to ensure that
$b^{ik}(\boldsymbol{x}_{S_{i}}(t),\boldsymbol{x}^{all}_{D}(t))\geq 0$ $\forall
t\geq 0$ and $\forall k\in\\{1,2,\cdots,m\\}$. This can be achieved by
requiring that

$\displaystyle\dot{b}^{ik}(\boldsymbol{x})+\gamma b^{ik}(\boldsymbol{x})\geq
0\hskip 14.22636pt\forall k\in\\{1,2,\cdots,m\\}$ (23)

where $\gamma>0$. This gives us a total of $m$ linear constraints on the
velocity of the dog robots for avoiding collisions with the $i^{th}$ sheep:

$\displaystyle A^{C}_{i}\boldsymbol{u}^{all}_{D}\leq b^{C}_{i}$ (24)

where,

$\displaystyle A^{C}_{i}$
$\displaystyle=\left[\begin{matrix}(\boldsymbol{x}_{S_{i}}-\boldsymbol{x}_{D_{1}})^{T}C_{1}\\\
\vdots\\\
(\boldsymbol{x}_{S_{i}}-\boldsymbol{x}_{D_{m}})^{T}C_{m}\end{matrix}\right]$
$\displaystyle b^{C}_{i}$
$\displaystyle=\left[\begin{matrix}\frac{\gamma}{2}b^{i1}+(\boldsymbol{x}_{S_{i}}-\boldsymbol{x}_{D_{1}})^{T}\boldsymbol{f}_{i}\\\
\vdots\\\
\frac{\gamma}{2}b^{im}+(\boldsymbol{x}_{S_{i}}-\boldsymbol{x}_{D_{m}})^{T}\boldsymbol{f}_{i}\end{matrix}\right]$
(25)

To ensure all collision avoidance with all $n$ sheep111inter-dog collision
avoidance constraints can also be added following a similar procedure., we
compose constraints (24) for all the herd as follows:

$\displaystyle\left[\begin{matrix}A^{C}_{1}\\\ \vdots\\\
A^{C}_{n}\end{matrix}\right]\boldsymbol{u}^{all}_{D}\leq\left[\begin{matrix}b^{C}_{1}\\\
\vdots\\\
b^{C}_{n}\end{matrix}\right]\implies\mathcal{A}^{C}\boldsymbol{u}^{all}_{D}\leq\boldsymbol{b}^{C}$
(26)

Given the defending (16) and collision avoidance (26) constraints on the dogs’
velocities, we compose them together using the following QP:

$\displaystyle\boldsymbol{u}^{*all}_{D}=\underset{\boldsymbol{u}^{all}_{D}}{\arg\min}\left\lVert\boldsymbol{u}^{all}_{D}\right\rVert^{2}$
(27) $\displaystyle\text{subject
to}\quad\mathcal{A}^{H}\boldsymbol{u}^{all}_{D}\leq\boldsymbol{b}^{H}$
$\displaystyle\mathcal{A}^{C}\boldsymbol{u}^{all}_{D}\leq\boldsymbol{b}^{C}$

Here $\boldsymbol{u}^{*all}_{D}$ are the optimal velocities for all the dog
robots to ensure both defending and collision avoidance simultaneously. The
cost function penalizes the total speed of the dog robots, thus encouraging
them to minimize their movement.

##  V Results

In this section, we show results of our approach by testing it on different
scenarios consisting of varying numbers of sheep and dog and varying their
initial positions. Additionally, we also run validate these results
experimentally. We perform several experiments with nonholonomic Khepera
robots and demonstrate how our algorithm find velocities for one dog to
simultaneously defend multiple protected zones from multiple sheep.

(a) Three dog robots v/s three sheep robots.

(b) Three dog robots v/s five sheep.

(c) Three dog robots v/s three sheep robots.

Figure 2: Preventing the breaching of the protected zone. In these
simulations, the dog is shown in blue and the sheep is shown in red. The green
disc represents the protected zone. The nominal task of the red agent is to go
straight towards its goal $\boldsymbol{x}_{G}$. However, since this would
result in infiltration of the protected zone, the dog intervenes using the
control algorithm presented in (27). In 2(c), we defend two protected zones
from three sheep.

(a) $t=0s$

(b) $t=3s$

(c) $t=20s$

(d) $t=55s$

Figure 3: Hardware experiment with one dog robot preventing one sheep from the
breaching of the protected zone. The dog robot is highlighted in blue and the
sheep in red. The goal position $x_{G}$ is at the center of the protected zone
and given as a black solid circle. The nominal task of the sheep is to go
straight towards its goal $\boldsymbol{x}_{G}$. However, since this would
result in infiltration of the protected zone, the dog intervenes using the
control algorithm presented in (IV). Video at https://tinyurl.com/2p9fjeft.

(a) $t=0s$

(b) $t=6s$

(c) $t=26s$

(d) $t=35s$

Figure 4: Hardware experiment with one dog robot preventing two sheep from the
breaching of the protected zone. The dog robot is highlighted with a blue box
and sheep using a red box. The goal position $\boldsymbol{x}_{G}$ is at the
center of the protected zone and shown as a solid black dot. Video at
https://tinyurl.com/37rduh43.

(a) $t=0s$

(b) $t=7s$

(c) $t=10s$

(d) $t=14s$

Figure 5: Hardware experiment with one dog robot preventing two sheep from the
breaching of two protected zones. The goal lies in the left most protected
zone. Video at https://tinyurl.com/ycuyhwe6.

###  V-A Numerical Simulation

We represent the protected zone using a circular disc with radius $R_{p}$ and
its center at the origin i.e. $\boldsymbol{x}_{P}=\boldsymbol{0}$. In our
simulations, we purposefully choose the agent’s goal
$\boldsymbol{x}_{G}=\boldsymbol{x}_{P}$ so that the sheep are motivated to
breach the protected zone should the dog robots not interfere. Thus, this is
an adversarial scenario. The initial position $\boldsymbol{x}_{S_{i}}(0)$ of
all the sheep are chosen such that they are all close to each other. This is
done to ensure that the sheep have enough time to stabilize/cohese as a flock
before interacting with the dog robots. The initial position
$\boldsymbol{x}^{all}_{D}(0)$ of the dog robots are chosen randomly within the
area of operation. The sheep’s velocities are calculated using (1). The values
of the gains in the sheep dynamics were taken as $k_{G}=1$, $k_{S}=0.3$ and
$k_{D}=0.08$.

The velocities of the dog robot was obtained using eqn. (27). The
hyperparameters $\alpha,\beta,\gamma$ are tuned satisfy the conditions on the
design parameters (8, 11) . Figure 2 shows three simulation results for this
behavior. In these simulations, we varied the initial position of the sheep
(blue), the dog (red), number of sheep and the number of dogs. As can be
noticed from the pictures, in all three scenarios, the dogs robots able to
successfully intercept the sheep and prevent them from entering the protected
zone while also avoiding collision with the sheep.

###  V-B Monte Carlo Simulations

We further study the performance of the proposed control strategy by using
Monte Carlo simulations with varying initial configurations and varying number
of sheep $n$ and dog robots $m$. The values of the constants in sheep dynamics
are $k_{G}=1$, $k_{S}=0.3$ and $k_{D}=0.08$. We vary $n$ and $m$ from 1 to 10
and for a given pair of ($n,m$) we run the simulation for a hundred times with
a random initialization of
$\boldsymbol{x}_{0}=(\boldsymbol{x}_{S_{1}}(0),\cdots,\boldsymbol{x}_{S_{n}}(0),\boldsymbol{x}_{D_{1}}(0),\cdots,\boldsymbol{x}_{D_{m}}(0))$
in every run. Table I reports these results. Each entry of this table reports
the percentage success rate i.e. in how many cases the sheep were diverted
away from the protected zone. As can be seen, almost all entries are 100,
which proves the success of our algorithm.

Table I: Performance of the proposed strategy with varying number of sheep and dog robots. Here, we did not consider collision avoidance constraints i.e. the dogs were allowed to run into the sheep.  $N_{S}$ $N_{D}$ |  2 |  4 |  6 |  8 |  10
---|---|---|---|---|---
2 |  100 |  100 |  100 |  100 |  100
4 |  100 |  100 |  100 |  100 |  100
6 |  100 |  98 |  100 |  100 |  100
8 |  100 |  98 |  100 |  100 |  98
10 |  100 |  98 |  98 |  100 |  96

Further, we considered the impact of including collision avoidance
constraints. These results are reported in Table II. Because of additional
constraints, it is possible that collision avoidance conflicts with the
defending constraint. As a result, we do not observe as good successes in this
case compared to when there are no collision avoidance constraints.

