# Out-of-distribution detection algorithms for robust insect classification

Mojdeh Saadati1, Aditya Balu2, Shivani Chiranjeevi2, Talukder Zaki Jubery2,
Asheesh K Singh3, Soumik Sarkar1,2, Arti Singh${}^{3\ast},$ Baskar
Ganapathysubramanian2∗

1Department of Computer Science, Iowa State University, Ames, IA, USA
2Department of Mechanical Engineering, Iowa State University, Ames, IA, USA
3Department of Agronomy, Iowa State University, Ames, IA, USA
∗To whom correspondence should be addressed; E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

Plants encounter a variety of beneficial and harmful insects during their
growth cycle. Accurate identification (i.e., detecting insects’ presence) and
classification (i.e., determining the type or class) of these insect species
is critical for implementing prompt and suitable mitigation strategies. Such
timely actions carry substantial economic and environmental implications. Deep
learning-based approaches have produced models with good insect classification
accuracy; Most of these models are conducive for application in controlled
environmental conditions. One of the primary emphasis of researchers is to
implement identification and classification models in the real agriculture
fields, which is challenging because input images that are wildly out of the
distribution (e.g., images like vehicles, animals, humans, or a blurred image
of an insect or insect class that is not yet trained on) can produce an
incorrect insect classification. Out-of-distribution (OOD) detection
algorithms provide an exciting avenue to overcome these challenge as it
ensures that a model abstains from making incorrect classification prediction
of non-insect and/or untrained insect class images.

We generate and evaluate the performance of state-of-the-art OOD algorithms on
insect detection classifiers. These algorithms represent a diversity of
methods for addressing an OOD problem. Specifically, we focus on extrusive
algorithms, i.e., algorithms that wrap around a well-trained classifier
without the need for additional co-training. We compared three OOD detection
algorithms: (i) Maximum Softmax Probability, which uses the softmax value as a
confidence score, (ii) Mahalanobis distance-based algorithm, which uses a
generative classification approach; and (iii) Energy-Based algorithm that maps
the input data to a scalar value, called energy. We performed an extensive
series of evaluations of these OOD algorithms across three performance axes:
(a) Base model accuracy: How does the accuracy of the classifier impact OOD
performance? (b) How does the level of dissimilarity to the domain impact OOD
performance? and (c) Data imbalance: How sensitive is OOD performance to the
imbalance in per-class sample size? Evaluating OOD algorithms across these
performance axes provides practical guidelines to ensure the robust
performance of well-trained models in the wild, which is a key consideration
for agricultural applications. Our results indicate that OOD detection
algorithms can significantly enhance user trust in insect pests classification
by abstaining from performing classification under uncertain conditions.

## Abbreviations

OOD |  Out-of-distribution
---|---
ID |  In-distribution
MSP |  Maximum Softmax Probability
MAH |  Mahalanobis Distance
EBM |  Energy-Based Models

##  1 Introduction

Insect pests infestation can be observed at all stages of growth in crop
plants, negatively affecting the quality and quantity of yields in agriculture
[1]. Accurate detection of insects is imperative for prompt, timely, and
optimal decision-making [2]. For instance, early detection of pest
infestations can allow farmers to take timely and appropriate action to
prevent or minimize crop damage [3, 4]. More importantly, accurate detection
allows farmers to identify the specific pest species that are causing damage
[5], enabling them to use targeted pest control methods instead of blanket
applications of pesticides. This reduces the risk of harm to beneficial
insects and other non-target organisms. Furthermore, accurate identification
of insect pests can result in effective pest control measures which reduce the
number of crop losses due to insect pests, increase the profitability of
farmer operations, and reduce the amount of chemical runoff into water bodies.
Finally, accurate and timely insect pest detection is also important for
compliance with regulatory requirements related to pesticide use and
environmental protection.

Traditionally, insect-pest identification (and quantification) [6, 7, 8] has
been performed by human experts and scouts. In the past decade, machine
learning approaches have increasingly been used for automating plant stress
identification, classification, and quantification (Singh etal 2016, singh
etal 2018). Researchers initially used classical machine learning algorithms
for insect identification and classification [9, 8, 10], however more
recently, deep learning approaches have been suggested [11, 12, 13]. In the
classical approaches, a set of features such as color and texture are first
extracted from images, and then a classifier learns a mapping from the feature
space to their corresponding label. For example, support vector machines (SVM)
have been commonly used for plant stress detection problems [14, 9]. However,
such approaches require extensive domain knowledge [7, 15], and tedious
approaches for feature extraction. In the past decade or so, researchers have
utilized deep-learning approaches to perform end-to-end classification,
without the need for manual feature extraction. In classifying 13 soybean
pests, Tetila et. al [16] compared the performance of 5 models (Inception-v3
[17], ResNet50 [18], VGG16 [19], VGG19 [19], and Xception [20]) across a
dataset containing 5000 samples. Li et al. [21] leveraged the GoogLeNet
architecture and achieved 98 percent accuracy in a 10-class insect
classification task using a dataset of around 7000 images. Manual labeling of
this moderately sized dataset required extensive effort by domain experts. The
need for manual annotation of large datasets for training deep learning
algorithms has become a major bottleneck to creating high accuracy
classifiers. Recent efforts to utilize self-supervised algorithms seek to
eliminate the need for large, annotated datasets. Using self-supervised
approaches, one can eliminate the need for labeling by utilizing specific
property of the data to design pretext task where model learns the underlying
structure and relationships between elements in the input data while
performing the pretext task. In a recent study by Kar et al. [13] the authors
utilized the Bootstrap Your Own Latent, a self-supervised method to create a
pest classifier algorithm. This insect classifier achieved an accuracy of 93
percent using approximately 15,000 images spanning 22 classes of
agriculturally relevant insects found in the US Midwest. A review of field
highlights a growing trends towards utilizing largerdatasets such as the
iNaturalist dataset (weblink) with $>$ 10 million images), along with weakly-
and self-supervised approaches to develop high accurate classifiers for a wide
range of agriculturally relevant insect-pests species. This progress opens up
the possibility for the practical deployment in agricultural decision support
systems.

However, the trustworthiness of these classifiers is a potential bottleneck
for their practical deployment in agricultural fields. For instance, while
classifiers (or well-trained ML models in general) can achieve high
classification accuracy, these models do not provide any measure of
(un)certainty in making predictions or abstaining from classification in the
case of uncertainty. This holds particularly important for autonomous
agricultural applications, inaccurate predictions may lead to wastage of
resources at the very least and in the worst-case scenario, substantial yield
losses. In the context of ML models for insect classification, practical
deployment requires distinguishing— or at least abstaining from making
predictions— when confronted with confusing or unseen images. This can occur
when the model is presented with images of insect species that do not fall
under any of the categories on which it has been trained. Such
misclassification could have disastrous consequences if the model mistakenly
identifies an unseen insect class, for instance, an invasive species of insect
as a harmless one. This may result in incorrect crop management decisions.
Although the sizes of such confusing classes may vary, the loss of scale
perspective in the image may contribute to these erroneous predictions. Thus,
there is a pressing need to integrate advances in model accuracy with enhanced
trustworthiness to ensure robust deployment in the field 111Trustworthiness
can be defined in several ways. See [22, 23, 24]. Increasing the
trustworthiness of ML models is an active area of research with several
promising approaches [25, 26, 27]. One approach is to identify images that do
not belong to the data distribution the model originally trained on. Such out-
of-distribution (OOD) approaches can ensure that the model can abstain from
making predictions when facing an image that does not belong to any of the
classes it is trained to predict. This is particularly attractive in the
context of insect pest detection as it allows human intervention in case of
uncertain model predictions. Fig. 1 illustrates how such OOD detection can be
used for the insect classification task.

Figure 1: Out-of-distribution visualization in insect classification. The OOD
classifier tries to define decision boundaries for each insect while ensuring
no non-insect images fall into any known insect classes. There is a spectrum
of OOD images. Here, we distinguish them as contextually near OOD (for
instance, leaf images) and contextually far OOD (for instance, car image)
based on their similarity to the in-distribution (ID) images.

OOD detection has been successfully used in several applications — like
autonomous systems [28, 29], medical diagnosis [30, 31, 32], robotics [33,
34], and social science [35]— to improve the reliability and safety of
systems. In autonomous systems, OOD detection can help self-driving cars avoid
any action or stop when they interface with an unknown object, which is
crucial for the safety of passengers and pedestrians. For deep learning
models, one of the earliest algorithms proposed for identifying out-of-
distribution data is maximum-softmax-probability (MSP) [36]. This algorithm
relies on the assumption that deep learning models are more confident in the
classification of in-distribution data rather than out-of-distribution data.
The algorithm uses the softmax value as a metric to measure the confidence of
predictions. Due to its simplicity and good performance, this algorithm has
been prevalent in addressing OOD detection. However, it has been shown in
practice that MSP can produce a high false positive rate during OOD detection.
Recently, Liu et. al. [37] showed (using both theory and empirical evidence)
that an energy-based model can be a great substitute for MSP as they are
aligned with the probability density of in-distribution data. The basic idea
here is to design a function that maps the input (or input features) to a
number, often called energy. This energy function is designed such that in-
distribution samples have low energy while out-of-distribution samples exhibit
higher energy. OOD boils down to comparing the energy of an unseen input
against a threshold.

In contrast to these discriminative OOD algorithms, which try to find the best
decision boundaries, there are also OOD algorithms based on generative models
that focus on estimating the in-distribution density [38, 39, 40, 41]. Among
generative model-based OOD algorithms, OOD detection based on Mahalanobis
distance is the most popular, with significant recent work. Denouden et.al.
[42] solves the OOD problem in the context of auto-encoder architecture, using
the observation that an auto-encoder is ineffective in encoding and
reconstructing OOD data in comparison to in-distribution ones. They
distinguish OOD data from in-distribution by defining a threshold based on the
Mahalanobis distance metric on reconstruction error. Lee et.al. [43] extracted
class conditional Gaussian distributions of deep learning features based on
Gaussian discriminant analysis, leading to a Mahalanobis distance-based
confidence score. Ren et.al. [44] commented that the latter algorithm suffered
from near-OOD detection and offered an adjustment to the previous algorithm.

All the algorithms described above are extrusive; i.e. they wrap around a
trained ML classifier without the need for additional (re)training of the
classifier. Another class of OOD algorithms are available that work by
introducing part of the OOD data during the classifier training process. These
intrusive approaches require co-training of the OOD detection along with
classifier. For instance, Hendrycks et.al. [45] modify the classifier’s cross-
entropy loss function and add an extra term to handle OOD data so that the
softmax distribution for OOD data is uniform. Researchers also incorporated
OOD detection into the classifier’s architecture and designed a hierarchical
outlier detection algorithm (HOD) to identify the outlier dermatological
conditions [46]. Fort et.al. [47] and Ren et.al. [40] also adjust the
classifier by adding an extra class to classify the OOD data during the
prediction.

We choose to focus on the extrusive algorithms from the agricultural
application point of view, for the following reasons: (a) The increasing
availability of well-trained models for various agricultural phenotyping
tasks. These models are often trained on huge datasets using significant
computational effort. Retraining them to incorporate OOD detection using
intrusive algorithms may be computationally infeasible for many practitioners.
(b) Additionally, extrusive algorithms allow efficient personalization of the
OOD detection for the specific application cases involved.

In this paper, we explored the utility of OOD models for agricultural
applications, using a specific application of insect pest classification. This
work addresses a gap to enhance trustworthiness in the agricultural domain,
particularly for insect pest classification. We perform OOD analysis on an
insect classifier that is able to distinguish between 142 agriculturally
relevant insect classes. This large class size makes OOD detection
particularly relevant. Our key contributions are as follows:

  * •

We incorporate the concept of OOD detection for insect-pest classification,
using a classifier that can distinguish between 142 agriculturally relevant
insect pest species trained on a large insect dataset (2 million images). This
differs from previous works, which conduct their analysis on benchmark
datasets such as CIFAR10 [48], CIFAR100 [48], and SVHN [49], which have
relatively smaller data sizes ( $\leq$ 100,000 samples).

  * •

We computationally explore the performance of three OOD methods (MSP,
Mahalanobis distance, and energy-based model) for insect-pest classification.
We report on OOD performance using different classifier architectures,
different training dataset distributions, and testing on different OOD
datasets.

The paper is structured as follows. Section 2 provides an overview of the
problem definition, including a discussion of the OOD detection methods, in-
distribution and out-of-distribution datasets, and classifier architectures.
Section 3 details the set of analyses that were conducted to evaluate OOD
detection algorithms through different performance axes. Finally, in Section
4, concluding remarks are provided, along with a discussion of future works.

