# Multiple Classifier for Concatenate-Designed Neural Network

Ka-Hou Chan  School of Applied Sciences, Macao Polytechnic Institute, Macao,
China<EMAIL_ADDRESS>,  Sio-Kei Im  Macao Polytechnic Institute, Macao,
China  and  Wei Ke  School of Applied Sciences, Macao Polytechnic Institute,
Macao, China<EMAIL_ADDRESS>
###### Abstract.

This article introduces a multiple classifier method to improve the
performance of concatenate-designed neural networks, such as ResNet and
DenseNet, with the purpose to alleviate the pressure on the final classifier.
We give the design of the classifiers, which collects the features produced
between the network sets, and present the constituent layers and the
activation function for the classifiers, to calculate the classification score
of each classifier. We use the $\mathit{L2}$ normalization method to obtain
the classifier score instead of the $\mathit{Softmax}$ normalization. We also
determine the conditions that can enhance convergence. As a result, the
proposed classifiers are able to improve the accuracy in the experimental
cases significantly, and show that the method not only has better performance
than the original models, but also produces faster convergence. Moreover, our
classifiers are general and can be applied to all classification related
concatenate-designed network models.

###### Key words and phrases:

Multiple Classifier and Convergence Enhancement and Concatenate-Designed
Neural Network and Softplus and $\mathit{L2}$ normalization

*Corresponding Author

##  1\. Introduction

Image classification is one of main topics of neural networks, starting from
the success of AlexNet [1]. The availability of object classification can lay
the foundation for advanced neural systems and is of great significance in the
research of perceiving media data, such as face recognition [2, 3, 4],
medicine image analysis [5, 6, 7] and autonomous vehicles [8, 9], etc. As we
know that image classification by using neural networks has good performance
in over 90.0% of cases, but there is the challenge of how to overcome the left
cases with significant variations of poor of illumination, image blurring and
occlusions [10]. Therefore, the current methods are hardly completely
applicable to those cases where errors are strictly intolerable, for instance
the autonomous vehicles often miss the traffic light when driving fast in the
midnight.

In order to increase the accuracy, studies on neural network architectures
mostly focus on several aspects. The easiest way is to improve the accuracy by
stacking numerous layers, but the increasing rate of this method presents a
logarithmic curve that the later impact is very small. Therefore, even if we
increase the number of AlexNet’s layers, the result is only slightly better
than nothing [11, 12]. Another aspect considers images as the perceptual data
that a learned model may describe random error or noise with redundancy,
instead of the strict underlying data distribution [13]. There are many data
augmentation and regularization approaches proposed in the preprocessing, such
as random cropping [1], flipping [14] and random erasing [15]. Data
augmentation is closely related to oversampling in data analysis that can
reduce overfitting when training machine learning models, by applying data
augmentation, one can achieve predictable accuracy improvement [16], but it
cannot overcome the shortcomings of the training model itself. The most
challenging aspect is to develop a new network model by combining layers
strategically. In order to obtain better performance, the state-of-the-art
deeper network models, such as GoogleNet [17], ResNet [18] and VGGNet [14],
etc., are designed with a large network architecture by stacking convolutional
layers, and these models have performed well for image classification. With
the in-depth study, it can be found that they all concatenate sets of similar
convolutional layers together, and connect one classifier at the end of the
model output, see Figure 1. They are realized by feedforward neural networks,
some features of similar images are always diluted during several
convolutions. For example, to determine the number ‘0’ and the letter ‘O’,
their main features are similar that can be identified with the entire shape.
Considering the Convolutional Neural Networks (CNNs), the convoluted results
will only retain the local features and discard the global information when
pooling is performed with decreasing resolutions, so the decision will be more
difficult after the convolution and pooling. However, no matter how good the
fitting of the network design, the final classifier will be under great
pressure to make decisions.

Figure 1. The brief architecture of VGG-16 and ResNet-18, both of the series
has concatenated with five sets of stacked layers about $\mathit{Conv}$ and
$\mathit{ResBlock}$, respectively, and one classifier is connected at the end
of the model output.

Below lists the main contribution of this paper.

  * •

In order to alleviate the pressure on the final classifier, we introduce
multiple classifiers in this process, and then make decisions based on their
results.