Table II: Performance of the proposed strategy with varying number of sheep and dog robots. Here we considered collision avoidance constraints in the dynamics of the dogs.  $N_{S}$ $N_{D}$ |  2 |  4 |  6 |  8 |  10
---|---|---|---|---|---
2 |  72 |  99 |  99 |  100 |  100
4 |  62 |  74 |  90 |  97 |  100
6 |  28 |  83 |  99 |  99 |  100
8 |  63 |  82 |  100 |  100 |  100
10 |  70 |  79 |  90 |  91 |  94

###  V-C Hardware Experiments

Finally, we tested our algorithm in robots in the multirobot test arena in our
lab. It consists of a 14ft $\times$ 7ft platform, several Khepera IV robots
and additionally eight Vicon cameras for motion tracking. All control inputs
are computed on a desktop and conveyed to the robots over WiFi. While we
developed our algorithms assuming that the dynamics of all agents are single-
integrator based, the robots have unicycle dynamics given by

$\displaystyle\begin{pmatrix}\dot{x}\\\ \dot{y}\\\
\dot{\theta}\end{pmatrix}=\begin{pmatrix}v\cos{\theta}\\\ v\sin{\theta}\\\
\omega\end{pmatrix}$ (28)

Thus, we do a minor adjustment to map the inputs computed from our algorithms
to the angular speed and forward translational speed of these robots. This is
done by considering a point at a distance $d$ on the $x_{b}$ axis of the body
frame of the robot:

$\displaystyle\boldsymbol{x}=\begin{pmatrix}x+d\cos{\theta}\\\
y+d\sin{\theta}\end{pmatrix}$ $\displaystyle\implies$
$\displaystyle\dot{\boldsymbol{x}}=\underbrace{\begin{pmatrix}\cos{\theta}&-\sin{\theta}\\\
\sin{\theta}&\cos{\theta}\end{pmatrix}\begin{pmatrix}1&0\\\
0&d\end{pmatrix}}_{M}\begin{pmatrix}v\\\
\omega\end{pmatrix}=\tilde{\boldsymbol{u}}$ $\displaystyle\implies$
$\displaystyle\begin{pmatrix}v\\\
\omega\end{pmatrix}=M^{-1}\tilde{\boldsymbol{u}}$ (29)

For the robots representing the sheep, $\tilde{\boldsymbol{u}}$ is obtained
from (1) while for the robots representing the dog, $\tilde{\boldsymbol{u}}$
is obtained from (IV). In Fig. 3, we have one sheep (in red box) and one dog
robot (in blue box). The protected zone is highlighted in green and the goal
and center of the protected zone are the black dot. We use (IV) to compute the
velocity of the dog robot and convert it to angular speed and forward
translational speed using (V-C). As can be noted from the snapshots, the dog
robot is able successfully defend the zone from the sheep. Next we consider
multiple sheep in Fig. 4. As can be seen from the snapshots, in this case, the
dog is able to defend the zone from both sheep. Finally, in Fig. 5 we
demonstrate that our approach is compositional i.e. we can have multiple
protected zones. In this figure, we purposefully kept the goal of the sheep in
the left most protected zone. This way, the sheep would be incentivized to
breach both the protected zones. Yet still, our algorithm is able to find
velocities for dogs to defend both the zones from both sheep.

##  VI Conclusions

In this paper, we developed a novel optimization-based control strategy for a
group of dog robots to prevent a herd of sheep agents from breaching a
protected zone. We have proven the feasibility of the algorithm for the single
dog v/s single sheep case. Empirical results show that our designed controller
can defend the protected zone from a flock of multiple sheep using multiple
dogs as well. The results also show that the algorithm is composable and
allows us to include multiple protected zones. Future work will focus on
finding design parameters of the constraints such that the velocities computed
by our controller does not exceed actuator limits. We also aim to perform
hardware experiments with a higher number of sheep and dogs using Khepera
robots. Further, in our current work, we assumed known dynamics of sheep. In
future, we plan to extend this to the case where we learn their dynamics
online while simultaneously performing defense of the protected zone.

## References

  * [1] R. D’Andrea, “Guest editorial: A revolution in the warehouse: A retrospective on kiva systems and the grand challenges ahead,” _IEEE Transactions on Automation Science and Engineering_ , vol. 9, no. 4, pp. 638–639, 2012.
  * [2] R. D’Andrea and G. E. Dullerud, “Distributed control design for spatially interconnected systems,” _IEEE Transactions on automatic control_ , vol. 48, no. 9, pp. 1478–1495, 2003.
  * [3] W. Kazmi, M. Bisgaard, F. Garcia-Ruiz, K. D. Hansen, and A. la Cour-Harbo, “Adaptive surveying and early treatment of crops with a team of autonomous vehicles,” in _Proceedings of the 5th European Conference on Mobile Robots ECMR 2011_ , 2011, pp. 253–258.
  * [4] M. Ji and M. Egerstedt, “Distributed coordination control of multiagent systems while preserving connectedness,” _IEEE Transactions on Robotics_ , vol. 23, no. 4, pp. 693–703, 2007.
  * [5] J. Lin, A. S. Morse, and B. D. Anderson, “The multi-agent rendezvous problem-the asynchronous case,” in _2004 43rd IEEE Conference on Decision and Control (CDC)(IEEE Cat. No. 04CH37601)_ , vol. 2. IEEE, 2004, pp. 1926–1931.
  * [6] C. W. Reynolds, “Flocks, herds and schools: A distributed behavioral model,” in _Proceedings of the 14th annual conference on Computer graphics and interactive techniques_ , 1987, pp. 25–34.
  * [7] Q. Gong, W. Kang, C. Walton, I. Kaminer, and H. Park, “Partial observability analysis of an adversarial swarm model,” _Journal of Guidance, Control, and Dynamics_ , vol. 43, no. 2, pp. 250–261, 2020.
  * [8] C. Walton, I. Kaminer, Q. Gong, A. Clark, T. Tsatsanifos _et al._ , “Defense against adversarial swarms with parameter uncertainty,” _arXiv preprint arXiv:2108.04205_ , 2021.
  * [9] T. Tsatsanifos, A. H. Clark, C. Walton, I. Kaminer, and Q. Gong, “Modeling and control of large-scale adversarial swarm engagements,” _arXiv preprint arXiv:2108.02311_ , 2021.
  * [10] J.-M. Lien, O. B. Bayazit, R. T. Sowell, S. Rodriguez, and N. M. Amato, “Shepherding behaviors,” in _IEEE International Conference on Robotics and Automation, 2004. Proceedings. ICRA’04. 2004_ , vol. 4. IEEE, 2004, pp. 4159–4164.
  * [11] A. Pierson and M. Schwager, “Controlling noncooperative herds with robotic herders,” _IEEE Transactions on Robotics_ , vol. 34, no. 2, pp. 517–525, 2017.
  * [12] R. Vaughan, N. Sumpter, J. Henderson, A. Frost, and S. Cameron, “Robot control of animal flocks,” in _Proceedings of the 1998 IEEE International Symposium on Intelligent Control (ISIC) held jointly with IEEE International Symposium on Computational Intelligence in Robotics and Automation (CIRA) Intell_. IEEE, 1998, pp. 277–282.
  * [13] ——, “Experiments in automatic flock control,” _Robotics and autonomous systems_ , vol. 31, no. 1-2, pp. 109–117, 2000.
  * [14] A. Pierson and M. Schwager, “Bio-inspired non-cooperative multi-robot herding,” in _2015 IEEE International Conference on Robotics and Automation (ICRA)_. IEEE, 2015, pp. 1843–1849.
  * [15] R. A. Licitra, Z. I. Bell, E. A. Doucette, and W. E. Dixon, “Single agent indirect herding of multiple targets: A switched adaptive control approach,” _IEEE Control Systems Letters_ , vol. 2, no. 1, pp. 127–132, 2017.
  * [16] R. A. Licitra, Z. D. Hutcheson, E. A. Doucette, and W. E. Dixon, “Single agent herding of n-agents: A switched systems approach,” _IFAC-PapersOnLine_ , vol. 50, no. 1, pp. 14 374–14 379, 2017.
  * [17] E. Sebastián and E. Montijano, “Multi-robot implicit control of herds,” in _2021 IEEE International Conference on Robotics and Automation (ICRA)_. IEEE, 2021, pp. 1601–1607.
  * [18] M. Bacon and N. Olgac, “Swarm herding using a region holding sliding mode controller,” _Journal of Vibration and Control_ , vol. 18, no. 7, pp. 1056–1066, 2012.
  * [19] J. S. Grover, C. Liu, and K. Sycara, “Parameter identification for multirobot systems using optimization based controllers (extended version),” _arXiv preprint arXiv:2009.13817_ , 2020.
  * [20] J. Grover, C. Liu, and K. Sycara, “Feasible region-based identification using duality (extended version),” _arXiv preprint arXiv:2011.04904_ , 2020.
  * [21] A. D. Ames, S. Coogan, M. Egerstedt, G. Notomista, K. Sreenath, and P. Tabuada, “Control barrier functions: Theory and applications,” in _2019 18th European Control Conference (ECC)_. IEEE, 2019, pp. 3420–3431.
  * [22] L. Wang, A. D. Ames, and M. Egerstedt, “Safety barrier certificates for collisions-free multirobot systems,” _IEEE Transactions on Robotics_ , vol. 33, no. 3, pp. 661–674, 2017.