##  2 Materials and Methods

To understand the characteristics of OOD algorithms in the context of insect-
pest classification, we selected three OOD algorithms - Maximum Softmax
Probability, Mahalanobis distance, and energy-based model. Maximum Softmax
Probability serves as a suitable baseline for comparison, while the
Mahalanobis distance-based and energy-based algorithms each represent distinct
families of OOD algorithms, discriminative and generative, respectively. We
chose extrusive OOD algorithms as they can be applied to any arbitrary insect
classifier as a wrapper. For the performance evaluation of these OOD detection
algorithms, we perform a comprehensive set of analyses, as illustrated in Fig.
2. We first prepared four OOD datasets and the ID dataset with 142 insect
classes. Next, we used various insect classifiers that used diverse
architectures and finally applied the OOD algorithms on the combination of
insect classifiers and datasets and measured their performance. We detail each
of these steps next (OOD algorithm description $\rightarrow$ classifier
architecture description $\rightarrow$ data description), starting with the
description of the algorithms; see also Fig. 3 for a schematic of these
algorithms. All our experiments were performed on the local high-performance
computing cluster, using one NVidia A100 80GB GPU card.

###  2.1 Out-of-distribution Methods

Out-of-distribution (OOD) detection refers to the process of identifying
whether a given input data point belongs to the same distribution as the
training data or not. In other words, it aims to identify when a model
encounters data that is significantly different from what it has been trained
on. We detail the three OOD algorithms considered:

Figure 2: The analysis process: Box 1 represents the data. We split the ID-
distribution data into three sub-datasets that are used for training the
insect classifier, training the Mahalanobis distance OOD algorithm, and
testing the results. We also utilize four different OOD datasets for testing
our results. Box 2 shows the three insect classifier architectures, along with
the three OOD algorithms used. Box 3 illustrates the workflow. The training
data (ID1) is used to train the insect classifier. The OOD algorithms (and ID2
data, if needed) are next trained. Finally, the test data (ID3, along with
OOD1, OOD2, OOD3, and OOD4) is used to test the OOD performance.

  * •

Maximum Softmax Probability(MSP) [36] is a simple yet effective method for OOD
detection in deep learning models. It is based on the fact that softmax output
probabilities (the last layer of the deep learning network before
classification) for in-distribution data tend to be more confident than those
for OOD data. Specifically, the MSP method computes the maximum value of the
softmax probabilities for a given input data point and compares it to a
threshold value. If the maximum value is below the threshold, the data point
is considered OOD. The threshold is usually determined based on a validation
set or using a predefined value.

MSP has been shown to perform well in detecting OOD data in various
applications, such as image classification, natural language processing, and
speech recognition [36]. It is also computationally efficient and easy to
implement as it only uses softmax value extracted from the last layer of the
neural network, making it a popular choice for OOD detection in practice [50,
51, 43]. However, MSP may not perform well when the distribution of in-
distribution data is significantly different from the training data or when
there is an overlap between in-distribution and OOD data distributions. In
such cases, more sophisticated methods, such as Mahalanobis distance or
energy-based models, may be more effective.

  * •

Mahalanobis distance-based algorithm [43] is another commonly used method for
OOD detection in deep learning models. Mahalanobis distance measures the
distance between a given input data point and the distribution of in-
distribution data in the feature space. It uses a distance metric that
considers the correlation between features and the covariance matrix of the
training data.

The feature space is simply the output of the penultimate layer of the
classifier, denoted as $f(x)$. Given a classifier with labels,
$\\{1,...,C\\}$, we first estimate the spread of the features for each class.
This is estimated by assuming that these features belong to multivariate
Gaussian distributions. Therefore, the probability of a feature vector,
$f(x)$, given an input, $x$ with a class label $y=c$ is given by
$P(f(x)|y=c)=\mathcal{N}(f(x)|\mu_{c},\hat{\Sigma})$ where $\mu_{c}$ and
$\hat{\Sigma}$ are respectively the mean and covariance.

The Mahalanobis distance-based (MAH) confidence score, $M(x)$, is then the
distance (of the feature, $f(x)$) of an unseen input $x$ to the closest class-
conditional Gaussian distribution:

$M(x)=\max\limits_{c}-(f(x)-\hat{\mu_{c}})^{T}\hat{\Sigma}^{-1}(f(x)-\hat{\mu_{c}})$
(1)

The Mahalanobis distance is the normalized distance, with the normalization
performed using the covariance matrix of the training data. This helps account
for correlations between features and results in a more accurate distance
measure. A threshold for this distance is computed using a given OOD training
sample $\\{(x_{1},y_{1}),...,(x_{N},y_{N})\\}$

  * •

Energy-based models (EBM) [37] are a type of probabilistic model that assigns
an energy score to the feature vector, $f(x)$, of each input, $x$. Using this
energy score, an EBM can be used to distinguish between in-distribution (ID)
and OOD samples. During inference, the energy score of an unseen input is
compared to the energy scores of the training data. If the energy score of the
input is significantly higher than the highest energy score of the training
data, it is likely an OOD sample. The energy model uses the softmax
representation, $f(x)$ of a trained classifier, and the energy function is
defined as

$E(x)=-Tlog\left(\sum_{i}^{C}e^{f_{i}(x)/T}\right)$ (2)

The form of the energy function is motivated by other energy-based models like
Boltzmann machines and variational autoencoders, where the energy function is
related to the Gibbs distribution and the associated free energy. In the above
formula, the parameter $T$ is called the temperature, which is set to $T=1$.

Figure 3: OOD detection process on MAH (top row), and MSP/EBM (bottom row).
(1) In the Mahalanobis distance algorithm, the trained insect classifier and
in-distribution data are given as input to the model, and the distribution of
the feature vector (here, the prelogit) is conditioned on each class
extracted. Then Mahalanobis distance of ID and OOD test data to the nearest
conditional feature vector is calculated, and a threshold is obtained. For MSP
and EBM instead of Mahalanobis distance, softmax, and energy are calculated,
respectively.

###  2.2 Insect Classifiers Methods

We next describe the architectures used to train the insect classifier. The
architecture of a classifier plays a critical role in determining the accuracy
and efficiency of the model, and careful consideration and experimentation are
often required to find the optimal design for a given task. We utilize three
diverse types of architectures that have found significant success in
practical image classification applications. These three network architecture
choices, along with the three OOD algorithms described in the previous sub-
section, provide a broad cross-section of approaches on which to evaluate OOD
performance for our specific insect classification problem.

We consider the following architectures:

  * •

ResNet50: ResNet50, a variant of the convolutional neural network (CNN) model
proposed by He et al. [18], has gained widespread popularity for its
exceptional performance in computer vision and image classification tasks. The
model owes its success to incorporating skip connections within its residual
blocks, effectively addressing the challenge of diminishing or exploding
gradients. Our implementation employs ResNet50, featuring 50 deep convolution
layers.

  * •

RegNetY32: RegNet is an optimized design space developed by Radosavovic et al.
[52] where they explore a diverse set of parameters of a network architecture
like width, depth, groups (commonly called as AnyNet, an initial space of
unconstrained models which uses models like ResNet50 [18] as its base). By
conducting numerous experiments with different parameter values for the design
space, they successfully developed the optimized RegNetX. The combination of
the well-known Squeeze-and-Excitation (SE) operation with ResNetX produced
RegNetY, which exhibited outstanding performance.[53] We employed the
RegNetY32 model in this study .

  * •

VGG11: VGG is a CNN introduced by the Visual Geometry Group (VGG) at the
University of Oxford [19]. This CNN architecture uses RELU activation function
and pretrained weights (trained on ImageNet) to reduce computation load. We
use VGG11 which is a variant of VGG. VGG11 is an architecture with 11 weighted
layers. It includes 9 convolutional layers and three fully connected layers.

###  2.3 Datasets

Figure 4: Dataset visualization.The Insecta dataset was used as In-
Distribution and ImageNet, Non-Insecta, OOD Insect, Human Face were used as
OOD datasets.

We curated two series of datasets, one for in-distribution (ID) and one for
OOD. We used ID data for three main purposes: (i) training the insect
classifier, (ii) training the generative OOD algorithm (Mahalanobis distance),
and (iii) evaluating the performance of OOD algorithms with respect to
distinguishing the ID data from OOD data. For these objectives, we curated an
insect dataset consisting of 142 agriculturally relevant species with a large
economic impact in North American agriculture. This dataset is a subset of the
publicly available iNaturalist dataset [54] and consists of 2 million insect
pictures. We split the dataset into train and validation sets with a ratio of
7 to 3, and use the training subset to train the classifier. We then divide
the validation folder into two equally sized smaller datasets. One was used
for training the OOD model (only for the generative OOD model), and the other
was used for evaluating the performance of OOD algorithms.

For OOD data, we utilized four datasets with different degrees of similarity
to the in-distribution insect data. An illustration for the datasets are
provided in Fig 4. We briefly describe each of these datasets.

  * •

ImageNet [55] (far OOD): We start with the ImageNet 2012 classification
dataset consisting of 1.2 million images across 1000 object categories. We
then took ten random samples from each class and excluded all insect-related
objects (e.g. bee, cricket, ant) from it, which resulted in 9730 samples.

  * •

Human Face [56] (far OOD): We use the dataset from the face mask recognition
dataset of the Kaggle
competition222https://www.kaggle.com/datasets/ashwingupta3012/human-faces,
which includes pictures of human faces with and without masks. The dataset
consists of 3059 images.

  * •

Non-Insecta [54] (near OOD): this dataset is a subset of iNaturalist
333https://github.com/visipedia/inat_comp/tree/master/2021, where we exclude
all the Insecta images from it. This dataset consists of 74740 images.

  * •

OODInsect (near OOD): This dataset includes all insect pictures that do not
belong to any of the 142 classes of ID data. This dataset is also collected
from the iNatrualist dataset. This dataset includes 56487 pictures of OOD
insects.

Taken together, these four datasets represent a diverse set of out-of-
distribution images, when compared to the dataset on which the insect
classifier is trained.

###  2.4 Metrices for OOD performance evaluation

We employed AUROC and FPR95 to evaluate the performance of OOD detection
algorithms. We briefly discuss each metric below:

AUROC: AUROC stands for the area under the receiver operating characteristics
(ROC) curve, which is a performance measure for classification tasks, in this
case distinguishing between ID or OOD. The ROC curve plots the true positive
rate (TPR) versus the false positive rate (FPR) at various classification
thresholds. The AUROC is the area under this curve, which ranges from 0 to 1.
A higher AUROC indicates better OOD detection performance, as it means that
the model has a higher TPR for a given FPR. AUROC is a useful metric for OOD
detection because it captures the overall performance of a model at different
operating points. By considering the full range of FPR and TPR values, AUROC
provides a comprehensive evaluation of a model’s ability to distinguish
between ID (in-distribution) and OOD samples. However, it is important to note
that AUROC does not provide information about the specific operating point at
which the model achieves its highest performance. For this reason, additional
metrics such as FPR95 are used to further evaluate the OOD detection
performance.

FPR95: FPR95, or false positive rate at 95% true positive rate, is a metric
used to evaluate OOD (Out-of-Distribution) detection algorithms. FPR95 is the
probability that a negative example (OOD) is wrongly classified as positive
(ID) when the true positive rate (TPR) is equal to 95%. A lower FPR95
indicates better OOD detection performance, as it means that the model is less
likely to misclassify OOD samples as in-distribution (ID) samples. FPR95 is an
important metric because it captures the performance of a model at a high TPR,
which is where OOD detection is most critical.

##  3 Results and Discussion

We performed a large number of computational experiments along three
performance axes: (i) classifier architectures and classifier accuracies,
detailed in Section. 3.1 (ii) different sample sizes for training data,
detailed in Section. 3.2, and (iii) the degree of similarity of OOD dataset
features to ID data, detailed in Section. 3.3.

These experiments indicate that EBM outperforms the other two (MAH and MSP)
OOD detection algorithms. Additionally, our results indicate that the
RegNetY32 architecture exhibits superior performance in insect detection
compared to the other two model architectures (ResNet50, VGG11). Taken
together, a combination of EBM with a RegNetY32 classifier produced an
excellent insect classifier.

We plot the AUROC curve for this classifier in Fig. 5(a), which exhibits an
aggregated OOD classifier performance over all possible energy level
thresholds. The result of our analysis indicated an AUROC of 0.72. We then
tried to find the threshold that gave us the most promising result. Based on
our analysis, we found that the threshold of 31.0 gave the OOD classifier
simultaneously the lowest FPR=0.31 and highest TPR=0.69, respectively ($FPR=0$
and $TPR=1$). Setting the energy level to this threshold resulted in an
accuracy of 68.97 for our OOD model.

(a)

(b)

Figure 5: The performance evaluation of EBM on RegNetY32. (a) AUROC curve of
binary classification between ID and OOD for the range of energy thresholds in
the proposed EBM OOD detector. (b) boxplot diagram of OOD and ID distribution.