  * •

We present the constituent layers and activation function of the proposed
classifiers, the purpose is to calculate the classification scores of each
classifier.

  * •

We introduce the $\mathit{L2}$ normalization method to obtain the classifier
scores instead of using the $\mathit{Softmax}$ normalization, and determine
the conditions under which the convergence can be improved.

  * •

The proposed method is applicable to the state-of-the-art network models and
achieves comparable results.

In particular, our approach is compatible with all discovered classification
neural networks, and can further be extended to deeper concatenate-designed
models. We conducted a comprehensive experiment on the CIFAR dataset [19] to
show the accuracy of VGGNet with and without our approach. We show that the
accuracy is obviously better with faster convergence. Similar phenomena are
also shown on the ResNet, which indicates that the optimization is effective,
and our approach is not just suitable to a particular network model.

The rest of this paper is organized as follows. section 2 goes through the
related work. section 3 present the design details and justification of the
multiple classifier performance. section 4 shows the experimental results.
Finally, section 5 concludes the paper.

##  2\. Related Work

The multiple classifier approach has attracted a lot of research attention and
has been widely used in many perceived media such as image classification and
speech recognition, etc. Researchers have studied multiple classifiers from
different aspects, including the classifier design and the concurrent rules.
Our work can be related to these three aspects, classifier design, type of
classifier outputs and architectures.

For the traditional design in neural networks, by combining linear layers and
activation functions, a classifier can have multiple output units and
categorizes a sample according to the class whose corresponding output gives
the highest value among the multiple outputs [20]. Also, [21] proposed an
error-correcting output code method to provide redundancy. Later, [22]
proposed an idea of using an additional single-layer perceptron neural network
to enhance the error-correcting capabilities. In particular, a CNN always
includes a number of convolutional and pooling layers which are optionally
accompanied by fully connected linear design. [23] proved the robustness of
their classifier by constructing seven CNNs. It allows to consider the average
error rate obtained as the best results. Moreover, Support vector machines
(SVM), considered as one of the strongest and robustest algorithm in machine
learning, was created by [24], and made use of [25]. It has become a well-
known approach exploited in many domains [26, 27], such as pattern
recognition, classification and image processing, and all of them obtained the
best performance. Later, [28] modified the CNN structure by replacing the
output layer of the fully connected with an SVM classifier. In recent years,
[29] involved multiple classifiers into their proposed model, then averaged
all the sets of the classifier outputs. In addition, in order to better
realize the function of the classifier in CNN, two state-of-art classifiers
Random Forest (RF) and Gradient Boosting (GB) have also been applied to deep
learning [30]. Meanwhile, for the types of classifier outputs, researchers
considered how to calculate the confidence of the classifier for each image
category, where each sample was represented by multiple images [31]. Each
classifier must be trained to increase the confidence of the corresponding
category, thus the category can be determined using the confidence of each
classifier [32]. Further, the results of the multiple classifiers can also be
used to determine in late fusion in multi-modal [33] or multi-voting [34]
classification. Specifically, the above method has a set of individual
classifiers, each classifier makes a decision on the input set individually,
then the method combines their decisions to form a composite result [35]. In
order to invoke the classifier in a concatenate-designed model, [36] proposed
convolutions as classifiers, instead of linear classifiers at the end of
ResNet. Further, [37] decided to combine the “long-term dependencies” and the
ResNet networks in one classifier to show that the accuracy was improved
significantly while maintaining a suitable interference time.

Meanwhile, the deeper convolutional architecture was the most important work,
demonstrating the powerful functions of concatenate-designed neural networks,
which showed that building a deeper network with a tiny convolution kernel was
effective in increasing the performance of the CNN-based network models. After
VGGNet [14], ResNet was first proposed by [18]. It greatly alleviated the
optimization difficulty and increased it to some hundreds of layers by using
skipping connections within their convolutional set. Since then, different
kinds of inner structures have been proposed, concentrating on various tasks
and consistently achieved the better performance in different areas [38, 39].
Further, [40] introduced the DenseNet which passed the input features to the
output through a densely connected path to concatenate the input features with
the convoluted output as the DenseBlock result. Take account to these networks
design, they aim to retain more original information to the classifier, so
they also tend to connect the input features directly to the output. However,
the width of their connection path increases linearly as the depth rises,
causing the number of training parameters to increase seriously. This limits
the building of deeper and wider networks that might further improve the
accuracy.