# Further understanding the nature of $\Omega(2012)$ within a chiral quark
model

Hui-Hua Zhong, Ru-Hui Ni, Mu-Yang Chen 111E-mail<EMAIL_ADDRESS>Xian-
Hui Zhong 222E-mail<EMAIL_ADDRESS>Department of Physics, Hunan Normal
University, and Key Laboratory of Low-Dimensional Quantum Structures and
Quantum Control of Ministry of Education, Changsha 410081, China  Synergetic
Innovation Center for Quantum Effects and Applications (SICQEA), Hunan Normal
University, Changsha 410081, China  Ju-Jun Xie 333E-mail:
<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  School of Nuclear Science and Technology,
University of Chinese Academy of Sciences, Beijing 101408, China  School of
Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001,
China

###### Abstract

In our previous works, we have analyzed the two-body strong decays of the low-
lying $\Omega$ baryon states within a chiral quark model. The results show
that the $\Omega(2012)$ resonance favors the three-quark state with
$J^{P}=3/2^{-}$ classified in the quark model. With this assignment, in the
present work we further study the three-body strong decay
$\Omega(2012)\to\Xi^{*}(1530)\bar{K}\to\Xi\pi\bar{K}$ and coupled-channel
effects on $\Omega(2012)$ from nearby channels $\Xi\bar{K}$, $\Omega\eta$ and
$\Xi^{*}(1530)\bar{K}$ within the chiral quark model as well. It is found that
the $\Omega(2012)$ resonance has a sizeable decay rate into the three-body
final state $\Xi\pi\bar{K}$. The predicted ratio
$R_{\Xi\bar{K}}^{\Xi\pi\bar{K}}=\mathcal{B}[\Omega(2012)\to\Xi^{*}(1530)\bar{K}\to\Xi\pi\bar{K}]/\mathcal{B}[\Omega(2012)\to\Xi\bar{K}]\simeq
12\%$ is close to the up limit $11\%$ measured by the Belle Collaboration in
2019, however, our predicted ratio is too small to be comparable with the
recent data $0.97\pm 0.31$. Furthermore, our results show that the coupled-
channel effects on the $\Omega(2012)$ is not large, its components should be
dominated by the bare three-quark state, while the proportion of the molecular
components is only $\sim 16\%$. To clarify the nature of $\Omega(2012)$, the
ratio $R_{\Xi\bar{K}}^{\Xi\pi\bar{K}}$ is expected to be tested by other
experiments.

##  I Introduction

In 2018, Belle Collaboration observed a new excited hyperon $\Omega(2012)^{-}$
decaying into $\Xi^{0}K^{-}$ and $\Xi^{-}\bar{K}^{0}$ with a mass of
$2012.4\pm 0.7(\mathrm{stat})\pm 0.6(\mathrm{syst})$ MeV and a width of
$\Gamma=6.4^{+2.5}_{-2.0}(\mathrm{stat})\pm 1.6(\mathrm{syst})$ MeV
Belle:2018mqs . In 2021, evidence of $\Omega(2012)^{-}$ was observed in the
$\Omega_{c}$ weak decay process
$\Omega_{c}\to\pi^{+}\Omega(2012)^{-}\to\pi^{+}(\Xi^{0}K^{-})$ at Belle
Belle:2021gtf  following the proposal in Ref. Zeng:2020och . According to the
previous mass spectrum predictions in various models and methods, such as the
Skyrme model Oh:2007cr , quark model Capstick:1986bm ; Faustov:2015eba ;
Loring:2001ky ; Liu:2007yi ; Chao:1980em ; Chen:2009de ; An:2013zoa ;
Kalman:1982ut ; Pervin:2007wa ; An:2014lga , lattice gauge theory Engel:2013ig
; Liang:2015bxr  and so on Carlson:2000zr ; Goity:2003ab ; Schat:2001xr ;
Matagne:2006zf ; Bijker:2000gq ; Aliev:2016jnp , the newly observed
$\Omega(2012)$ may be a good candidate of the first orbital ($1P$) excitations
of $\Omega(1672)$.

The discovery of $\Omega(2012)$ immediately attracted a great deal of
attention from the hadron physics community in the literature. Our group
analyzed the Okubo-Zweig-Iizuka (OZI)-allowed two body strong decays of the
low-lying $P$\- and $D$-wave $\Omega$ baryon states within the chiral quark
model Liu:2019wdr ; Xiao:2018pwe  and ${}^{3}P_{0}$ model Wang:2018hmi  by
combining the mass spectrum analysis, and found that the $\Omega(2012)^{-}$
resonance favors the assignment of the $1P$-wave state with $J^{P}=3/2^{-}$.
Furthermore, the newly measured ratio
$\mathcal{B}[\Omega_{c}\to\Omega(2012)\pi^{+}\to(\Xi\bar{K})^{-}\pi^{+}]/\mathcal{B}[\Omega_{c}\to\Omega\pi^{+}]$
at Belle can be well understood as well within the three quark picture in a
recent work Wang:2022zja . Our conclusion is consistent with that based on the
framework of QCD sum rules Aliev:2018yjo ; Aliev:2018syi , the constituent
quark model by including relativistic corrections Arifi:2022ntc , and the
flavour SU(3) analysis Polyakov:2018mow . However, in the literature the
$\Omega(2012)$ was interpreted as a hadronic molecule Hu:2022pae ;
Ikeno:2022jpe ; Lin:2019tex ; Lu:2020ste ; Ikeno:2020vqv ; Xie:2021dwe ;
Valderrama:2018bmv ; Pavao:2018xub ; Huang:2018wth ; Gutsche:2019eoh . In the
hadronic molecular picture, the decay rate into the three-body final state
$\Xi\pi\bar{K}$ is predicted to be similar to that of the two-body final state
$\Xi\bar{K}$. While in the three-quark picture, the decay rates for OZI-
allowed two-body final state $\Xi\bar{K}$ should be dominant, the decay rate
of $\Omega(2012)^{-}\to\Xi^{*}(1530)\bar{K}\to\Xi\pi\bar{K}$ should be
suppressed by the intermediate hadron state $\Xi^{*}(1530)$.

To test the decay properties of $\Omega(2012)$ predicted within different
pictures, the three-body channel $\Xi\pi\bar{K}$ through the decay process
$\Omega(2012)^{-}\to\Xi^{*}(1530)\bar{K}\to\Xi\pi\bar{K}$ was observed by the
Belle Collaboration. The first observation was carried out in 2019
Belle:2019zco . The collaboration observed no significant $\Omega(2012)$
signals in the three-body final state $\Xi\pi\bar{K}$, and set an upper limit
at 90 confidence level on the branching fraction ratio
$R_{\Xi\bar{K}}^{\Xi\pi\bar{K}}\equiv\mathcal{B}[\Omega(2012)^{-}\to\Xi^{*}(1530)\bar{K}\to\Xi\pi\bar{K}]/\mathcal{B}[\Omega(2012)^{-}\to\Xi\bar{K}]<11\%$,
which is consistent with expectation in the three-quark picture. However, in
the recent search, they observed significant $\Omega(2012)$ signals in the
three-body decay process
$\Omega(2012)^{-}\to\Xi^{*}(1530)^{0}K^{-}\to\Xi^{-}\pi^{+}K^{-}$
Belle:2022mrg . A rather large ratio $R_{\Xi\bar{K}}^{\Xi\pi\bar{K}}=0.97\pm
0.24\pm 0.07$ was extracted from the observations, which seems to be
consistent with expectation in the hadronic molecular picture.