In order to interpret the threshold and AUROC results, the energy distribution
of ID and OOD data were further investigated. Fig. 5(b) captures the boxplot
illustration of OOD and ID data distributions. They exhibit the following
inter-quantiles: In-distribution inter-quantile of {min: -11.23, IQ1: 29.00,
median: 36.68, IQ3: 44.71, max: 676.69}, and Out-of-distribution inter-
quantile of {min: -43.73, IQ1: 16.33, median: 23.69, IQ3: 34.30, max:
2445.05}.

Majority of ID data were retained while an acceptable portion of OOD was
excluded when using an energy threshold level of 31.0. The wide scattering in
the energy level of OOD data indicated the diversity of OOD images. Difficulty
in OOD classification was also evident, as there was a partial overlap of low
energy values on both ID and OOD.

###  3.1 Impact of classifier accuracy (and architecture) on OOD detection

We explored the interplay between the accuracy of the insect classifier and
the performance of the OOD algorithm. This question is motivated by a recent
empirical observation [57] that suggests that improving the accuracy of the
classifier is correlated with its ability to identify when an input does not
belong to any of its classes. This observation is particularly appealing
because one could invest effort in improving the accuracy of the classifier
(via improved data collection, augmentation, hyperparameter tuning, and label
smoothing) and get the added benefit of improved OOD detection.

To explore this, we created a sequence of 15 trained insect classifiers with
increasing accuracy for each of the three types of network architectures
(ResNet50, RegNetY32, and VGG11). These models are trained using cross-entropy
loss function and AdamW optimizer with a learning rate of $1e-3$ and a batch
size of $256$. Their performance – on the validation dataset – ranged from an
accuracy of 80% to 97.5% across these 45 models, providing a statistically
robust number of trained models. Next, we ran all three OOD algorithms (MSP,
MAH, and EBM) on these 45 trained insect classifiers. For evaluating the OOD
performance, we consider the non-Insecta dataset as the OOD dataset. We
ensured both ID and OOD test datasets have the same size by random sampling of
74740 (the size of the Non-Insecta dataset) from ID data. The result of our
analysis is shown in Fig. 6.

Figure 6: The trend of AUROC for the three OOD algorithms (MSP, MAH, EBM) over
the increment in insect classifier’s accuracy for the three architectures
(ResNet50, RegNetY32, VGG11). Each diagram specifies the performance of one
architecture, and diagrams are sorted from left to right based on the increase
in the number of weight parameters ($mp$ stands for million parameters).

We observe several trends. First, the performance of the EBM algorithm is
(almost always) better than the other two OOD algorithms across all three
architectures. However, notice that improving the accuracy of the classifier
does not improve the EBM-based OOD detection performance, especially for
architectures with a very large number of trainable parameters; VGG11 has 133
million trainable parameters, and RegNetY32 has 145 million trainable
parameters. For architecture with a moderate number of parameters (i.e.
Resnet50 with 23 million trainable parameters), the observation of [57] holds.
On the other hand, the other two OOD algorithms (MSP and MAH) show slight to
no improvement in OOD performance as classifier accuracy is increased.

###  3.2 Impact of degree of out-of-domainness on OOD detection

In this sub-section, we explore how OOD performance is affected by the type of
OOD datasets considered. This is particularly important to understand for
agricultural applications, where the adoption of these tools may be
compromised if the classifier makes obvious classification mistakes, for
instance, classifying a human face as an insect.

We considered four datasets – Human face, ImageNet, Non-Insecta, and OODInsect
– that exhibit increasing contextual variations from the insect dataset used
to train the classifiers. The non-Insecta and OOD Insect datasets share many
similar features to the in-domain dataset, such as leaves or trees in the
image background; on the other hand, ImageNet and HumanFace share less similar
features to the ID insect dataset. These datasets are visualized in Fig. 4.
Furthermore, to avoid potential bias from different sizes of datasets, we took
a random sampling of 3059 (our smallest OOD dataset size) from each of these
datasets. We then measured the performance of the three OOD algorithms on our
best-performing classifier.

Figure 7: OOD detection algorithms performance across OOD datasets with
different degrees of similarity to ID. Two metrics of AUROC and FPR95 are
chosen to show two performance aspects. (Right) True Positive rate at False
Positive rate 95%, the lower the FPR95 the better performance of OOD detector.

Fig. 7 lays out the results of this analysis. We observe that EBM has the best
performance on all four OOD datasets – exhibiting consistently good AUROC, as
well as low FPR95. MSP exhibited high FPR95, independent of the type of OOD
data. This has been well documented [37]. EBM and MAH performed better than
MSP on all OOD datasets. In particular, better performance was noted in the
ImageNet dataset and OOD Insects. This could be because the classifier is pre-
trained on Instagram images, which share similar features to the ImageNet
dataset, and is trained on insect data that is similar to the OOD insect
dataset. This makes the OOD detector more robust on these datasets, and
suggests that the similarity of features between the images the classifier is
trained (or pre-trained on) and the OOD samples can help improve the OOD
detection. This observation raises the hypothesis that using models pre-
trained on diverse/large datasets may produce especially robust OOD detection.

###  3.3 Effect of data imbalance on OOD detection

Here, we explore how data imbalance in the training data impacts OOD
performance. We consider three subsets of ID data to train the OOD algorithms.
We call these three datasets balanced, unbalanced, and unbalanced-uniform. For
consistency, we ensured that each dataset had the same number of images (58362
images). In the balanced subset, all classes have the same number of images
(411 images per class). The unbalanced subset, on the other hand, exhibits the
same data distribution as the original 2M image dataset used to train the
classifier. Finally, in unbalanced-uniform, we considered a data distribution
that is completely random, i.e. the size of each class is an independent
uniform random variable. This allows us to explore the impact (or lack
thereof) of specific data distributions exhibited by the unbalanced vs the
unbalanced-uniform datasets. We compared these three ID subsets with the Non-
Insecta OOD dataset on OOD detection performance on our best classifier.

The result of this analysis is shown in Fig. 8. EBM – a discriminative OOD
algorithm – is robust to data imbalance, and performs very well with high
AUROC. On the other hand, the generative OOD algorithm, MAH, shows sensitivity
to data imbalance. Over all, we observe that using EBM is particularly
promising for agricultural applications, where there is often the possibility
of data imbalance.

This analysis is especially important in agricultural applications,
particularly for the insect classification problem, due to the possibility of
significant data imbalance. For instance, charismatic species like the monarch
butterfly are typically heavily imaged, while rare (or uninteresting) species
can have very few samples in a dataset. One of the largest insect datasets,
the iNaturalist dataset, consists of 13 million insect images. This dataset
exhibits a large data imbalance between classes, with some classes containing
fewer than 100 images and others containing more than 100,000 images. Such
data imbalances are rather common in agricultural applications, and a clear
understanding of the impact of such data imbalance on OOD performance is
useful for practical deployment.

Figure 8: OOD performance across different levels of equilibrium of ID
dataset. Only the MAH algorithm shows a significant difference in performance
between these three subsets.

##  4 Conclusions

Automated insect pest identification is an economically critical agricultural
task. It is important that well-trained models, when deployed in the wild,
abstain from making predictions when encountering data that is out of their
training distribution. We explored and quantified the performance of several
OOD approaches applied to insect pests classification. Our insect pest
classifier was trained on over 2 million images to accurately classify 142
agriculturally relevant insect species. For this particular application, the
energy-based model (EBM) algorithm for OOD detection was the best-performing
model. This work is the first step towards robustifying the performance of
classifiers deployed for agricultural applications.

We performed an extensive set of computational experiments to evaluate the
performance of OOD algorithms, which provided several useful observations to
the agricultural community. First, we find that for classifiers with fewer
model parameters, there is a correlation between classifier performance and
OOD detection accuracy. However, this correlation disappears for more complex
models. Second, it was observed that the EBM OOD detector has superior
performance when OOD datasets are more similar to the training and pretraining
datasets. Third, empirical observations indicate that both discriminatively
selected OOD detection algorithms, EBMs and MSP, exhibit insensitivity to data
imbalance, in contrast to their generative counterpart algorithm, MAH, method.

The OOD approach described in this paper allows the classifier to abstain from
making predictions when the input data does not belong to the training
distribution. This provides a systematic approach for autonomous agricultural
applications to request human intervention, thus improving the reliability and
trustworthiness of these tools. We anticipate this approach to be applicable
to a wide range of agriculturally relevant classification tasks, such as
scouting and identification of biotic (disease) and abiotic stresses (nutrient
deficiency), as well as providing assistance in agricultural decision-making
processes.

## Author Contributions

M.S., A.S., and B.G. conceived the project. Insect classifier training and
data pre-processing were conducted by M.S. and S.C. OOD analysis and training
was conducted by M.S. M.S., S.C, A.B., Z.J., and B.G wrote the manuscript
draft. All authors reviewed and edited the manuscript.

## Acknowledgements

This work was supported by the AI Institute for Resilient Agriculture (USDA-
NIFA #2021-67021-35329), COALESCE: COntext Aware LEarning for Sustainable
CybEr-Agricultural Systems (NSF CPS Frontier #1954556), FACT: A Scalable Cyber
Ecosystem for Acquisition, Curation, and Analysis of Multispectral UAV Image
Data (USDA-NIFA #2019-67021-29938), Smart Integrated Farm Network for Rural
Agricultural Communities (SIRAC) (NSF S&CC #1952045), and USDA CRIS Project
IOW04714. Support was also provided by the Plant Sciences Institute.

## Competing Interests

The authors declare that there is no conflict of interest regarding the
publication of this article.