In this work, inspired by the connection from the original information to the
classifier, we aim to adopt the proposed classifier following each layer set.
These classifiers make interim decisions instead of one decision of the last
classifier. Based on these decisions, we then propose a novel combination
method and add it to the state-of-the-art concatenate-designed network that
can achieve higher accuracy.

##  3\. Multiple Classifier Strategy

Looking into concatenate-designed networks, there are several sets connected
in order, as shown in Figure 1, composing multiple convolutional layers and
various types of blocks. We use $h_{t}$ to denote the output feature of
$\mathit{Set}_{t}$ at the $t$-th step with attributes
$\left[\mathit{batch},\mathit{channel},\mathit{width},\mathit{height}\right]$,
and $h_{0}$ is the original image. For each step, $\mathit{Set}_{t}$ refers to
the feature extraction function that takes the previous feature as input and
output the extracted information,

(3.1) $h_{t}=\mathit{Set}_{t}\left(h_{t-1}\right).$

Then, the classification function $\mathit{Classifier}$ transforms the last
output feature $h_{-1}$ to a specific dimension vector $\vec{c}$,

(3.2) $\vec{c}=\mathit{Classifier}\left(h_{-1}\right).$

We further transform $\vec{c}$ into a probability vector through the
$\mathit{Softmax}$ function. In Equation 3.1 and Equation 3.2, we encapsulate
the network rule of various concatenate-designed architectures in a
generalized way. This observation shows that the connection path is
essentially an extensible higher order function that extracts information from
the previous states. However, the feature size is reduced and the number of
channels is increased when a feature passes through the sets. Although it can
capture the major part of the classification target, the overall structure of
the image is somewhat lost. In addition, some useful information in the later
sets may also be discarded during the extraction from the earlier sets. These
problems become more obvious as the number of connections increases.
Therefore, the concatenation usually connects up to five sets, and the final
classifier will be under great pressure to make decisions.

In order to address this issue, let’s revisit the network and divide it by the
pooling layers. In this view, the architecture can be considered as a simple
CNN network if $t=1$, similar to LeNet-5 [41] if $t=2$, and similar to AlexNet
[1] if $t=3$, etc. Particularly, the most advanced CNN network also satisfies
with the $t=5$ case [18, 14, 40]. Thus, with the advancement of hardware
performance, CNN models in different eras can be summarized as this series of
neural networks with different $t$ values. From the above analysis, we observe
that the number of sets can increase indefinitely. In practical applications,
we only consider the part before the classifier, i.e., $\exists
t\bullet\mathit{Classifier}_{t}\left(h_{t}\right)\equiv\vec{c}$. Based on this
idea, we propose to employ the number of $t$ classifiers as

(3.3) $\forall
t\bullet\sum\mathit{Classifier}_{t}\left(h_{t}\right)=\sum\vec{c}_{t}\equiv\vec{c}.$

Meanwhile, such a decision contributing strategy makes it possible to
compromise the global structure and the local pattern of an image.
Comparatively, the former one gets more original information and the latter
one obtains the extracted information. All classifiers make decisions based on
their recent acquired features independently. This strategy can provide more
references for decision making, especially in controversial cases such as ‘0’
and ‘O’, ‘1’ and ‘l’. Multiple classifiers can alleviate the perplexity,
leading to high redundancy.

Figure 2. A general procedure of classifier: often consists of one
$\mathit{Pooling}$ layer and multiple $\mathit{Linear}$ layers. These Linear
layers are often collectively referred to as Fully-Connected (FC) layers.

###  3.1. Classifier Design

Before we present our classifier design, we first analyze those classifiers
used in the current neural networks. It can be found that the procedure of
their stacked layers is the same (see Figure 2). The classifier always
receives the features produced by the last set, and must consider following
dynamic attributes:

$\mathit{width}\times\mathit{height}$,:

the size of the image varies with different applications, but the Fully-
Connected (FC) layer only accepts a fixed size. Classifiers have to pool a fix
size in order to facilitate feature extraction from all sizes of $h_{-1}$.

$channel$,:

different from the image size, this attribute of the original image $h_{0}$
always equals to 3, corresponding to the RGB channels, and then increases to
64, 128, 256 and 512 (up to 1024 in DenseNet) through the various predefined
set.