In the previous works Liu:2019wdr ; Xiao:2018pwe , we did not study the three-
body decay process $\Omega(2012)^{-}\to\Xi^{*}(1530)\bar{K}\to\Xi\pi\bar{K}$.
It is unclear how large the decay rate of the $\Xi\pi\bar{K}$ mode within our
three-quark picture. Furthermore, the $\Omega(2012)$ as a conventional three-
quark state, it may strongly couple to the $\Xi(1530)\bar{K}$ channel, then
the $\Omega(2012)$ may contain significant molecular components due to
coupled-channel effects. To uncover these puzzles and better understand the
nature of $\Omega(2012)$, in the three quark picture we further study the
three-body decays of $\Omega(2012)\to\Xi\pi\bar{K}$ and the coupled-channel
effects from nearby channels $\Xi\bar{K}$, $\Omega\eta$ and
$\Xi^{*}(1530)\bar{K}$. For self-consistency, we adopt the same framework,
i.e., the chiral quark model Manohar:1983md , as our previous works for the
$\Omega(2012)$ studies Liu:2019wdr ; Xiao:2018pwe . In this model an effective
chiral Lagrangian is introduced to account for the quark-meson coupling at the
baryon-meson interaction vertex. The light pseudoscalar mesons, i.e., $\pi,K$,
and $\eta$, are treated as Goldstone bosons. Since the quark-meson coupling is
invariant under the chiral transformation, some of the low-energy properties
of QCD are retained Li:1994cy ; Li:1997gd ; Zhao:2002id .

This paper is organized as follows. In Sec. II, a brief review of the chiral
quark model is given, then by using this model the two-body and three body
strong decays of $\Omega(2012)$ are estimated. In Sec. III, the coupled-
channel effects on the $\Omega(2012)$ are evaluated by combining the simplest
version of the coupled-channel model with the chiral quark model. Finally, we
give a short discussion and summary in Sec. IV.

Table 1: The baryon wave functions involved in our calculations. The
$\psi_{n_{\rho}l_{\rho}m_{\rho}}^{\rho}$ and
$\psi_{n_{\lambda}l_{\lambda}m_{\lambda}}^{\lambda}$ represent the radial wave
functions of $\rho$-mode and $\lambda$-mode, respectively, while the
$\chi_{SS_{z}}^{\sigma}$ and $\phi_{\Omega,\Xi^{*},\Xi}^{\sigma}$ expressed as
spin and flavor wave functions of the baryon system, respectively. The details
can be found in Ref. Xiao:2013xi .
$\Omega^{*}\left|1^{2}P_{3/2^{-}}\right\rangle~{}=\frac{1}{\sqrt{2}}\left\\{\psi_{000}^{\rho}\psi_{01m_{\lambda}}^{\lambda}\chi_{\frac{1}{2}S_{z}}^{\lambda}\phi_{\Omega}^{s}+\psi_{01m_{\rho}}^{\rho}\psi_{000}^{\lambda}\chi_{\frac{1}{2}S_{z}}^{\rho}\phi_{\Omega}^{s}\right\\}$
---
$\Omega\left|1^{4}S_{3/2^{+}}\right\rangle~{}~{}=\psi_{000}^{\rho}\psi_{000}^{\lambda}\chi_{\frac{3}{2}S_{z}}^{s}\phi_{\Omega}^{s}$
$\Xi^{*}(1530)\left|1^{4}S_{3/2^{+}}\right\rangle~{}=\psi_{000}^{\rho}\psi_{000}^{\lambda}\chi_{\frac{3}{2}S_{z}}^{s}\phi_{\Xi^{*}}^{s}$
$\Xi\left|1^{2}S_{1/2^{+}}\right\rangle~{}~{}~{}=\frac{1}{\sqrt{2}}\left\\{\psi_{000}^{\rho}\psi_{000}^{\lambda}\chi_{\frac{1}{2}\frac{1}{2}}^{\rho}\phi_{\Xi}^{\rho}+\psi_{000}^{\rho}\psi_{000}^{\lambda}\chi_{\frac{1}{2}\frac{1}{2}}^{\lambda}\phi_{\Xi}^{\lambda}\right\\}$
Table 2: Masses (MeV) of the hadrons adopted in this work. Hadron | $J^{P}$ | Mass (MeV)
---|---|---
$\pi^{0},\pi^{\pm}$ | $0^{-}$ | $135.0,139.6$
$\bar{K}^{0},K^{\pm}$ | $0^{-}$ | $497.6,493.6$
$\eta$ | $0^{-}$ | $547.9$
$\Xi^{-},\Xi^{0}$ | $1/2^{+}$ | $1321.7,1314.9$
$\Xi^{*-},\Xi^{*0}$ | $3/2^{+}$ | $1535.0,1531.8$
$\Omega^{-}$ | $3/2^{+}$ | $1671.7$
$\Omega(2012)$ | $3/2^{-}$ | $2012.4$

##  II strong Decay

###  II.1 two-body decay

In the chiral quark model Manohar:1983md , the low-energy quark-pseudoscalar-
meson interactions in the SU(3) flavor basis are represented by the effective
Lagrangian Li:1994cy ; Li:1997gd ; Zhao:2002id

$\displaystyle\mathbf{{\cal L}}$ $\displaystyle=$
$\displaystyle\sum_{j}\frac{1}{f_{m}}\bar{\psi}_{j}\gamma^{j}_{\mu}\gamma^{j}_{5}\psi_{j}\partial^{\mu}\phi_{m}.$
(1)

In the above effective Lagrangians, $\psi_{j}$ represents the $j$th quark
field in the hadron, $\phi_{m}$ is the pseudoscalar meson field, $f_{m}$ is
the pseudoscalar meson decay constant. To match the nonrelativistic wave
functions of the initial and final hadron states, we adopt the nonrelativistic
form of the Lagrangians in following form,

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

where $\omega_{m}$ and $\textbf{q}_{m}$ are the energy and three momentum of
the final state pseudoscalar meson, respectively; $E_{i}$ and $M_{i}$ are the
energy and mass of the initial heavy hadron, respectively; $E_{f}$ and $M_{f}$
represent the energy and mass of the final state heavy hadron, respectively;
$\textbf{p}_{j}$ is the internal momentum operator of the $j$th quark in the
baryon system rest frame; $\mbox{\boldmath$\sigma$\unboldmath}_{j}$ is the
spin operator for the $j$th quark of the baryon system; and $\mu_{q}$ is a
reduced mass expressed as $1/\mu_{q}=1/m_{j}+1/m^{\prime}_{j}$ with $m_{j}$
and $m^{\prime}_{j}$ for the masses of the $j$th quark in the initial and
final baryons, respectively. The plane wave part of the emitted light meson is
$\varphi_{m}=e^{-i\textbf{q}\cdot\textbf{r}_{j}}$, and $I_{j}$ is the flavor
operator defined for the transitions in the SU(3) flavor space Li:1997gd ;
Zhong:2007gp ; Zhong:2008kd . $\delta$ as a global parameter accounts for the
strength of the quark-meson couplings. Here, we take the same value as that
determined in Refs. Xiao:2013xi ; Xiao:2018pwe ; Liu:2019wdr , i.e.,
$\delta=0.576$.

For a excited baryon state, within the chiral quark model its two-body OZI-
allowed strong decay amplitudes can be described by

$\displaystyle\mathcal{M}\left[\mathcal{B}\rightarrow\mathcal{B}^{\prime}\mathbb{M}\right]=\left\langle\mathcal{B}^{\prime}\left|\mathbf{{\cal
H}}_{m}^{nr}\right|\mathcal{B}\right\rangle,$ (3)

where $\left|\mathcal{B}\right\rangle$ and
$\left|\mathcal{B^{\prime}}\right\rangle$ are the wave functions of the
initial and final baryon states, respectively. With derived decay amplitudes
from Eq.(3), the partial decay width for the
$\mathcal{B}\rightarrow\mathcal{B}^{\prime}\mathbb{M}$ process can be
calculated with

$\displaystyle\Gamma=\frac{1}{8\pi}\frac{\left|\mathbf{q}_{m}\right|}{M_{i}^{2}}\frac{1}{2J_{i}+1}\sum_{J_{iz}J_{fz}}\left|\mathcal{M}_{J_{iz}J_{fz}}\right|^{2},$
(4)

where $J_{iz}$ and $J_{fz}$ represent the third components of the total
angular momenta of the initial and final baryons, respectively.

The OZI-allowed two-body strong decay for the $\Omega$ resonances have been
evaluated under the frame of the chiral quark model in our previous works
Liu:2019wdr ; Xiao:2018pwe . It is found that the newly observed
$\Omega(2012)$ resonance favors the assignment of the $1P$-wave state with
$J^{P}=3/2^{-}$ (i.e. $\Omega^{*}|1P_{3/2^{-}}\rangle$ listed in Table 1) in
the $\Omega$ baryon family Liu:2019wdr . Considering the $\Omega(2012)$ as
$\Omega^{*}|1P_{3/2^{-}}\rangle$, the $\Xi\bar{K}$ is the only OZI allowed
two-body decay channel, which is shown in Fig. 1, while the
$\Omega(2012)\to\Xi^{*}(1530)\bar{K}$ decay process is forbidden since the
mass of $\Omega(2012)$ lies below the $\Xi^{*}(1530)\bar{K}$ threshold. Thus,
the total width should be nearly saturated by the $\Xi\bar{K}$ channel.