## References

  * [1] Sandra Skendžić et al.  “The impact of climate change on agricultural insect pests”  In _Insects_ 12.5 MDPI, 2021, pp. 440
  * [2] David Dent and Richard H Binks  “Insect pest management”  Cabi, 2020
  * [3] Roslyn D Noar, Chelsea J Jahant-Miller, Sherrie Emerine and Rosemary Hallberg  “Early warning systems as a component of integrated pest management to prevent the introduction of exotic pests”  In _Journal of Integrated Pest Management_ 12.1 Oxford University Press US, 2021, pp. 16
  * [4] Luca Heeb, Emma Jenner and Matthew JW Cock  “Climate-smart pest management: building resilience of farms and landscapes to changing pest threats”  In _Journal of Pest Science_ 92.3 Springer, 2019, pp. 951–969
  * [5] Kil-Nam Kim, Qiu-Ying Huang and Chao-Liang Lei  “Advances in insect phototaxis and application to pest management: a review”  In _Pest management science_ 75.12 Wiley Online Library, 2019, pp. 3135–3143
  * [6] Sapan Naik and Bankim Patel  “Machine vision based fruit classification and grading-a review”  In _International Journal of Computer Applications_ 170.9, 2017, pp. 22–34
  * [7] Arti Singh, Baskar Ganapathysubramanian, Asheesh Kumar Singh and Soumik Sarkar  “Machine learning for high-throughput stress phenotyping in plants”  In _Trends in plant science_ 21.2 Elsevier, 2016, pp. 110–124
  * [8] Denan Xia et al.  “Insect detection and classification based on an improved convolutional neural network”  In _Sensors_ 18.12 MDPI, 2018, pp. 4169
  * [9] Thenmozhi Kasinathan, Dakshayani Singaraju and Srinivasulu Reddy Uyyala  “Insect classification and detection in field crops using modern machine learning techniques”  In _Information Processing in Agriculture_ 8.3 Elsevier, 2021, pp. 446–457
  * [10] Yanping Chen et al.  “Flying insect detection and classification with inexpensive sensors”  In _JoVE (Journal of Visualized Experiments)_ , 2014, pp. e52111
  * [11] Toke T Høye et al.  “Deep learning and computer vision will transform entomology”  In _Proceedings of the National Academy of Sciences_ 118.2 National Acad Sciences, 2021, pp. e2002545117
  * [12] Benjamin Feuer et al.  “Zero-Shot Insect Detection via Weak Language Supervision”, 2023
  * [13] Soumyashree Kar et al.  “Self-Supervised Learning Improves Agricultural Pest Classification”  In _AI for Agriculture and Food Systems_ , 2021
  * [14] MA Ebrahimi, Mohammad Hadi Khoshtaghaza, Saeid Minaei and Bahareh Jamshidi  “Vision-based pest detection based on SVM classification method”  In _Computers and Electronics in Agriculture_ 137 Elsevier, 2017, pp. 52–58
  * [15] Asheesh Kumar Singh, Baskar Ganapathysubramanian, Soumik Sarkar and Arti Singh  “Deep learning for plant stress phenotyping: trends and future perspectives”  In _Trends in plant science_ 23.10 Elsevier, 2018, pp. 883–898
  * [16] Everton Castelão Tetila et al.  “Detection and classification of soybean pests using deep learning with UAV images”  In _Computers and Electronics in Agriculture_ 179 Elsevier, 2020, pp. 105836
  * [17] Christian Szegedy et al.  “Rethinking the inception architecture for computer vision”  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2016, pp. 2818–2826
  * [18] Kaiming He, Xiangyu Zhang, Shaoqing Ren and Jian Sun  “Deep residual learning for image recognition”  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2016, pp. 770–778
  * [19] Karen Simonyan and Andrew Zisserman  “Very deep convolutional networks for large-scale image recognition”  In _arXiv preprint arXiv:1409.1556_ , 2014
  * [20] François Chollet  “Xception: Deep learning with depthwise separable convolutions”  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2017, pp. 1251–1258
  * [21] Yanfen Li et al.  “Crop pest recognition in natural scenes using convolutional neural networks”  In _Computers and Electronics in Agriculture_ 169 Elsevier, 2020, pp. 105174
  * [22] James Duncan et al.  “VeridicalFlow: a Python package for building trustworthy data science pipelines with PCS”  In _Journal of Open Source Software_ 7.69, 2022, pp. 3895
  * [23] Ehsan Toreini et al.  “The relationship between trust in AI and trustworthy machine learning technologies”  In _Proceedings of the 2020 conference on fairness, accountability, and transparency_ , 2020, pp. 272–283
  * [24] Ujwal Gadiraju and Jie Yang  “What can crowd computing do for the next generation of AI systems?”  In _CSW@ NeurIPS_ , 2020, pp. 7–13
  * [25] Umang Bhatt et al.  “Uncertainty as a form of transparency: Measuring, communicating, and using uncertainty”  In _Proceedings of the 2021 AAAI/ACM Conference on AI, Ethics, and Society_ , 2021, pp. 401–413
  * [26] Thai Quang Le  “Trustworthy Machine Learning: Learning Under Security, Explainability and Uncertainty Constraints”  The Pennsylvania State University, 2022
  * [27] Angelos Chatzimparmpas et al.  “The state of the art in enhancing trust in machine learning models with the use of visualizations”  In _Computer Graphics Forum_ 39.3, 2020, pp. 713–756  Wiley Online Library
  * [28] Gregory P Meyer and Niranjan Thakurdesai  “Learning an uncertainty-aware object detector for autonomous driving”  In _2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_ , 2020, pp. 10521–10527  IEEE
  * [29] Feiyang Cai and Xenofon Koutsoukos  “Real-time out-of-distribution detection in learning-enabled cyber-physical systems”  In _2020 ACM/IEEE 11th International Conference on Cyber-Physical Systems (ICCPS)_ , 2020, pp. 174–183  IEEE
  * [30] Tianshi Cao, Chin-Wei Huang, David Yu-Tung Hui and Joseph Paul Cohen  “A benchmark of medical out of distribution detection”  In _arXiv preprint arXiv:2007.04250_ , 2020
  * [31] Davood Karimi and Ali Gholipour  “Improving calibration and out-of-distribution detection in medical image segmentation with convolutional neural networks”  In _arXiv preprint arXiv:2004.06569_ , 2020
  * [32] Jasper Linmans, Jeroen Laak and Geert Litjens  “Efficient Out-of-Distribution Detection in Digital Pathology Using Multi-Head Convolutional Neural Networks.”  In _MIDL_ , 2020, pp. 465–478
  * [33] Alec Farid, Sushant Veer and Anirudha Majumdar  “Task-driven out-of-distribution detection with statistical guarantees for robot learning”  In _Conference on Robot Learning_ , 2022, pp. 970–980  PMLR
  * [34] Michael Yuhas et al.  “Embedded out-of-distribution detection on an autonomous robot platform”  In _Proceedings of the Workshop on Design Automation for CPS and IoT_ , 2021, pp. 13–18
  * [35] Philip Boyer, David Burns and Cari Whyne  “Out-of-distribution detection of human activity recognition with smartwatch inertial sensors”  In _Sensors_ 21.5 MDPI, 2021, pp. 1669
  * [36] Dan Hendrycks and Kevin Gimpel  “A baseline for detecting misclassified and out-of-distribution examples in neural networks”  In _arXiv preprint arXiv:1610.02136_ , 2016
  * [37] Weitang Liu, Xiaoyun Wang, John Owens and Yixuan Li  “Energy-based out-of-distribution detection”  In _Advances in Neural Information Processing Systems_ 33, 2020, pp. 21464–21475
  * [38] Hyunsun Choi and Eric Jang  “Generative ensembles for robust anomaly detection”, 2018
  * [39] Eric Nalisnick et al.  “Do deep generative models know what they don’t know?”  In _arXiv preprint arXiv:1810.09136_ , 2018
  * [40] Jie Ren et al.  “Likelihood ratios for out-of-distribution detection”  In _Advances in neural information processing systems_ 32, 2019
  * [41] Joan Serrà et al.  “Input complexity and out-of-distribution detection with likelihood-based generative models”  In _arXiv preprint arXiv:1909.11480_ , 2019
  * [42] Taylor Denouden et al.  “Improving reconstruction autoencoder out-of-distribution detection with Mahalanobis distance”  In _arXiv preprint arXiv:1812.02765_ , 2018
  * [43] Kimin Lee, Kibok Lee, Honglak Lee and Jinwoo Shin  “A simple unified framework for detecting out-of-distribution samples and adversarial attacks”  In _Advances in neural information processing systems_ 31, 2018
  * [44] Jie Ren et al.  “A simple fix to mahalanobis distance for improving near-ood detection”  In _arXiv preprint arXiv:2106.09022_ , 2021
  * [45] Dan Hendrycks, Mantas Mazeika and Thomas Dietterich  “Deep anomaly detection with outlier exposure”  In _arXiv preprint arXiv:1812.04606_ , 2018
  * [46] Abhijit Guha Roy et al.  “Does your dermatology classifier know what it doesn’t know? detecting the long-tail of unseen conditions”  In _Medical Image Analysis_ 75 Elsevier, 2022, pp. 102274
  * [47] Stanislav Fort, Jie Ren and Balaji Lakshminarayanan  “Exploring the limits of out-of-distribution detection”  In _Advances in Neural Information Processing Systems_ 34, 2021, pp. 7068–7081
  * [48] Alex Krizhevsky and Geoffrey Hinton  “Learning multiple layers of features from tiny images”  Toronto, ON, Canada, 2009
  * [49] Yuval Netzer et al.  “Reading digits in natural images with unsupervised feature learning”, 2011
  * [50] Petra Bevandić, Ivan Krešo, Marin Oršić and Siniša Šegvić  “Discriminative out-of-distribution detection for semantic segmentation”  In _arXiv preprint arXiv:1808.07703_ , 2018
  * [51] Jiefeng Chen et al.  “Atom: Robustifying out-of-distribution detection using outlier mining”  In _Machine Learning and Knowledge Discovery in Databases. Research Track: European Conference, ECML PKDD 2021, Bilbao, Spain, September 13–17, 2021, Proceedings, Part III 21_ , 2021, pp. 430–445  Springer
  * [52] Ilija Radosavovic et al.  “Designing network design spaces”  In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , 2020, pp. 10428–10436
  * [53] Jie Hu, Li Shen and Gang Sun  “Squeeze-and-excitation networks”  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2018, pp. 7132–7141
  * [54] Grant Van Horn et al.  “Benchmarking Representation Learning for Natural World Image Collections”  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2021, pp. 12884–12893
  * [55] Olga Russakovsky et al.  “ImageNet Large Scale Visual Recognition Challenge”  In _International Journal of Computer Vision (IJCV)_ 115.3, 2015, pp. 211–252  DOI: 10.1007/s11263-015-0816-y
  * [56] Zhongyuan Wang et al.  “Masked face recognition dataset and application”  In _arXiv preprint arXiv:2003.09093_ , 2020
  * [57] Sagar Vaze, Kai Han, Andrea Vedaldi and Andrew Zisserman  “Open-set recognition: A good closed-set classifier is all you need”  In _arXiv preprint arXiv:2110.06207_ , 2021


# Understanding the newly observed $\Omega_{c}$ states through their decays

Kai-Lei Wang1, Li-Ye Xiao1, Xian-Hui Zhong1,3 111E-mail<EMAIL_ADDRESS>Qiang Zhao2,3,4 222E-mail<EMAIL_ADDRESS>1) Department of Physics, Hunan
Normal University, and Key Laboratory of Low-Dimensional Quantum Structures
and Quantum Control of Ministry of Education, Changsha 410081, China  2)
Institute of High Energy Physics and Theoretical Physics Center for Science
Facilities, Chinese Academy of Sciences, Beijing 100049, China  3) Synergetic
Innovation Center for Quantum Effects and Applications (SICQEA), Hunan Normal
University, Changsha 410081, China  4) School of Physical Sciences, University
of Chinese Academy of Sciences, Beijing 100049, China

###### Abstract

The strong and radiative decay properties of the low-lying $\Omega_{c}$ states
are studied in a constituent quark model. We find that the newly observed
$\Omega_{c}$ states by the LHCb Collaboration can fit in well the decay
patterns. Thus, their spin-parity can be possibly assigned as the following:
(i) The $\Omega_{c}(3000)$ has $J^{P}=1/2^{-}$ and corresponds to the narrow
$1P$ mixed state $|1^{2}P_{\lambda}\frac{1}{2}^{-}\rangle_{1}$, its partner
$|1^{2}P_{\lambda}\frac{1}{2}^{-}\rangle_{2}$ should be a broad state with a
width of $\sim 100$ MeV. (ii) The $\Omega_{c}(3050)$ and $\Omega_{c}(3066)$
can be assigned to be two $J^{P}=3/2^{-}$ states,
$|1^{4}P_{\lambda}\frac{3}{2}^{-}\rangle$ and
$|1^{2}P_{\lambda}\frac{3}{2}^{-}\rangle$, respectively. (iii) The
$\Omega_{c}(3090)$ can be assigned as the
$|1^{4}P_{\lambda}\frac{5}{2}^{-}\rangle$ state with $J^{P}=5/2^{-}$. (iv) The
$\Omega_{c}(3119)$ might correspond to one of the two $2S$ states of the first
radial excitations, i.e. $|2^{2}S_{\lambda\lambda}\frac{1}{2}^{+}\rangle$ or
$|2^{4}S_{\lambda\lambda}\frac{3}{2}^{+}\rangle$.

###### pacs:

12.39.Jh, 13.30.-a, 14.20.Lq

##  I Introduction

Although the existence of $\Omega_{c}$ states have been predicted by the quark
model for a long time, experimental information about the $\Omega_{c}$
spectrum has been extremely limited during the past several decades. The
status about the $\Omega_{c}$ spectrum can be found in the recent reviews
Chen:2016spr ; Klempt:2009pi ; Crede:2013sze ; Cheng:2015iom . Very recently,
five new narrow $\Omega_{c}$ states, $\Omega_{c}(3000)$, $\Omega_{c}(3050)$,
$\Omega_{c}(3066)$, $\Omega_{c}(3090)$ and $\Omega_{c}(3119)$, were observed
in the $\Xi_{c}^{+}K^{-}$ channel by the LHCb Collaboration Aaij:2017nav .
This observation can be regarded as a significant progress towards a better
understanding of the $\Omega_{c}$ spectrum and immediately attracts a lot of
attention from the hadron physics community. Together with the two established
ground states, $\Omega_{c}(2695)1/2^{+}$ and $\Omega_{c}(2770)3/2^{+}$
Olive:2016xmw , the $\Omega_{c}$ spectrum, for the first time, allows a more
quantitative analysis of the internal structures, quantum numbers, and decay
modes for higher excited states.

These newly observed $\Omega_{c}$ states are good candidates for the low-lying
$\Omega_{c}$ resonances. Since the $\Omega_{c}$ contains a heavy $c$ quark and
two relatively light $s$ quarks, the low-lying internal excitations will favor
excitations of the so-called “$\lambda$-mode” in one orbital excitation in a
Jacobi coordinate between the light quarks and the heavy $c$ quark. Such a
structure is illustrated in Fig. 1. According to the mass spectrum from
various theoretical studies Ebert:2011kk ; Maltman:1980er ; Roberts:2007ni ;
Valcarce:2008dr ; Ebert:2007nw ; Garcilazo:2007eh ; Shah:2016nxi ;
Bali:2015lka ; Yoshida:2015tia ; Chen:2016phw ; Chen:2015kpa ; Wang:2017goq ,
these newly observed $\Omega_{c}$ states can be organized into the first
orbital excitations ($1P$ states with $J^{P}=1/2^{-},\ 3/2^{-},\ 5/2^{-}$) and
the first radial excitations ($2S$ states with $J^{P}=1/2^{+},\ 3/2^{+}$) of
the $\lambda$ mode, which have been summarized in Table 1. Stimulated by the
newly observed $\Omega_{c}$ states from LHCb, some groups have discussed their
nature and possible quantum numbers Agaev:2017jyt ; Chen:2017sci ;
Karliner:2017kfm ; Padmanath:2017lng ; Agaev:2017lip ; Kim:2017jpx ;
Aliev:2017led ; Chen:2017gnu ; Wang:2017zjw ; Huang:2017dwn ; Cheng:2017ove ;
Zhao:2017fov ; Wang:2017vnc ; Yang:2017rpg . The possible spin-parity quantum
numbers suggested in the literature are collected in Table 2 and there are
still different views on their properties.