By using the wave function calculated from the potential model Liu:2019wdr ,
within the chiral quark model the partial width of $\Omega(2012)\to\Xi\bar{K}$
are predicted to be

$\displaystyle\Gamma_{\Xi\bar{K}}=\Gamma_{\Xi^{0}K^{-}}+\Gamma_{\Xi^{-}\bar{K}^{0}}\simeq(3.0+2.7)\
\ \mathrm{MeV},$ (5)

which is consistent with the data $\Gamma_{exp}=6.4^{+2.5}_{-2.0}\pm 1.6$ MeV
ParticleDataGroup:2020ssz . It should be mentioned that in the calculations,
to consist with the mass spectrum predictions in the potential model
Liu:2019wdr , the constituent quark masses of $u/d$ and $s$ quarks are adopted
to be $m_{u}=m_{d}=350$ MeV and $m_{s}=600$ MeV. The meson decay constants for
$\pi$, $K$ and $\eta$ are taken with $f_{\pi}=132$ MeV and
$f_{K}=f_{\eta}=160$ MeV. The masses of the final and initial states are taken
the measured values from experiments ParticleDataGroup:2020ssz , which have
been collected in Table 2. The spatial wave functions of the baryons appearing
in the initial and final states are adopted the harmonic oscillator form. For
the $\Xi$ and $\Xi^{*}(1530)$ states, the harmonic oscillator strength
parameter $\alpha_{\rho}$ is taken as $\alpha_{\rho}=400$ MeV, while the
parameter $\alpha_{\lambda}$ for the $\lambda$ oscillator is related to
$\alpha_{\rho}$ with
$\alpha_{\lambda}=\sqrt[4]{3m_{u}/(2m_{s}+m_{u})}\alpha_{\rho}$. For the
$\Omega^{*}|1P_{3/2^{-}}\rangle$ state, we take
$\alpha_{\rho}=\alpha_{\lambda}=411$ MeV, which is fitted by reproducing the
root-mean-square radius of the $\rho$-mode excitations from the potential
model Liu:2019wdr .

Figure 1: The OZI-allowed two-body decay process $\Omega(2012)\to\Xi K$ at (i) the hadronic level and (ii) the quark level. Figure 2: The cascade three-body decay process $\Omega(2012)\to\Xi^{*}(1530)K\to[\Xi\pi]K$ at (i) the hadronic level and (ii) the quark level. Table 3: The partial widths (MeV) of three-body final states for the $\Omega(2012)$ resonance. $\Gamma_{i}$ stands for the results by considering the isospin breaking effects of the final states. While $\overline{\Gamma}_{i}$ stands for the results without the isospin breaking effects, the mass of a hadron in the final states is adopted the average value of different charged states. | Channel | $\Xi^{-}\pi^{+}K^{-}$ | $\Xi^{0}\pi^{0}K^{-}$ | $\Xi^{-}\pi^{0}\bar{K}^{0}$ | $\Xi^{0}\pi^{-}\bar{K}^{0}$ | $\mathbf{Sum}$
---|---|---|---|---|---|---
|  $\Gamma_{i}$ (MeV) | $0.25$ | $0.18$ | $0.07$ | $0.17$ | $\mathbf{0.67}$
|  $\overline{\Gamma}_{i}$ (MeV) | $0.21$ | $0.11$ | $0.11$ | $0.21$ | $\mathbf{0.64}$

###  II.2 three-body decay

Recently, the Belle Collaboration observed a rather large three-body decay
rate in the $\Omega(2012)^{-}\to\Xi^{*}(1530)^{0}K^{-}\to\Xi^{-}\pi^{+}K^{-}$
process. This cascade decay process can be evaluated in the chiral quark model
with the same parameter set as well. For the cascade decay process
$\Omega(2012)\to\Xi^{*}(1530)K\to[\Xi\pi]\bar{K}$ as shown in Fig. 2, the
decay amplitude can be expressed as

$\displaystyle\mathcal{M}_{\Omega(2012)\rightarrow\Xi\pi\bar{K}}=\frac{\mathcal{M}_{\Omega(2012)\to\Xi^{*}\bar{K}}\mathcal{M}_{\Xi^{*}\to\Xi\pi}}{2M_{\Xi^{*}}\left(M_{\Xi\pi}-M_{\Xi^{*}}+i\tilde{\Gamma}_{\Xi^{*}\to\Xi\pi}/2\right)},$
(6)

where $\mathcal{M}_{\Omega(2012)\to\Xi^{*}(1530)\bar{K}}$ and
$\mathcal{M}_{\Xi^{*}\rightarrow\Xi\pi}$ are the amplitudes defined in Eq.(3).
It should be emphasized that the $\Xi^{*}(1530)$ baryon mass $M_{\Xi^{*}}$ in
$\mathcal{M}_{\Omega(2012)\to\Xi^{*}K}$ and $\mathcal{M}_{\Xi^{*}\to\Xi\pi}$
is replaced with the invariant mass $M_{\Xi\pi}$ of $\Xi\pi$ system since the
unstable baryon $\Xi^{*}(1530)$ can be slightly off-shell. Adopting the wave
functions listed in Table 1, one can work out the
$\mathcal{M}_{\Omega(2012)\to\Xi^{*}(1530)\bar{K}}$ and
$\mathcal{M}_{\Xi^{*}\rightarrow\Xi\pi}$ within the chiral quark model.
Moreover, $\tilde{\Gamma}_{\Xi^{*}(1530)\rightarrow\Xi\pi}$ also depends on
the invariant mass of $M_{\Xi\pi}$, and it can be obtained by Eq.(4).

Then the three-body decay width of the $\Omega(2012)$ resonance in their rest
frame was obtained as

$\displaystyle
d\Gamma_{\Xi\pi\bar{K}}=\frac{M_{\Xi\pi}^{2}}{M_{\Omega^{*}}^{2}}\frac{|\mathbf{q}_{K}|}{4\pi^{2}}\frac{\tilde{\Gamma}_{\Xi^{*}\rightarrow\Xi\pi}\overline{\left|\mathcal{M}_{\Omega(2012)\rightarrow\Xi^{*}\bar{K}}\right|^{2}}}{4M_{\Xi^{*}}^{2}\left[\left(M_{\Xi\pi}-M_{\Xi^{*}}\right)^{2}+\tilde{\Gamma}_{\Xi^{*}\rightarrow\Xi\pi}^{2}/4\right]}dM_{\Xi\pi},$
(7)

where the absolute value of the momentum for the final $K$ meson is give by

$\displaystyle|\mathbf{q}_{K}|=\frac{\sqrt{\left[M_{\Omega^{*}}^{2}-\left(m_{K}+M_{\pi\Xi}\right)^{2}\right]\left[M_{\Omega^{*}}^{2}-\left(m_{K}-M_{\pi\Xi}\right)^{2}\right]}}{2M_{\Omega^{*}}}.$
(8)

Integrating the Eq.(7) from $M_{\Xi}+m_{\pi}$ to $M_{\Omega^{*}}-m_{K}$, one
can easily work out the three-body decay width of the cascade decay process
$\Omega(2012)\to\Xi^{*}(1530)\bar{K}\to[\Xi\pi]\bar{K}$. Our results have been
listed in Table 3.

The $\Omega(2012)$ as the $\Omega^{*}|1P_{3/2^{-}}\rangle$ state has a
sizeable partial decay width into the three-body final state $\Xi\pi\bar{K}$,

$\displaystyle\Gamma_{\Xi\pi\bar{K}}\simeq 0.67\ \ \mathrm{MeV},$ (9)

combined it with the two-body decay width predicted in Eq.(5), the total width
is predicted to be

$\displaystyle\Gamma_{total}\simeq 6.37\ \ \mathrm{MeV},$ (10)

which is in good agreement with the data $\Gamma_{exp}=6.4^{+2.5}_{-2.0}\pm
1.6$ MeV ParticleDataGroup:2020ssz . The predicted ratio

$\displaystyle
R_{\Xi\bar{K}}^{\Xi\pi\bar{K}}\equiv\frac{\mathcal{B}[\Omega(2012)\to\Xi^{*}(1530)\bar{K}\to\Xi\pi\bar{K}]}{\mathcal{B}[\Omega(2012)\to\Xi\bar{K}]}\simeq
12\%,$ (11)

is slightly larger than the previous data $(6.0\pm 3.7\pm 1.3)\%$ measured by
the Belle Collaboration in 2020 Belle:2019zco , however, is a factor of $\sim
4-8$ smaller than their recent data $0.97\pm 0.31$ with improved selection
criteria Belle:2022mrg . In the recent work Arifi:2022ntc , the three body
decay of $\Omega(2012)\to\Xi\pi\bar{K}$ was also studied within a constituent
quark model by including relativistic corrections, a small ratio
$R_{\Xi\bar{K}}^{\Xi\pi\bar{K}}\simeq 4.5\%$ is obtained with the
$\Omega^{*}|1P_{3/2^{-}}\rangle$ assignment. It should be mentioned that the
isospin breaking effects on the partial widths of the three-body decays are
obvious. The results in Table 3 show that there are significant differences
between the partial widths with isospin breaking effects and that without
isospin breaking effects.

The recent measured ratio $R_{\Xi\bar{K}}^{\Xi\pi\bar{K}}$ at Belle
Belle:2022mrg  seems to be compatible with the molecular interpretation for
the $\Omega(2012)$ proposed in Refs. Valderrama:2018bmv ; Pavao:2018xub ;
Huang:2018wth ; Gutsche:2019eoh  where similar branching fractions
$\Omega(2012)$ decays into $\Xi\pi\bar{K}$ and $\Xi\bar{K}$ were predicted.

In the following, we will further explore the molecular components of
$\Omega(2012)$ due to coupled-channel effects.

Figure 3:  The hadronic loop for a bare baryon state $|A\rangle$ coupling to
two-hadron continuum $BC$ at (i) the hadronic level and (ii) the quark level.

##  III Coupled-channel effects

First, we give a brief review of the coupled-channel model adopted in this
work. A bare baryon state $|A\rangle$ predicted in the quark model can couple
to the two-hadron continuum $BC$ via hadronic loops, as shown in Fig. 3. In
the simplest version of the coupled-channel model Morel:2002vk ; Lu:2017hma ;
Kalashnikova:2005ui ; Liu:2011yp ; Lu:2016mbb , the wave function of the
physical state is given by

$\displaystyle|\Psi\rangle=c_{A}|A\rangle+\sum_{BC}\int
c_{BC}(\mathbf{p})d^{3}\mathbf{p}|BC,\mathbf{p}\rangle,$ (12)

where $\mathbf{p}=\mathbf{p}_{B}=-\mathbf{p}_{C}$ is final two-hadron relative
momentum in the initial hadron static system, $c_{A}$ and $c_{BC}(\mathbf{p})$
denote the probability amplitudes of the bare valence state $|A\rangle$ and
$|BC,\mathbf{p}\rangle$ continuum, respectively.

The coupling between the bare state $|A\rangle$ and the continuum components
sectors $|BC,\mathbf{p}\rangle$ is achieved by creating light quark pairs.
This coupling, as an effective coupling for the quark-meson interactions, can
be described with the chiral interactions given in Eq.(II.1) in the chiral
quark model. Thus, the full Hamiltonian of the physical state $|\Psi\rangle$
can be written as

$\displaystyle\mathbf{{\cal H}}=\left(\begin{matrix}\mathbf{{\cal
H}}_{0}~{}~{}~{}~{}~{}~{}\mathbf{{\cal H}}_{m}^{nr}\\\ \mathbf{{\cal
H}}_{m}^{nr}~{}~{}~{}~{}~{}~{}\mathbf{{\cal H}}_{c}\end{matrix}\right),$ (13)

where $\mathbf{{\cal H}}_{0}$ is the Hamiltonian of the bare state $|A\rangle$
in the potential model, while $\mathbf{{\cal H}}_{c}$ is the Hamiltonian for
the continuum state $|BC,\mathbf{p}\rangle$. Neglecting the interaction
between the hadrons $B$ and $C$, one has

$\displaystyle\mathbf{{\cal H}}_{c}|BC,\mathbf{p}\rangle$
$\displaystyle=E_{BC}|BC,\mathbf{p}\rangle,$ (14)

where $E_{BC}=\sqrt{m_{B}^{2}+p^{2}}+\sqrt{m_{C}^{2}+p^{2}}$ represents the
energy of $BC$ continuum.

The Schrödinger equation of a mixed system can be written as

$\displaystyle\left(\begin{matrix}\mathbf{{\cal
H}}_{0}~{}~{}~{}~{}~{}~{}\mathbf{{\cal H}}_{m}^{nr}\\\ \mathbf{{\cal
H}}_{m}^{nr}~{}~{}~{}~{}~{}~{}\mathbf{{\cal
H}}_{c}\end{matrix}\right)~{}\left(\begin{matrix}c_{A}|A\rangle\\\
\sum_{BC}\int
c_{BC}(\mathbf{p})d^{3}\mathbf{p}|BC,\mathbf{p}\rangle\end{matrix}\right)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(15)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=M\left(\begin{matrix}c_{A}|A\rangle\\\
\sum_{BC}\int
c_{BC}(\mathbf{p})d^{3}\mathbf{p}|BC,\mathbf{p}\rangle\end{matrix}\right).$

From Eq.( 15), we have

$\displaystyle\langle A|\mathbf{{\cal H}}|\Psi\rangle$ $\displaystyle=$
$\displaystyle c_{A}M$ (16) $\displaystyle=$ $\displaystyle
c_{A}M_{A}+\sum_{BC}\int c_{BC}(\mathbf{p})d^{3}\mathbf{p}\langle
A|\mathbf{{\cal H}}_{m}^{nr}|BC,\mathbf{p}\rangle,$ $\displaystyle\langle
BC,\mathbf{p}|\mathbf{{\cal H}}|\Psi\rangle$ $\displaystyle=$ $\displaystyle
c_{BC}(\mathbf{p})M$ (17) $\displaystyle=c_{BC}(\mathbf{p})E_{BC}+c_{A}\langle
BC,\mathbf{p}|\mathbf{{\cal H}}_{m}^{nr}|A\rangle.$

Deriving $c_{BC}(\mathbf{p})$ from Eq.(17), and substituting it into Eq.(16),
we get a coupled-channel equation

$\displaystyle M=M_{A}+\Delta M(M),$ (18)

where the mass shift $\Delta M(M)$ is given by

$\displaystyle\Delta M(M)$
$\displaystyle=\mathrm{Re}\sum_{BC}\int_{0}^{\infty}\frac{|\langle
BC,\mathbf{p}|\mathbf{{\cal
H}}_{m}^{nr}|A\rangle|^{2}}{(M-E_{BC})}p^{2}dpd\Omega_{p},$ (20)
$\displaystyle=\mathrm{Re}\sum_{BC}\int_{0}^{\infty}\frac{\overline{|\mathcal{M}_{A\to
BC}(\mathbf{p})|^{2}}}{(M-E_{BC})}p^{2}dp,$

and $M_{A}$ is the bare mass of the baryon state $|A\rangle$ obtained from the
potential model. From Eqs.(18) and (20), the physical mass $M$ and the bare
state mass shift $\Delta M$ can be determined simultaneously. It should be
mentioned that when we calculate the mass shift $\Delta M$ by using the
Eq.(20), the contribution in the higher $\mathbf{p}$ region may be nonphysical
because the quark pair production rates via the non-perturbative interaction
$\mathbf{{\cal H}}_{m}^{nr}$ should be strongly suppressed Morel:2002vk . In
the chiral quark model, the chiral interaction $\mathbf{{\cal H}}_{m}^{nr}$ is
only applicable to the low $\mathbf{p}$ region. To eliminate the nonphysical
contributions, in our calculations, we cut off the momentum $p$ at the
inflection point of $\Delta M(\mathbf{p})$ function as followed by our
previous work Ni:2021pce . It should be noted that due to the various
locations of the inflection point, the cut-off momentum for each channel
varies.

Table 4: The proportions of different components and mass shift for the $\Omega(2012)$ resonance in the coupled-channel model. Channel | $\Xi\bar{K}$ | $\Xi^{*}(1530)\bar{K}$ | $\Omega\eta$ | Baryon core | $\mathbf{Total}$
---|---|---|---|---|---
$\Delta M$ (MeV) | $-2.2$ | $-9.4$ | $-3.0$ | $\mathbf{...}$ | $\mathbf{-14.6}$
Proportion $(\%)$ | $1.9$ | $13.7$ | $0.8$ | $\mathbf{83.6}$ | $\mathbf{100}$

To estimate the component of baryon core and baryon-meson continuum components
in the physical state, for below-threshold states, deriving
$c_{BC}(\mathbf{p})$ from Eq.(17), and substituting it into normalized
constants containing two Fock components,

$\displaystyle|c_{A}|^{2}+\sum_{BC}\int|c_{BC}(\mathbf{p})|^{2}d^{3}\mathbf{p}=1,$
(21)

we get the probability of hadron $A$ component,

$P_{A}=|c_{A}|^{2}=\left(1+\sum_{BC}\int\frac{\overline{|\mathcal{M}_{A\to
BC}(\mathbf{p})|^{2}}}{(M-E_{BC})^{2}}p^{2}dp\right)^{-1}.$ (22)

Further, combining Eq.(17) and Eq.(22), we can obtain the probability of the
$|BC,\mathbf{p}\rangle$ continuum components,

$\displaystyle P_{BC}=|c_{A}|^{2}\int\frac{\overline{|\mathcal{M}_{A\to
BC}(\mathbf{p})|^{2}}}{(M-E_{BC})^{2}}p^{2}dp.$ (23)

The Eqs.(22) and (23) can give the probability of the bare baryon components
and $BC$ hadron pair components, in the coupled-channel effects. However, the
integral in Eqs.(22) and (23) is only well defined for $M<m_{B}+m_{C}$. For
states above the $BC$ hadron pair threshold, as suggested by Ref. Xie:2021dwe
, we replace the real $M$ with its complex counterpart, $M+i\Gamma/2$, for
coupling channel calculation, where $\Gamma$ is the decay width of the
physical state.