Figure 1:  (Color online) $ssc$ system with $\lambda$\- or $\rho$-mode
excitations. $\rho$ and $\lambda$ are the Jacobi coordinates defined as
$\mbox{\boldmath$\rho$\unboldmath}=\frac{1}{\sqrt{2}}(\mathbf{r}_{1}-\mathbf{r}_{2})$
and
$\mbox{\boldmath$\lambda$\unboldmath}=\frac{1}{\sqrt{6}}(\mathbf{r}_{1}+\mathbf{r}_{2}-2\mathbf{r}_{3})$.
$q_{1}$ and $q_{2}$ stand for the light $s$ quarks, and $Q_{3}$ stands for the
heavy $c$ quark.  Figure 2:  (Color online) Energy level of the $1P$\- and
$2S$-wave $\Omega_{c}$ states and their possible decay modes considered in
present work. The solid and dotted arrows stand for the possible decay modes
with one-photon and one-kaon emissions, respectively.

It should be noted that most of these low-lying $\Omega_{c}$ states have
masses in the vicinity of the $\Xi_{c}^{+}K^{-}$ and $\Xi_{c}^{\prime}\bar{K}$
threshold, to which the strong decay will almost saturate their total decay
widths. Meanwhile, for these states, their decays will be dominated by the
leading constituent quark model wavefunctions instead of detailed structures,
e.g. due to hyperfine splittings, because of their relatively small mass
differences. In other words, we anticipate that without detailed information
about the mass orderings in their classification, one can still possibly
identify the predominant feature of their strong decay patterns for given
quantum numbers. This makes it possible for us to determine their quantum
numbers based on the present available experimental information on the partial
and total widths. In addition to the hadronic decay, we also show that the
electromagnetic (EM) transitions are useful for providing further information
about their internal structures. For the low-lying $\Omega_{c}$ hadronic and
radiative decays, the allowed decay channels are plotted in Fig. 2 as an
illustration.

As follows, in Sec. II we first give a brief introduction to the quark model
description of the strong and radiative decay of the $ssc$ system. The
numerical results are presented and discussed in Sec. III. Finally, a summary
is given in Sec. IV.

Table 1: The spectrum of $1P$ and $2S$-wave $\Omega_{c}$ states in the constituent quark model. The total wave function of a $\Omega_{c}$ state is denoted by $|N^{2S+1}L_{\sigma}J^{P}\rangle$. The Clebsch-Gordan series for the spin and angular-momentum addition $|N^{2S+1}L_{\sigma}J^{P}\rangle=\sum_{L_{z}+S_{z}=J_{z}}\langle LL_{z},SS_{z}|JJ_{z}\rangle^{N}\Psi^{\sigma}_{LL_{z}}\chi_{S_{z}}\phi_{\Omega_{c}}$ has been omitted. The details of the wavefunctions can be found in our previous work Zhong:2007gp . The unit of mass is MeV in the table. State | Wave | Predicted | predicted | predicted | predicted | predicted | predicted | Observed
---|---|---|---|---|---|---|---|---
$|N^{2S+1}L_{\sigma}$ $J^{P}\rangle$ | function | mass Ebert:2011kk  | mass Maltman:1980er  | mass Roberts:2007ni  | mass Shah:2016nxi  | mass Yoshida:2015tia  | mass Bali:2015lka  | state
$|0\ ^{2}S$ $\frac{1}{2}^{+}\rangle$ | ${}^{0}\Psi^{S}_{00}\chi^{\lambda}_{S_{z}}\phi_{\Omega_{c}}$ | 2698 | 2745 | 2718 | 2695 | 2731 | 2648(28) | $\Omega_{c}(2695)$
$|0\ ^{4}S$ $\frac{3}{2}^{+}\rangle$ | ${}^{0}\Psi^{S}_{00}\chi^{s}_{S_{z}}\phi_{\Omega_{c}}$ | 2768 | 2805 | 2776 | 2767 | 2779 | 2709(32) | $\Omega_{c}(2770)$
$|1\ ^{2}P_{\lambda}$ $\frac{1}{2}^{-}\rangle$ | ${}^{1}\Psi^{\lambda}_{1L_{z}}\chi^{\lambda}_{S_{z}}\phi_{\Omega_{c}}$ | 3055 | 3015 | 2977 | 3011 | 3030 | 2995(46) |
$|1\ ^{2}P_{\lambda}$ $\frac{3}{2}^{-}\rangle$ |  | 3029 | 3030 | 2986 | 2976 | 3033 | 3016(69) |  $\Omega_{c}(3066)$?
$|1\ ^{4}P_{\lambda}$ $\frac{1}{2}^{-}\rangle$ |  | 2966 | 3040 | 2990 | 3028 | 3048 |  |
$|1\ ^{4}P_{\lambda}$ $\frac{3}{2}^{-}\rangle$ | ${}^{1}\Psi^{\lambda}_{1L_{z}}\chi^{s}_{S_{z}}\phi_{\Omega_{c}}$ | 3054 | 3065 | 2994 | 2993 | 3056 |  |  $\Omega_{c}(3050)$?
$|1\ ^{4}P_{\lambda}$ $\frac{5}{2}^{-}\rangle$ |  | 3051 | 3050 | 3014 | 2947 | 3057 |  |  $\Omega_{c}(3090)$?
$|2\ ^{2}S_{\lambda\lambda}$ $\frac{1}{2}^{+}\rangle$ | ${}^{2}\Psi^{\lambda\lambda}_{00}\chi^{\lambda}_{S_{z}}\phi_{\Omega_{c}}$ | 3088 | 3020 | 3152 | 3100 |  |  |  $\Omega_{c}(3119)$?
$|2\ ^{4}S_{\lambda\lambda}$ $\frac{3}{2}^{+}\rangle$ | ${}^{2}\Psi^{\lambda\lambda}_{00}\chi^{s}_{S_{z}}\phi_{\Omega_{c}}$ | 3123 | 3090 | 3190 | 3126 |  |  |  $\Omega_{c}(3119)$?
Table 2: Spin-parity ($J^{P}$) numbers of the newly observed $\Omega_{c}$ states suggested in various works. State |  Agaev:2017jyt  |  Chen:2017sci  |  Karliner:2017kfm  |  Padmanath:2017lng  | Chen:2017gnu  | Cheng:2017ove  | Wang:2017zjw  | Zhao:2017fov  | Agaev:2017lip  | Huang:2017dwn  | This work
---|---|---|---|---|---|---|---|---|---|---|---
$\Omega_{c}(3000)$ |  | $1/2^{-}$ |  $1/2^{-}$ ($3/2^{-}$) | $1/2^{-}$ | $1/2^{-}$ | $1/2^{-}$ | $1/2^{-}$ |  $1/2^{+}$ or $3/2^{+}$ | $1/2^{-}$ |  | $1/2^{-}$
$\Omega_{c}(3050)$ |  | $1/2^{-}$ |  $1/2^{-}$ ($3/2^{-}$) | $1/2^{-}$ | $5/2^{-}$ | $3/2^{-}$ | $1/2^{-}$ |  $5/2^{+}$ or $7/2^{+}$ | $3/2^{-}$ |  | $3/2^{-}$
$\Omega_{c}(3066)$ | $1/2^{+}$ |  $1/2^{+}$ or $1/2^{-}$ |  $3/2^{-}$($5/2^{-}$) | $3/2^{-}$ | $3/2^{-}$ | $5/2^{-}$ | $3/2^{-}$ | $3/2^{-}$ | $1/2^{+}$ |  | $3/2^{-}$
$\Omega_{c}(3090)$ |  |  |  $3/2^{-}$ ($1/2^{+}$) | $3/2^{-}$ | $1/2^{-}$ | $1/2^{+}$ | $3/2^{-}$ | $5/2^{-}$ | $1/2^{+}$ |  | $5/2^{-}$
$\Omega_{c}(3119)$ | $3/2^{+}$ | $3/2^{+}$ |  $5/2^{-}$ ($3/2^{+}$) | $5/2^{-}$ | $3/2^{-}$ | $3/2^{+}$ | $5/2^{-}$ |  $5/2^{+}$ or $7/2^{+}$ | $3/2^{+}$ | $1/2^{-}$ |  $1/2^{+}$ or $3/2^{+}$

##  II The model

We apply the chiral quark model Manohar:1983md  to the study of the hadronic
decays of the low-lying $\Omega_{c}$ states for rather empirical reasons. For
instance, it was shown in Refs. Xiao:2014ura ; Zhong:2010vq ; Zhong:2008kd ;
Zhong:2009sk ; Liu:2012sj ; Zhong:2007gp ; Xiao:2013xi ; Nagahiro:2016nsx ,
that the hadronic decays of heavy-light mesons and baryons can be reasonably
described by treating the light pseudoscalar mesons, i.e. $\pi$, $K$ and
$\eta$, as a fundamental state in the chiral quark model. Then, the decay
patterns of those low-lying heavy-light mesons and baryons can be described.
The chiral quark model has also been broadly applied to various processes
involving light pseudoscalar meson productions Li:1995si ; Li:1995vi ;
Li:1998ni ; Zhao:2002id ; Li:1994cy ; Li:1997gd ; Saghai:2001yd ; Zhao:2000iz
; He:2008ty ; He:2008uf ; Xiao:2013hca ; Zhong:2008km ; Zhong:2013oqa ;
Xiao:2016dlf ; Zhong:2007fx ; Xiao:2015gra ; Zhong:2011ti ; Zhong:2011ht . In
this model, the low energy quark-pseudoscalar-meson interactions in the SU(3)
flavor basis are described by the effective Lagrangian Li:1994cy ; Li:1997gd ;
Zhao:2002id

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

where $\psi_{j}$ represents the $j$th quark field in the hadron; $\phi_{m}$ is
the pseudoscalar meson field, $f_{m}$ is the pseudoscalar meson decay
constant, and $\hat{I}_{j}$ is the isospin operator associated with the
pseudoscalar meson.

Meanwhile, to treat the radiative decay of a hadron we apply the constituent
quark model which has been successfully applied to study the radiative decays
of $c\bar{c}$ and $b\bar{b}$ systems Deng:2016stx ; Deng:2016ktl . In this
model, the quark-photon EM coupling at the tree level is adopted as
Brodsky:1968ea

$\displaystyle
H_{e}=-\sum_{j}e_{j}\bar{\psi}_{j}\gamma^{j}_{\mu}A^{\mu}(\mathbf{k},\mathbf{r}_{j})\psi_{j},$
(2)

where $A^{\mu}$ represents the photon field with 3-momentum $\mathbf{k}$.
$e_{j}$ and $\mathbf{r}_{j}$ stand for the charge and coordinate of the
constituent quark $\psi_{j}$, respectively.

To match the non-relativistic harmonic oscillator wave functions adopted in
our calculations, we should provide the quark-pseudoscalar and quark-photon EM
couplings in a nonrelativistic form. In the initial-hadron-rest system, the
nonrelativistic form of the quark-photon EM coupling can be written as
Deng:2016stx ; Deng:2016ktl ; Brodsky:1968ea ; Li:1997gd ; Zhao:2002id ;
Li:1994cy

$H_{e}^{nr}=\sum_{j}\left[e_{j}\mathbf{r}_{j}\cdot\mbox{\boldmath$\epsilon$\unboldmath}-\frac{e_{j}}{2m_{j}}\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot(\mbox{\boldmath$\epsilon$\unboldmath}\times\hat{\mathbf{k}})\right]e^{-i\textbf{k}\cdot\textbf{r}_{j}},$
(3)

while the nonrelativistic form of the quark-pseudoscalar-meson coupling can be
written as

$\displaystyle
H_{m}^{nr}=\sum_{j}\left[A\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot\textbf{q}+\frac{\omega_{m}}{2\mu_{q}}\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot\textbf{p}_{j}\right]I_{j}e^{-i\mathbf{q}\cdot\mathbf{r}_{j}},$
(4)

where $A\equiv-(1+\frac{\omega_{m}}{E_{f}+M_{f}})$;
$\mbox{\boldmath$\sigma$\unboldmath}_{j}$ and $\textbf{p}_{j}$ stand for the
Pauli spin vector and internal momentum operator for the $j$th quark of the
initial hadron; $\mathbf{q}$ is three momentum of the emitted light meson;
$I_{j}$ is the flavor operator defined for the transitions in the SU(3) flavor
space Li:1997gd ; Zhao:2002id ; and $\mu_{q}$ is a reduced mass given by
$1/\mu_{q}=1/m_{j}+1/m^{\prime}_{j}$ with $m_{j}$ and $m^{\prime}_{j}$ for the
masses of the $j$th quark in the initial and final hadrons, respectively.