By using the above coupled-channel model, we study the coupled-channel effects
on the $\Omega(2012)$. The bare state $|A\rangle$ is considered to be the
$1P$-wave state with $J^{P}=3/2^{-}$ (i.e. $\Omega^{*}|1P_{3/2^{-}}\rangle$)
predicted within the quark model. Three nearby channels $\Xi\bar{K}$,
$\Omega\eta$ and $\Xi^{*}(1530)\bar{K}$ are considered. The strong coupling
amplitudes $\langle\Xi K/\Xi^{*}(1530)\bar{K}/\Omega\eta|\mathbf{{\cal
H}}_{m}^{nr}|\Omega^{*}1P_{3/2^{-}}\rangle$ can be easily obtained within the
chiral quark model by using the same parameter set. With which, we further
extract out the mass shift of the bare state $|\Omega^{*}1P_{3/2^{-}}\rangle$
due to the coupled-channel effects and the proportion of various components.
Our results have been given in Table 4.

It is found that, the coupled-channel effects on the $\Omega(2012)$ is not
large. There is a small mass shift of $\Delta\simeq-15$ MeV, which mainly
contributed by the $S$-wave channel $\Xi^{*}(1530)\bar{K}$. The components of
$\Omega(2012)$ are dominated by the bare quark model state
$\Omega^{*}|1P_{3/2^{-}}\rangle$, its proportion can reach up to about $84\%$;
the proportion of the continuum components is only $\sim 16\%$. In other words
if the $\Omega(2012)$ resonance is a molecular dominant state, it cannot
originate from the three quark state $\Omega^{*}|1P_{3/2^{-}}\rangle$.

##  IV Discussion and Summary

In our quark model study, with the $\Omega^{*}|1P_{3/2^{-}}\rangle$
assignment, both the observed mass and width of $\Omega(2012)$ can be well
explained Liu:2019wdr ; Wang:2018hmi ; Xiao:2018pwe . Furthermore, the newly
measured ratio
$\mathcal{B}[\Omega_{c}\to\Omega(2012)\pi^{+}\to(\Xi\bar{K})^{-}\pi^{+}]/\mathcal{B}[\Omega_{c}\to\Omega\pi^{+}]$
at Belle can be well understood as well in our recent work Wang:2022zja . The
decay width is nearly saturated by two-body decay channel $\Xi K$. In the
present work, to further understand the nature of $\Omega(2012)$, in the three
quark picture we study the three-body decays of $\Omega(2012)\to\Xi\pi\bar{K}$
and coupled-channel effects from nearby channels $\Xi\bar{K}$, $\Omega\eta$
and $\Xi^{*}(1530)\bar{K}$.

It is found that considering the $\Omega(2012)$ resonance as a conventional
$1P$-wave state $\Omega^{*}|1P_{3/2^{-}}\rangle$, there are sizeable decay
rates into the three-body final state $\Xi\pi\bar{K}$. The partial width and
branching fraction is predicted to be $\Gamma_{\Xi\pi\bar{K}}\simeq 0.67$ MeV
and $\mathcal{B}[\Omega(2012)\to\Xi\pi\bar{K}]\simeq 11\%$. The predicted
ratio
$R_{\Xi\bar{K}}^{\Xi\pi\bar{K}}=\mathcal{B}[\Omega(2012)^{-}\to\Xi^{*}(1530)\bar{K}\to\Xi\pi\bar{K}]/\mathcal{B}[\Omega(2012)^{-}\to\Xi\bar{K}]\simeq
0.12$ is close to the previous data $(6.0\pm 3.7\pm 1.3)\%$ measured by the
Belle Collaboration in 2019 Belle:2019zco , however, is too small to be
comparable with recent data $0.97\pm 0.31$ Belle:2022mrg .

The newly measured ratio $R_{\Xi\bar{K}}^{\Xi\pi\bar{K}}=0.97\pm 0.31$ at
Belle seems to be consistent with the molecular interpretation for the
$\Omega(2012)$ proposed in Refs. Valderrama:2018bmv ; Pavao:2018xub ;
Huang:2018wth ; Gutsche:2019eoh . However, our results show that the coupled-
channel effects on the $\Omega(2012)$ is not large, if it originates from the
$1P$-wave three-quark state $\Omega^{*}|1P_{3/2^{-}}\rangle$. In this case,
the components of $\Omega(2012)$ are dominated by the bare quark model state
$\Omega^{*}|1P_{3/2^{-}}\rangle$, its proportion can reach up to about $84\%$.

The latest observations at Belle Belle:2022mrg  make the situation more
complicated for understanding the nature of $\Omega(2012)$. The ratio
$R_{\Xi\bar{K}}^{\Xi\pi\bar{K}}$ is expected to be tested by other
experiments. Furthermore, the radiative decay of $\Omega(2012)\to\Omega\gamma$
is also expected to be measured in future experiments. In the three quark
picture, the branching fraction is predicted to be
$\mathcal{B}[\Omega(2012)\to\Omega\gamma]\simeq 1.7\times 10^{-3}$ Liu:2019wdr
, which can be used to test the nature of $\Omega(2012)$.

## Acknowledgement

This work is supported by the National Natural Science Foundation of China
(Grants No.12175065, No. 12005060, No. 12075288). Ju-Jun Xie is also supported
by the Youth Innovation Promotion Association CAS.

## References

  * [1] J. Yelton et al. [Belle], Observation of an Excited $\Omega^{-}$ Baryon, Phys. Rev. Lett. 121, 052003 (2018).
  * [2] Y. Li et al. [Belle], Evidence for the decay $\Omega_{c}^{0}\to\pi^{+}\Omega(2012)^{-}\to\pi^{+}(\bar{K}\Xi)^{-}$, Phys. Rev. D 104, 052005 (2021).
  * [3] C. H. Zeng, J. X. Lu, E. Wang, J. J. Xie and L. S. Geng, Theoretical study of the $\Omega(2012)$ state in the $\Omega_{c}^{0}\to\pi^{+}\Omega(2012)^{-}\to\pi^{+}(\bar{K}\Xi)^{-}$ and $\pi^{+}(\bar{K}\Xi\pi)^{-}$ decays, Phys. Rev. D 102, 076009 (2020).
  * [4] Y. Oh, $\Xi$ and $\Omega$ baryons in the Skyrme model, Phys. Rev. D 75, 074002 (2007).
  * [5] S. Capstick and N. Isgur, Baryons in a Relativized Quark Model with Chromodynamics, Phys. Rev. D 34, 2809 (1986).
  * [6] R. N. Faustov and V. O. Galkin, Strange baryon spectroscopy in the relativistic quark model, Phys. Rev. D 92, 054005 (2015).
  * [7] U. Loring, B. C. Metsch and H. R. Petry, The Light baryon spectrum in a relativistic quark model with instanton induced quark forces: The Strange baryon spectrum, Eur. Phys. J. A 10, 447 (2001).
  * [8] J. Liu, R. D. McKeown and M. J. Ramsey-Musolf, Global Analysis of Nucleon Strange Form Factors at Low $Q^{2}$, Phys. Rev. C 76, 025202 (2007).
  * [9] K. T. Chao, N. Isgur and G. Karl, Strangeness -2 and -3 Baryons in a Quark Model With Chromodynamics, Phys. Rev. D 23, 155 (1981).
  * [10] Y. Chen and B. Q. Ma, Light flavor baryon spectrum with higher order hyperfine interactions, Nucl. Phys. A 831, 1 (2009).
  * [11] C. S. An, B. C. Metsch and B. S. Zou, Mixing of the low-lying three- and five-quark $\Omega$ states with negative parity, Phys. Rev. C 87, 065207 (2013).
  * [12] C. S. Kalman, $P$ Wave Baryons in a Consistent Quark Model With Hyperfine Interactions, Phys. Rev. D 26, 2326 (1982).
  * [13] M. Pervin and W. Roberts, Strangeness -2 and -3 baryons in a constituent quark model, Phys. Rev. C 77, 025202 (2008).
  * [14] C. S. An and B. S. Zou, Low-lying $\Omega$ states with negative parity in an extended quark model with Nambu-Jona-Lasinio interaction, Phys. Rev. C 89, 055209 (2014).
  * [15] G. P. Engel et al. [BGR Collaboration], QCD with Two Light Dynamical Chirally Improved Quarks: Baryons, Phys. Rev. D 87, 074504 (2013).
  * [16] J. Liang et al. [CLQCD Collaboration], Spectrum and Bethe-Salpeter amplitudes of $\Omega$ baryons from lattice QCD, Chin. Phys. C 40, 041001 (2016).
  * [17] C. E. Carlson and C. D. Carone, Predictions for decays of radially excited baryons, Phys. Lett. B 484, 260 (2000).
  * [18] J. L. Goity, C. Schat and N. N. Scoccola, Analysis of the $[56,2^{+}]$ baryon masses in the 1 / N(c) expansion, Phys. Lett. B 564, 83 (2003).
  * [19] C. L. Schat, J. L. Goity and N. N. Scoccola, Masses of the 70- baryons in large $N_{c}$ QCD, Phys. Rev. Lett. 88, 102002 (2002).
  * [20] N. Matagne and F. Stancu, Masses of $[70,l^{+}]$ Baryons in the $1/N_{c}$ Expansion, Phys. Rev. D 74, 034014 (2006).
  * [21] R. Bijker, F. Iachello and A. Leviatan, Algebraic models of hadron structure. 2. Strange baryons, Annals Phys. 284, 89 (2000).
  * [22] T. M. Aliev, K. Azizi and H. Sundu, Radial Excitations of the Decuplet Baryons, Eur. Phys. J. C 77, 222 (2017).
  * [23] M. S. Liu, K. L. Wang, Q. F. Lü and X. H. Zhong, $\Omega$ baryon spectrum and their decays in a constituent quark model, Phys. Rev. D 101, no.1, 016002 (2020).
  * [24] L. Y. Xiao and X. H. Zhong, Possible interpretation of the newly observed $\Omega$(2012) state, Phys. Rev. D 98, 034004 (2018).
  * [25] Z. Y. Wang, L. C. Gui, Q. F. Lü, L. Y. Xiao and X. H. Zhong, Newly observed $\Omega(2012)$ state and strong decays of the low-lying $\Omega$ excitations, Phys. Rev. D 98, 114023 (2018).
  * [26] K. L. Wang, Q. F. Lü, J. J. Xie and X. H. Zhong, Toward discovering the excited $\Omega$ baryons through nonleptonic weak decays of $\Omega_{c}$, [arXiv:2203.04458 [hep-ph]].
  * [27] T. M. Aliev, K. Azizi, Y. Sarac and H. Sundu, Nature of the $\Omega(2012)$ through its strong decays, Eur. Phys. J. C 78, 894 (2018).
  * [28] T. M. Aliev, K. Azizi, Y. Sarac and H. Sundu, Interpretation of the newly discovered $\Omega$(2012), Phys. Rev. D 98, 014031 (2018).
  * [29] A. J. Arifi, D. Suenaga, A. Hosaka and Y. Oh, Strong decays of multistrangeness baryon resonances in the quark model, Phys. Rev. D 105, 094006 (2022).
  * [30] M. V. Polyakov, H. D. Son, B. D. Sun and A. Tandogan, $\Omega(2012)$ through the looking glass of flavour SU(3), Phys. Lett. B 792, 315-319 (2019).
  * [31] X. Hu and J. Ping, Analysis of $\Omega(2012)$ as a molecule in the chiral quark model, [arXiv:2207.05598 [hep-ph]].
  * [32] N. Ikeno, W. H. Liang, G. Toledo and E. Oset, Interpretation of the $\Omega_{c}\to\pi^{+}\Omega(2012)\to\pi^{+}(\bar{K}\Xi)$ relative to $\Omega_{c}\to\pi^{+}\bar{K}\Xi$ from the $\Omega(2012)$ molecular perspective, [arXiv:2204.13396 [hep-ph]].
  * [33] Y. H. Lin, F. Wang and B. S. Zou, Reanalysis of the newly observed $\Omega^{*}$ state in a hadronic molecule model, Phys. Rev. D 102, 074025 (2020).
  * [34] J. X. Lu, C. H. Zeng, E. Wang, J. J. Xie and L. S. Geng, Revisiting the $\Omega(2012)$ as a hadronic molecule and its strong decays, Eur. Phys. J. C 80, 361 (2020).
  * [35] N. Ikeno, G. Toledo and E. Oset, Molecular picture for the $\Omega(2012)$ revisited, Phys. Rev. D 101, 094016 (2020).
  * [36] J. M. Xie, M. Z. Liu and L. S. Geng, $D_{s0}(2590)$ as a dominant c$\bar{s}$ state with a small $D^{*}K$ component, Phys. Rev. D 104, 094051 (2021).
  * [37] M. P. Valderrama, $\Omega(2012)$ as a hadronic molecule, Phys. Rev. D 98, 054009 (2018).
  * [38] R. Pavao and E. Oset, Coupled channels dynamics in the generation of the $\Omega(2012)$ resonance, Eur. Phys. J. C 78, 857 (2018).
  * [39] Y. Huang, M. Z. Liu, J. X. Lu, J. J. Xie and L. S. Geng, Strong decay modes $\bar{K}\Xi$ and $\bar{K}\Xi\pi$ of the $\Omega(2012)$ in the $\bar{K}\Xi(1530)$ and $\eta\Omega$ molecular scenario, Phys. Rev. D 98, 076012 (2018).
  * [40] T. Gutsche and V. E. Lyubovitskij, Strong decays of the hadronic molecule $\Omega^{\ast}(2012)$, J. Phys. G 48, 025001 (2020).
  * [41] S. Jia et al. [Belle], Search for $\Omega(2012)\to K\Xi(1530)\to K\pi\Xi$ at Belle, Phys. Rev. D 100, 032006 (2019).
  * [42] [Belle], Observation of $\Omega(2012)^{-}\to\Xi(1530)\bar{K}$ and measurement of the effective couplings of $\Omega(2012)^{-}$ to $\Xi(1530)\bar{K}$ and $\Xi\bar{K}$, [arXiv:2207.03090 [hep-ex]].
  * [43] A. Manohar and H. Georgi, Chiral Quarks and the Nonrelativistic Quark Model, Nucl. Phys. B 234, 189-212 (1984)
  * [44] Z. P. Li, The Threshold pion photoproduction of nucleons in the chiral quark model, Phys. Rev. D 50, 5639-5646 (1994).
  * [45] Z. P. Li, H. X. Ye and M. H. Lu, An Unified approach to pseudoscalar meson photoproductions off nucleons in the quark model, Phys. Rev. C 56, 1099-1113 (1997).
  * [46] Q. Zhao, J. S. Al-Khalili, Z. P. Li and R. L. Workman, Pion photoproduction on the nucleon in the quark model, Phys. Rev. C 65, 065204 (2002).
  * [47] X. H. Zhong and Q. Zhao, Charmed baryon strong decays in a chiral quark model, Phys. Rev. D 77, 074008 (2008).
  * [48] X. h. Zhong and Q. Zhao, Strong decays of heavy-light mesons in a chiral quark model, Phys. Rev. D 78, 014029 (2008).
  * [49] L. Y. Xiao and X. H. Zhong, $\Xi$ baryon strong decays in a chiral quark model, Phys. Rev. D 87, 094002 (2013).
  * [50] P. A. Zyla et al. [Particle Data Group], Review of Particle Physics, PTEP 2020, 083C01 (2020).
  * [51] D. Morel and S. Capstick, Baryon meson loop effects on the spectrum of nonstrange baryons, [arXiv:nucl-th/0204014 [nucl-th]].
  * [52] Y. S. Kalashnikova, Coupled-channel model for charmonium levels and an option for $X(3872)$, Phys. Rev. D 72, 034010 (2005).
  * [53] Y. Lu, M. N. Anwar and B. S. Zou, How Large is the Contribution of Excited Mesons in Coupled-Channel Effects?, Phys. Rev. D 95, 034018 (2017).
  * [54] J. F. Liu and G. J. Ding, Bottomonium Spectrum with Coupled-Channel Effects, Eur. Phys. J. C 72, 1981 (2012).
  * [55] Y. Lu, M. N. Anwar and B. S. Zou, Coupled-Channel Effects for the Bottomonium with Realistic Wave Functions, Phys. Rev. D 94, 034021 (2016).
  * [56] R. H. Ni, Q. Li and X. H. Zhong, Mass spectra and strong decays of charmed and charmed-strange mesons, Phys. Rev. D 105, 056006 (2022).