For a light pseudoscalar meson emission in a hadron strong decays, the partial
decay width can be calculated with Zhong:2008kd ; Zhong:2007gp

$\Gamma_{m}=\left(\frac{\delta}{f_{m}}\right)^{2}\frac{(E_{f}+M_{f})|\textbf{q}|}{4\pi
M_{i}(2J_{i}+1)}\sum_{J_{fz},J_{iz}}|\mathcal{M}_{J_{fz},J_{iz}}|^{2},$ (5)

while for a photon emission in a hadron radiative decays, the partial decay
width can be calculated with Deng:2016stx ; Deng:2016ktl

$\Gamma_{\gamma}=\frac{|\mathbf{k}|^{2}}{\pi}\frac{2}{2J_{i}+1}\frac{M_{f}}{M_{i}}\sum_{J_{fz},J_{iz}}|\mathcal{A}_{J_{fz},J_{iz}}|^{2},$
(6)

where $\mathcal{M}_{J_{fz},J_{iz}}$ and $\mathcal{A}_{J_{fz},J_{iz}}$
correspond to the strong and radiative transition amplitudes, respectively.
The quantum numbers $J_{iz}$ and $J_{fz}$ stand for the third components of
the total angular momenta of the initial and final heavy baryons,
respectively. $\delta$ as a global parameter accounts for the strength of the
quark-meson couplings. It has been determined in our previous study of the
strong decays of the charmed baryons and heavy-light mesons Zhong:2007gp ;
Zhong:2008kd . Here, we fix its value the same as that in Refs. Zhong:2008kd ;
Zhong:2007gp , i.e. $\delta=0.557$.

In the calculation, the standard quark model parameters are adopted. Namely,
we set $m_{s}=450$ MeV, and $m_{c}=1480$ MeV for the constituent quark masses.
The harmonic oscillator parameter $\alpha_{\rho}$ in the wave function
$\psi^{n}_{lm}=R_{nl}Y_{lm}$ of the $\rho$-mode excitation between the two $s$
quarks is taken as $\alpha_{\rho}=0.44$ GeV, which is slightly larger than
that of the $\rho$-mode excitation between the two light nonstrange quarks
($\alpha_{\rho}=0.40$ GeV ) adopted in our previous work Zhong:2007gp .
Another harmonic oscillator parameter $\alpha_{\lambda}$ can be related to
$\alpha_{\rho}$ with the relation
$\alpha_{\lambda}=[3m_{c}/(2m_{s}+m_{c})]^{1/4}\alpha_{\rho}$ Zhong:2007gp .
The kaon decay constant is taken as $f_{K}=160$ MeV. The masses of the well-
established hadrons used in the calculations are adopted from the Particle
Data Group (PDG) Olive:2016xmw . With these parameters, the strong decay
properties of most of the heavy-light mesons and charmed baryons can be
reasonably well described Liu:2012sj ; Zhong:2007gp ; Xiao:2013xi ;
Xiao:2014ura ; Zhong:2010vq ; Zhong:2008kd ; Zhong:2009sk .

Figure 3:  The strong decay partial width as a function of mass. The dashed
and dotted lines stand for the partial width of $\Xi_{c}\bar{K}$ and
$\Xi_{c}^{\prime}\bar{K}$ channels, respectively. The solid lines stand for
the total width of these two channels. Figure 4:  The radiative decay partial
widths of the $1P$-wave states as a function of mass. In the figure, the
$\Omega_{c}$ and $\Omega_{c}^{*}$ stand for the $\Omega_{c}(2695)$ and
$\Omega_{c}(2770)$ states, respectively. Figure 5:  The strong decay partial
width of the $\Xi_{c}\bar{K}$ channel for the $J^{P}=1/2^{-}$ mixed states as
a function of mixing angle $\phi$. The solid curves stand for the partial
width of the mixed state $|1P_{\lambda}1/2\rangle_{1}$ with a mass of 3000
MeV. The dashed and dotted curves stand for the partial width of the mixed
state $|1P_{\lambda}1/2\rangle_{2}$ with masses of 3020 and 2980 MeV,
respectively.

##  III RESULTS AND DISCUSSIONS

One important feature arising from the hadronic decays of the low-lying
$\Omega_{c}$ states is that their hadronic decay properties are determined by
their dominant quark excitations. Within a local mass region containing
several states, their decays are not sensitive to the local mass orderings
which are determined by more detailed dynamics such as spin-dependent forces.
Namely, the decay pattern should not change if the mass of such states vary
within a small mass range. As a natural assumption for these observed
$\Omega_{c}$ states that they are most likely the $1P$ and $2S$ excited
states, one can easily check that in most models the relative partial decay
widths will not change dramatically if their masses change within 100 MeV (see
figs. 3 and 4). In such a sense, the pattern arising from the relative partial
widths should be more selective to their quantum numbers instead of their
masses.

In Table 3 the calculations of the partial decay widths for the $P$-wave
states into $\Xi\bar{K}$, $\Xi^{\prime}\bar{K}$, and radiative decay channels
are listed. It can be seen that although there are still some uncertainties
with both the experimental and theoretical results, the magnitudes of the
partial decay widths have indicated patterns determined by the three-body
quark model wavefunctions.

###  III.1 $\Omega_{c}(3000)$

To be more specific, the relatively low mass of $\Omega_{c}(3000)$ make it a
good candidate for the $J^{P}=1/2^{-}$ states as the first orbital excitation
states of $|1^{2}P_{\lambda}\frac{1}{2}^{-}\rangle$ or
$|1^{4}P_{\lambda}\frac{1}{2}^{-}\rangle$. However, the quark model predicts
rather broad widths for both $|1^{2}P_{\lambda}\frac{1}{2}^{-}\rangle$ and
$|1^{4}P_{\lambda}\frac{1}{2}^{-}\rangle$ (see fig. 3) and suggest more
profound configurations with the physical state. It could happen that these
two $J^{P}=1/2^{-}$ states $|1^{2}P_{\lambda}\frac{1}{2}^{-}\rangle$ and
$|1^{4}P_{\lambda}\frac{1}{2}^{-}\rangle$ can have significant mixings for the
presence of the spin-orbit interaction. Thus, we further consider the
$\Omega_{c}(3000)$ as a mixed state of
$|1^{2}P_{\lambda}\frac{1}{2}^{-}\rangle$ and
$|1^{4}P_{\lambda}\frac{1}{2}^{-}\rangle$ by the following mixing scheme

$\displaystyle\left|1P_{\lambda}\frac{1}{2}^{-}\right\rangle_{1}$
$\displaystyle=$
$\displaystyle+\cos(\phi)\left|1^{2}P_{\lambda}\frac{1}{2}^{-}\right\rangle+\sin(\phi)\left|1^{4}P_{\lambda}\frac{1}{2}^{-}\right\rangle,$
(7) $\displaystyle\left|1P_{\lambda}\frac{1}{2}^{-}\right\rangle_{2}$
$\displaystyle=$
$\displaystyle-\sin(\phi)\left|1^{2}P_{\lambda}\frac{1}{2}^{-}\right\rangle+\cos(\phi)\left|1^{4}P_{\lambda}\frac{1}{2}^{-}\right\rangle,$
(8)

where $\phi$ is the mixing angle.

Taking the $\Omega_{c}(3000)$ as the mixed state
$|1P_{\lambda}\frac{1}{2}^{-}\rangle_{1}$, we plot the strong decay width into
the $\Xi_{c}\bar{K}$ channel as a function of the mixing angle $\phi$ in fig.
5. It shows that with a mixing angle $\phi\simeq 24^{\circ}$ or $47^{\circ}$,
the measured decay width $\Gamma\simeq 4.5\pm 0.9$ MeV of $\Omega_{c}(3000)$
can be well explained. Note that an intrinsic sign between
$|1^{2}P_{\lambda}\frac{1}{2}^{-}\rangle$ and
$|1^{4}P_{\lambda}\frac{1}{2}^{-}\rangle$ is included which introduces the
cancelation between the two transition amplitudes from these two
configurations. In Ref. Cheng:2017ove  a similar mixing mechanism for
obtaining a narrow width for the $1/2^{-}$ state is also discussed in the
basis of heavy quark spin symmetry (HQSS).

If $\Omega_{c}(3000)$ corresponds to the mixed state
$|1P_{\lambda}\frac{1}{2}^{-}\rangle_{1}$ indeed, the other mixed state
$|1P_{\lambda}\frac{1}{2}^{-}\rangle_{2}$ should be a broad state with a width
much larger than these observed states. At this moment we do not intend to
determine the mass of the broad state but only discuss its width range near
the mass of $\Omega_{c}(3000)$. It is found that with the mass of 2980 MeV,
the decay width of $|1P_{\lambda}\frac{1}{2}^{-}\rangle_{2}$ is about $\sim
70$ MeV, while with the mass of 3020 MeV the width is about $\sim 110$ MeV.
This presumably suggests the difficulty of identifying it from the background
in the present data sets. One notices that in the LHCb data Aaij:2017nav
there are events excesses below $\Omega_{c}(3000)$ which are noted as the
feed-down events from higher partially constructed $\Omega_{c}(X)$ states. It
would be interesting to have more elaborate analysis of these event excesses
to look for signals of the broad $1/2^{-}$. Moreover, it shows that that the
lineshape of $\Omega_{c}(3000)$ has been distorted at the higher energy side
and a broad structure is present below the narrow $\Omega_{c}(3000)$. further
analysis of the $\Xi_{c}^{+}K^{-}$ invariant mass spectrum may help clarify
the status of the broad partner of $\Omega_{c}(3000)$.

The assignment of $\Omega_{c}(3000)$ as the narrow $1/2^{-}$ state naturally
leads to the dominance of the $E1$ transition in the EM transitions of
$\Omega_{c}(3000)\to\Omega_{c}(2695)\gamma$. It predicts an EM transition
partial width of $200-360$ keV, which is quite significant and can be searched
for in experiment as further evidence for its assignment. We also take the
ratio between the EM and hadronic decays as a guidance for its future studies:

$\frac{\Gamma[\Omega_{c}(2695)\gamma]}{\Gamma(\Xi_{c}\bar{K})}\simeq 5\sim
9\%.$ (9)

The EM transition of the state to $\Omega_{c}(2770)$ can also be calculated.
Taking into account the phase space factor, it predicts a rather small partial
decay width of about 10s keV, which is much smaller than that for
$\Omega_{c}(3000)\to\Omega_{c}(2695)\gamma$.

###  III.2 $\Omega_{c}(3050)$

The $\Omega_{c}(3050)$ is most likely to be the $J^{P}=3/2^{-}$ state. It
corresponds to $|1^{4}P_{\lambda}\frac{3}{2}^{-}\rangle$. If we assign the
$\Omega_{c}(3050)$ as $|1^{4}P_{\lambda}\frac{3}{2}^{-}\rangle$, the two main
decay channels will be $\Xi_{c}\bar{K}$ and $\Omega_{c}(2770)\gamma$ with the
latter dominated by the $E1$ transition. The partial decay width of
$\Omega_{c}(3050)\to\Xi_{c}\bar{K}$ is estimated to be about 0.61 MeV and the
EM transition width of $0.33$ MeV. The EM transition of
$\Omega_{c}(3050)\to\Omega_{c}(2695)\gamma$ in the assignment of
$|1^{4}P_{\lambda}\frac{3}{2}^{-}\rangle$ will be suppressed by the $M2$
transition which leads to a small partial width of $1.12\times 10^{-3}$ MeV.
The total width reads about $0.94$ MeV which is also nearly saturated by the
hadronic and EM decays. This value is consistent with the data. The large
branching ratio of the radiative transition of $\Omega_{c}(3050)$ into the
$\Omega_{c}(2770)\gamma$ channel

$Br[\Omega_{c}(3050)\to\Omega_{c}(2770)\gamma]\simeq 35\%,$ (10)

indicates that the radiative transition of
$\Omega_{c}(3050)\to\Omega_{c}(2770)\gamma$ should be accessible in future
experiment.

###  III.3 $\Omega_{c}(3066)$

The $\Omega_{c}(3066)$ can be assigned to another $J^{P}=3/2^{-}$ state,
$|^{2}P_{\lambda}\frac{3}{2}^{-}\rangle$. As the result, the
$\Omega_{c}(3066)$ will mainly decay into $\Xi_{c}\bar{K}$ and
$\Omega_{c}(2695)\gamma$, while its decays into $\Omega_{c}(2770)\gamma$ will
be suppressed due to the spin-flipping $M2$ transition. Our results have been
listed in Table 3. One can see that the total width $\Gamma\simeq 4.96$ MeV is
in good agreement with the experimental data of $3.5\pm 0.4\pm 0.2$ MeV. In
this scenario, the branching ratio of the radiative transition
$\Omega_{c}(3066)\to\Omega_{c}(2695)\gamma$ is predicted to be a fairly large
value:

$Br[\Omega_{c}(3066)\to\Omega_{c}(2695)\gamma]\simeq 7\%.$ (11)

This makes the experimental measurement of the radiative decay of
$\Omega_{c}(3066)\to\Omega_{c}(2695)\gamma$ a possible way to further test its
configuration.

###  III.4 $\Omega_{c}(3090)$

The $\Omega_{c}(3090)$ can be assigned to the $J^{P}=5/2^{-}$ state,
$|1^{4}P_{\lambda}\frac{5}{2}^{-}\rangle$. Its decays are governed by the
strong decay channel $\Xi_{c}\bar{K}$, and the predicted partial width is

$\displaystyle\Gamma[\Omega_{c}(3090)\to\Xi_{c}\bar{K}]\simeq\ 9.32\
\mathrm{MeV}.$ (12)

Following this scenario, its radiative decay rate into the
$\Omega_{c}(2770)\gamma$ is expected to be sizeable with a ratio of

$\displaystyle\frac{\Gamma[\Omega_{c}(3090)\to\Omega_{c}(2770)\gamma]}{\Gamma[\Omega_{c}(3090)\to\Xi_{c}\bar{K}]}\simeq
2\%.$ (13)

The total width is nearly saturated by the $\Xi_{c}\bar{K}$ channel and with
the EM transition the total width, $\Gamma\simeq 9.5$ MeV, is in good
agreement with the measured width $8.7\pm 1.0\pm 0.8$ MeV. To confirm the
nature of $\Omega_{c}(3090)$, experimental measurements of the radiative decay
$\Omega_{c}(3090)\to\Omega_{c}(2770)\gamma$ are strongly recommended.

###  III.5 $\Omega_{c}(3119)$

The $\Omega_{c}(3119)$ has the highest mass among these five states but has a
narrow width of $1.1\pm 0.8\pm 0.4$ MeV. It may be assigned to be one of the
first radially excited states, i.e. either
$|2^{2}S_{\lambda\lambda}\frac{1}{2}^{+}\rangle$ or
$|2^{4}S_{\lambda\lambda}\frac{3}{2}^{+}\rangle$. These $2S$ radial excitation
states are found to usually have a very narrow decay width, which is about 1
MeV (see Table 4). Considering the $\Omega_{c}(3119)$ as the
$|2^{2}S_{\lambda\lambda}\frac{1}{2}^{+}\rangle$ state, we find that
$\Omega_{c}(3119)$ should have two main decay channels, i.e. $\Xi_{c}\bar{K}$
and $\Xi_{c}^{\prime}\bar{K}$, of which the calculated partial widths are
listed in Table 4. By summing up these dominant partial widths, the total
width amounts to about 1.2 MeV, which is consistent with the central value of
the experimental data. In this assignment, one notices that partial decay
widths of the $\Xi_{c}\bar{K}$ and $\Xi_{c}^{\prime}\bar{K}$ channels are
compatible.

In contrast, by assigning $\Omega_{c}(3119)$ as the
$|2^{4}S_{\lambda\lambda}\frac{3}{2}^{+}\rangle$ state, we find that it mainly
decays into $\Xi_{c}\bar{K}$ and $\Xi_{c}^{\prime}\bar{K}$ channels. Also, the
partial width into $\Xi_{c}\bar{K}$ will be much larger (about a factor of 6)
than into the $\Xi_{c}^{\prime}\bar{K}$ channel. The calculated partial decay
widths in this assignment are also listed in Table 4. The measurement of the
partial decay widths into these two channels should allow a determination of
the quantum number and structure of the $\Omega_{c}(3119)$.

###  III.6 $\Omega_{c}(2770)$

As a byproduct, we also study the radiative decay process
$\Omega_{c}(2770)\to\Omega_{c}(2695)\gamma$ as a test of our simple model. Our
predicted partial width is

$\Gamma[\Omega_{c}(2770)\to\Omega_{c}(2695)\gamma]\simeq\ 0.89\ \mathrm{keV},$
(14)

which is in good agreement with other predictions in Refs. Aliev:2014bma ;
Majethiya:2009vx ; Dey:1994qi ; Wang:2009cd . Interestingly, one notices that
the lattice QCD simulation yields a rather small value for this quantity at
nearly physical pion mass Bahtiyar:2015sga , which is about an order of
magnitude smaller than phenomenological model calculations. Finally, it should
be mentioned that the decay widths of the low-lying $S$ and $P$-wave charmed
baryons, such as $\Sigma_{c}(2455,2520)$, $\Lambda_{c}(2593,2625)$ and
$\Xi_{c}(2645,2815)$, predicted within our nonrelativistic constituent quark
model Zhong:2007gp ; Liu:2012sj  are in good agreement with the relativistic
quark model predictions Ivanov:1999bk ; Lyubovitskij:2003pn  and the
experimental data Olive:2016xmw , which indicates that the relativistic
effects are relatively small for these processes.

Table 3: The decay properties of the $P$-wave states compared with the observations. $\Gamma_{\mathrm{total}}^{\mathrm{th}}$ stands for the total decay width calculated in present work, while $\Gamma_{\mathrm{total}}^{\mathrm{exp}}$ stands for the total width obtained from the LHCb experiments. The unit of mass and width is MeV in the table.  state | Mass | $\Gamma(\Xi_{c}\bar{K})$ | $\Gamma(\Xi_{c}^{\prime}\bar{K})$ | $\Gamma[\Omega_{c}(2695)\gamma]$ | $\Gamma[\Omega_{c}(2770)\gamma]$ | $\Gamma_{\mathrm{total}}^{\mathrm{th}}$ | $\Gamma_{\mathrm{total}}^{\mathrm{exp}}$ | Possible assignment
---|---|---|---|---|---|---|---|---
$|1P_{\lambda}\frac{1}{2}^{-}\rangle_{1}$ | 3000 | 4.0 | / | 0.36/0.20 | $0.02/0.08$ | 4.38/4.28 | $4.5\pm 0.9$ | $\Omega_{c}(3000)$
$|1^{4}P_{\lambda}\frac{3}{2}^{-}\rangle$ | 3050 | 0.61 | / | $1.12\times 10^{-3}$ | 0.33 | 0.94 | $0.8\pm 0.3$ | $\Omega_{c}(3050)$
$|1^{2}P_{\lambda}\frac{3}{2}^{-}\rangle$ | 3066 | 4.61 | / | $0.35$ | $5.68\times 10^{-4}$ | 4.96 | $3.5\pm 0.4$ | $\Omega_{c}(3066)$
$|1^{4}P_{\lambda}\frac{5}{2}^{-}\rangle$ | 3090 | 9.32 | 0.03 | $1.00\times 10^{-4}$ | $0.18$ | 9.53 | $8.7\pm 1.8$ | $\Omega_{c}(3090)$
Table 4: The decay properties of $\Omega_{c}(3119)$ as the $2S$ states. $\Gamma_{\mathrm{total}}^{\mathrm{th}}$ stands for the total decay width calculated in present work, while $\Gamma_{\mathrm{total}}^{\mathrm{exp}}$ stands for the total width obtained from the LHCb experiments. The unit of mass and width is MeV in the table.  state | Mass | $\Gamma(\Xi_{c}\bar{K})$ | $\Gamma(\Xi_{c}^{\prime}\bar{K})$ | $\Gamma[\Omega_{c}(2695)\gamma]$ | $\Gamma[\Omega_{c}(2770)\gamma]$ | $\Gamma_{\mathrm{total}}^{\mathrm{th}}$ | $\Gamma_{\mathrm{total}}^{\mathrm{exp}}$ |
---|---|---|---|---|---|---|---|---
$|2^{2}S_{\lambda\lambda}\frac{1}{2}^{+}\rangle$ | 3119 | $0.60$ | $0.45$ | $2.9\times 10^{-3}$ | $6.4\times 10^{-4}$ | 1.15 | $1.1\pm 0.8\pm 0.4$ |
state | Mass | $\Gamma(\Xi_{c}K)$ | $\Gamma(\Xi_{c}^{\prime}K)$ | $\Gamma[\Omega_{c}(2695)\gamma]$ | $\Gamma[\Omega_{c}(2770)\gamma]$ | $\Gamma_{\mathrm{total}}^{\mathrm{th}}$ | $\Gamma_{\mathrm{total}}^{\mathrm{exp}}$ |
$|2^{4}S_{\lambda\lambda}\frac{3}{2}^{+}\rangle$ | 3119 | $0.60$ | $0.11$ | $1.0\times 10^{-3}$ | $8.1\times 10^{-4}$ | 0.73 | $1.1\pm 0.8\pm 0.4$ |

##  IV summary

In this work we have studied the strong and radiative decay properties of the
newly observed $\Omega_{c}$ states, i.e. $\Omega_{c}(3000)$,
$\Omega_{c}(3050)$, $\Omega_{c}(3066)$, $\Omega_{c}(3090)$ and
$\Omega_{c}(3119)$, by LHCb Collaboration in a constituent chiral quark model.
It shows that these low-lying states can be accommodated into the quark model
with the consideration of proper internal excitations. In particular, the
excitations of the $\lambda$ mode in the Jacobi coordinate (Fig. 1) will give
rise to the main configurations of these observed states.

It is also found that for these low-lying states with masses close to each
other, their relative magnitudes of partial decay widths are a more selective
observable for the determination of their quantum numbers. In contrast, the
mass ordering patterns, which are determined by more detailed dynamics, may
not be an ideal quantity for classifying their quantum numbers at the present
stage.

As a conclusion of this investigation, the following assignments seem to be
favored in the quark model: (i) The $\Omega_{c}(3000)$ has $J^{P}=1/2^{-}$ and
corresponds to the narrow $1P$ mixed state
$|1^{2}P_{\lambda}\frac{1}{2}^{-}\rangle_{1}$. Its partner
$|1^{2}P_{\lambda}\frac{1}{2}^{-}\rangle_{2}$ should be a broad state, which
is worthy looking for in the future experiments. The lineshape distortion
under the peak of $\Omega_{c}(3000)$ may be a signal for its presence. (ii)
Both $\Omega_{c}(3050)$ and $\Omega_{c}(3066)$ have $J^{P}=3/2^{-}$ and
correspond to the $1P$ states $|1^{4}P_{\lambda}\frac{3}{2}^{-}\rangle$ and
$|1^{2}P_{\lambda}\frac{3}{2}^{-}\rangle$, respectively. The
$\Omega_{c}(3050)$ is expected to have large radiative decay rates into the
$\Omega_{c}(2770)\gamma$ channel, while $\Omega_{c}(3066)$ has large radiative
decay rates into the $\Omega_{c}(2695)\gamma$ channel. (iii) The
$\Omega_{c}(3090)$ should correspond to the $J^{P}=5/2^{-}$ state
$|1^{4}P_{\lambda}\frac{5}{2}^{-}\rangle$. The radiative decay rate of
$\Omega_{c}(3050)\to\Omega_{c}(2770)\gamma$ is is expected to be large as
well. (iv) The $\Omega_{c}(3119)$ may correspond to one of the first radially
excited $2S$ states, i.e. either
$|2^{2}S_{\lambda\lambda}\frac{1}{2}^{+}\rangle$ or
$|2^{4}S_{\lambda\lambda}\frac{3}{2}^{+}\rangle$. The relative partial decay
width fraction $\Gamma(\Xi_{c}\bar{K})/\Gamma(\Xi_{c}^{\prime}\bar{K})$ can
distinguish these two different assignments with either $J^{P}=1/2^{+}$ or
$3/2^{+}$.

Finally, we emphasize that the EM transitions appear to be useful for
determining the quantum numbers of these $\Omega_{c}$ states in this analysis.
Future experiments measuring their radiative decay widths are strongly
recommended.

##  Acknowledgements

We thank Y. X. Yao for checking some of the calculations. This work is
supported, in part, by the National Natural Science Foundation of China (Grant
Nos. 11375061, 11425525, and 11521505), DFG and NSFC funds to the Sino-German
CRC 110 Symmetries and the Emergence of Structure in QCD (NSFC Grant No.
11261130311), and National Key Basic Research Program of China under Contract
No. 2015CB856700.

## References

  * (1) H. X. Chen, W. Chen, X. Liu, Y. R. Liu and S. L. Zhu, A review of the open charm and open bottom systems, arXiv:1609.08928 [hep-ph].
  * (2) H. Y. Cheng, Charmed baryons circa 2015, Front. Phys. (Beijing) 10, 101406 (2015).
  * (3) V. Crede and W. Roberts, Progress towards understanding baryon resonances, Rept. Prog. Phys. 76, 076301 (2013).
  * (4) E.Klempt and J.M.Richard, Baryon spectroscopy, Rev. Mod. Phys. 82, 1095 (2010).
  * (5) R. Aaij et al. [LHCb Collaboration], Observation of five new narrow $\Omega_{c}^{0}$ states decaying to $\Xi_{c}^{+}K^{-}$, Phys. Rev. Lett. 118, 182001 (2017).
  * (6) C. Patrignani et al. [Particle Data Group], Review of Particle Physics, Chin. Phys. C 40, 100001 (2016).
  * (7) D. Ebert, R. N. Faustov and V. O. Galkin, Spectroscopy and Regge trajectories of heavy baryons in the relativistic quark-diquark picture, Phys. Rev. D 84, 014025 (2011).
  * (8) K. Maltman and N. Isgur, Baryons with strangeness and charm in a quark model with chromodynamics, Phys. Rev. D 22, 1701 (1980).
  * (9) W. Roberts and M. Pervin, Heavy baryons in a quark model, Int. J. Mod. Phys. A 23, 2817 (2008).
  * (10) D. Ebert, R. N. Faustov and V. O. Galkin, Masses of excited heavy baryons in the relativistic quark model, Phys. Lett. B 659, 612 (2008).
  * (11) A. Valcarce, H. Garcilazo and J. Vijande, Towards an understanding of heavy baryon spectroscopy, Eur. Phys. J. A 37, 217 (2008).
  * (12) H. Garcilazo, J. Vijande and A. Valcarce, Faddeev study of heavy baryon spectroscopy, J. Phys. G 34, 961 (2007).
  * (13) Z. Shah, K. Thakkar, A. K. Rai and P. C. Vinodkumar, Mass spectra and Regge trajectories of $\Lambda_{c}^{+}$, $\Sigma_{c}^{0}$, $\Xi_{c}^{0}$ and $\Omega_{c}^{0}$ baryons, Chin. Phys. C 40, 123102 (2016).
  * (14) P. P rez-Rubio, S. Collins and G. S. Bali, Charmed baryon spectroscopy and light flavor symmetry from lattice QCD, Phys. Rev. D 92, 034504 (2015).
  * (15) T. Yoshida, E. Hiyama, A. Hosaka, M. Oka and K. Sadato, Spectrum of heavy baryons in the quark model, Phys. Rev. D 92, 114029 (2015).
  * (16) H. X. Chen, Q. Mao, A. Hosaka, X. Liu and S. L. Zhu, $D$-wave charmed and bottomed baryons from QCD sum rules, Phys. Rev. D 94, 114016 (2016).
  * (17) H. X. Chen, W. Chen, Q. Mao, A. Hosaka, X. Liu and S. L. Zhu, $P$-wave charmed baryons from QCD sum rules, Phys. Rev. D 91, 054034 (2015).
  * (18) Z. Y. Wang, K. W. Wei, J. J. Qi and X. H. Guo, Spectra of charmed and bottom baryons with hyperfine interaction, arXiv:1701.04524 [hep-ph].
  * (19) S. S. Agaev, K. Azizi and H. Sundu, On the nature of the newly discovered $\Omega_{c}^{0}$ states, arXiv:1703.07091 [hep-ph].
  * (20) H. X. Chen, Q. Mao, W. Chen, A. Hosaka, X. Liu and S. L. Zhu, Decay properties of $P$-wave charmed baryons from light-cone QCD sum rules, Phys. Rev. D 95, 094008 (2017).
  * (21) M. Karliner and J. L. Rosner, Very narrow excited $\Omega_{c}$ baryons, arXiv:1703.07774 [hep-ph].
  * (22) G. Yang and J. Ping, The structure of pentaquarks $\Omega_{c}^{0}$ in the chiral quark model, arXiv:1703.08845 [hep-ph].
  * (23) M. Padmanath and N. Mathur, Quantum numbers of recently discovered $\Omega^{0}_{c}$ baryons from lattice QCD, arXiv:1704.00259 [hep-ph].
  * (24) W. Wang and R. L. Zhu, Interpretation of the newly observed $\Omega_{c}^{0}$ resonance, arXiv:1704.00179 [hep-ph].
  * (25) H. Y. Cheng and C. W. Chiang, Quantum Numbers of $\Omega_{c}$ States and Other Charmed Baryons, Phys. Rev. D 95, 094018 (2017).
  * (26) H. Huang, J. Ping and F. Wang, Investigating the excited $\Omega^{0}_{c}$ states through $\Xi_{c}K$ and $\Xi^{{}^{\prime}}_{c}K$ decay channels, arXiv:1704.01421 [hep-ph].
  * (27) Z. G. Wang, Analysis of the $\Omega_{c}(3000)$, $\Omega_{c}(3050)$, $\Omega_{c}(3066)$, $\Omega_{c}(3090)$ and $\Omega_{c}(3119)$ with QCD sum rules, Eur. Phys. J. C 77, 325 (2017).
  * (28) Z. Zhao, D. D. Ye and A. Zhang, Hadronic decay properties of newly observed $\Omega_{c}$ baryons, arXiv:1704.02688 [hep-ph].
  * (29) B. Chen and X. Liu, Six $\Omega_{c}^{0}$ states discovered by LHCb as new members of $1P$ and $2S$ charmed baryons, arXiv:1704.02583 [hep-ph].
  * (30) T. M. Aliev, S. Bilmis and M. Savci, Are the new excited $\Omega_{c}$ baryons negative parity states?, arXiv:1704.03439 [hep-ph].
  * (31) H. C. Kim, M. V. Polyakov and M. Prasza?owicz, On a possibility of charmed exotica, arXiv:1704.04082 [hep-ph].
  * (32) S. S. Agaev, K. Azizi and H. Sundu, Interpretation of the new $\Omega_{c}^{0}$ states via their mass and width, arXiv:1704.04928 [hep-ph].
  * (33) A. Manohar and H. Georgi, Chiral Quarks and the Nonrelativistic Quark Model, Nucl. Phys. B 234, 189 (1984).
  * (34) X. H. Zhong and Q. Zhao, Strong decays of newly observed $D_{sJ}$ states in a constituent quark model with effective Lagrangians, Phys. Rev. D 81, 014031 (2010).
  * (35) X. H. Zhong and Q. Zhao, Strong decays of heavy-light mesons in a chiral quark model, Phys. Rev. D 78, 014029 (2008).
  * (36) X. H. Zhong, Strong decays of the newly observed $D(2550)$, $D(2600)$, $D(2750)$, and $D(2760)$, Phys. Rev. D 82, 114014 (2010).
  * (37) L. Y. Xiao and X. H. Zhong, Strong decays of higher excited heavy-light mesons in a chiral quark model, Phys. Rev. D 90, 074029 (2014).
  * (38) L. Y. Xiao and X. H. Zhong, $\Xi$ baryon strong decays in a chiral quark model, Phys. Rev. D 87, 094002 (2013). [arXiv:1302.0079 [hep-ph]].
  * (39) X. H. Zhong and Q. Zhao, Charmed baryon strong decays in a chiral quark model, Phys. Rev. D 77, 074008 (2008).
  * (40) L. H. Liu, L. Y. Xiao and X. H. Zhong, Charm-strange baryon strong decays in a chiral quark model, Phys. Rev. D 86, 034024 (2012).
  * (41) H. Nagahiro, S. Yasui, A. Hosaka, M. Oka and H. Noumi, Structure of charmed baryons studied by pionic decays, Phys. Rev. D 95, 014023 (2017).
  * (42) Z. P. Li, The Kaon photoproduction of nucleons in the chiral quark model, Phys. Rev. C 52, 1648 (1995).
  * (43) Z. P. Li, The $\eta$ photoproduction of nucleons and the structure of the resonance $S_{11}(1535)$ in the quark model, Phys. Rev. D 52, 4961 (1995).
  * (44) Z. P. Li and B. Saghai, Study of the baryon resonances structure via eta photoproduction, Nucl. Phys. A 644, 345 (1998).
  * (45) 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).
  * (46) Z. P. Li, The threshold pion photoproduction of nucleons in the chiral quark model, Phys. Rev. D 50, 5639 (1994).
  * (47) Z. P. Li, H. X. Ye and M. H. Lu, A unified approach to pseudoscalar meson photoproductions off nucleons in the quark model, Phys. Rev. C 56, 1099 (1997).
  * (48) B. Saghai and Z. P. Li, Quark model study of the eta photoproduction: Evidence for a new $S_{11}$ resonance?, Eur. Phys. J. A 11, 217 (2001).
  * (49) Q. Zhao, B. Saghai and Z. P. Li, Quark model approach to the eta meson electroproduction on the proton, J. Phys. G 28, 1293 (2002).
  * (50) J. He, B. Saghai and Z. P. Li, Study of $\eta$ photoproduction on the proton in a chiral constituent quark approach via one-gluon-exchange model, Phys. Rev. C 78, 035204 (2008).
  * (51) J. He and B. Saghai, Combined study of $\gamma p\to\eta p$ and $\pi^{-}p\to\eta n$ in a chiral constituent quark approach, Phys. Rev. C 80, 015207 (2009).
  * (52) L. Y. Xiao and X. H. Zhong, Low-energy $K^{-}p\to\Lambda\eta$ reaction and the negative parity $\Lambda$ resonances, Phys. Rev. C 88, 065201 (2013).
  * (53) X. H. Zhong and Q. Zhao, The $K^{-}p\to\Sigma^{0}\pi^{0}$ reaction at low energies in a chiral quark model, Phys. Rev. C 79, 045202 (2009).
  * (54) X. H. Zhong and Q. Zhao, Low energy reactions $K^{-}p\rightarrow\Sigma^{0}\pi^{0}$, $\Lambda\pi^{0}$, $\bar{K}^{0}n$ and the strangeness $S=-1$ hyperons, Phys. Rev. C 88, 015208 (2013).
  * (55) L. Y. Xiao, F. Ouyang, Kai-Lei Wang and X. H. Zhong, Combined analysis of the $\pi^{-}p\rightarrow K^{0}\Lambda$, $\eta n$ reactions in a chiral quark model, Phys. Rev. C 94, 035202 (2016).
  * (56) X. H. Zhong, Q. Zhao, J. He and B. Saghai, Study of $\pi^{-}p\to\eta n$ at low energies in a chiral constituent quark model, Phys. Rev. C 76, 065205 (2007).
  * (57) L. Y. Xiao, X. Cao and X. H. Zhong, Neutral pion photoproduction on the nucleon in a chiral quark model, Phys. Rev. C 92, 035202 (2015).
  * (58) X. H. Zhong and Q. Zhao, $\eta$ photoproduction on the quasi-free nucleons in the chiral quark model, Phys. Rev. C 84, 045207 (2011).
  * (59) X. H. Zhong and Q. Zhao, $\eta^{\prime}$ photoproduction on the nucleons in the quark model, Phys. Rev. C 84, 065204 (2011).
  * (60) W. J. Deng, H. Liu, L. C. Gui and X. H. Zhong, Spectrum and electromagnetic transitions of bottomonium, arXiv:1607.04696 [hep-ph].
  * (61) W. J. Deng, H. Liu, L. C. Gui and X. H. Zhong, Charmonium spectrum and their electromagnetic transitions with higher multipole contributions, Phys. Rev. D 95, 034026 (2017).
  * (62) S. J. Brodsky and J. R. Primack, The Electromagnetic Interactions of Composite Systems, Annals Phys. (N.Y.) 52, 315 (1969).
  * (63) T. M. Aliev, K. Azizi and H. Sundu, Radiative $\Omega_{Q}^{*}\rightarrow\Omega_{Q}\gamma$ and $\Xi_{Q}^{*}\rightarrow\Xi^{\prime}_{Q}\gamma$ transitions in light cone QCD, Eur. Phys. J. C 75, 14 (2015).
  * (64) A. Majethiya, B. Patel and P. C. Vinodkumar, Radiative decays of single heavy flavour baryons, Eur. Phys. J. A 42, 213 (2009)
  * (65) J. Dey, V. Shevchenko, P. Volkovitsky and M. Dey, Radiative decays of S wave charmed baryons, Phys. Lett. B 337, 185 (1994).
  * (66) Z. G. Wang, Analysis of the vertexes $\Omega_{Q}^{*}\Omega_{Q}\phi$ and radiative decays $\Omega_{Q}^{*}\to\Omega_{Q}\gamma$, Phys. Rev. D 81, 036002 (2010).
  * (67) H. Bahtiyar, K. U. Can, G. Erkol and M. Oka, $\Omega_{c}\gamma\rightarrow\Omega_{c}^{\ast}$ transition in lattice QCD, Phys. Lett. B 747, 281 (2015).
  * (68) M. A. Ivanov, J. G. Korner, V. E. Lyubovitskij and A. G. Rusetsky, Strong and radiative decays of heavy flavored baryons, Phys. Rev. D 60, 094002 (1999)
  * (69) V. E. Lyubovitskij, A. Faessler, T. Gutsche, M. A. Ivanov and J. G. Korner, Heavy baryons in the relativistic quark model, Prog. Part. Nucl. Phys. 50, 329 (2003).