Therefore, the number of input features of an FC layer is
$\mathit{channel}\times\mathit{width}\times\mathit{height}$, and a vector
$\vec{c}$ is produced as the result. According to this design, when applied to
our classifier that satisfies Equation 3.3, there are multiple classifiers to
collect every output feature from $h_{0}$ to $h_{-1}$. The number of
classifiers is the same as the number of concatenated sets, and we use
$\mathit{Classifier}_{t}$ to denote the $t$-th classifier we employed.
Corresponding to the order of $\mathit{Set}_{t}$, the first problem is to fix
the channel differences, because the concatenation operation used in Equation
3.3 is not viable when the size of the features changes. Thus, our classifier
makes use of one convolutional layer to adjust the number of channel.
Regardless of the output feature $h_{t}$, we also aim to increase their
channels to match the last feature $h_{-1}$ with a small convolutional kernel
size $\left(3\times 3\right)$, each side of the input is zero-padded by one
pixel to keep the feature size fixed. Then we use an adaptive maximum pooling
layer to downsample the convoluted results into a fixed size $\left(1\times
1\right)$, followed by a batch normalization [42] as a transition layer
between the pooling and FC layers. Since the feature size has been downsampled
to $\left(1\times 1\right)$, the number of FC inputs will equal to the size of
the channel merely. For simplicity, we only use one linear layer to achieve
the FC transformation in this work, followed by the type of rectifier
activation function to ensure that the score is positive. We do not recommend
using the $\mathit{ReLU}$ because they become inactive for essentially all
inputs smaller than zero. In this state, no gradients flow backward through
the neuron, and so the neuron falls into a permanent inactive state, becoming
a dead neuron, and will not be conducive to score any category. In view of
this, our design tends to use the $\mathit{Softplus}$ that can be viewed as a
smooth version of $\mathit{ReLU}$, which is monotonic and differentiable,
having a positive first order derivative in $\mathbb{R}$. By stacking the
above layers, we have completed the feature extraction within the proposed
classifiers, and each classifier generates a score vector for each category.

###  3.2. Score Normalization

In addition to feature extraction, we must also normalize the score vector to
collect the result of each classifier. We design that the classifier should
provide the confidence rate (score) rather than making a classification
decision. The result is regarded as the scores for each category. Usually,
this part can be achieved directly through the $\mathit{Softmax}$ function,
but the convergence is slow in practice, so we introduce the method of using
the $\mathit{L2}$ normalization to enhance the convergence of this part.

We first review the general form of the $\mathit{L1}$ and $\mathit{L2}$
normalization, for any $f_{1}\left(x_{i}\right)$ and
$f_{2}\left(x_{i}\right)$,

$\mathit{L1}\left(f_{1}\left(x_{i}\right)\right)=\frac{f_{1}\left(x_{i}\right)}{\sum_{k}f_{1}\left(x_{k}\right)},$
$\mathit{L2}\left(f_{2}\left(x_{i}\right)\right)=\frac{f_{2}\left(x_{i}\right)}{\sqrt{\sum_{k}f^{2}_{2}\left(x_{k}\right)}},$

where $N$ denote the number of categories, and the respect partial derivatives
are,

$\frac{\partial\mathit{L1}\left(f_{1}\left(x_{i}\right)\right)}{\partial
x_{j}}=\frac{1}{\sum_{k}f_{1}\left(x_{k}\right)}\left(-\frac{f_{1}\left(x_{i}\right)}{\sum_{k}f_{1}\left(x_{k}\right)}\right)\frac{\partial
f_{1}\left(x_{j}\right)}{\partial x_{j}},$
$\frac{\partial\mathit{L2}\left(f_{2}\left(x_{i}\right)\right)}{\partial
x_{j}}=\frac{1}{\sqrt{\left(\sum_{k}f^{2}_{2}\left(x_{k}\right)\right)}}\left(-\frac{f_{2}\left(x_{i}\right)f_{2}\left(x_{j}\right)}{\sum_{k}f^{2}_{2}\left(x_{k}\right)}\right)\frac{\partial
f_{2}\left(x_{j}\right)}{\partial x_{j}}.$

It is worth noting that the $\mathit{L1}\left(f_{1}\left(x\right)\right)$
becomes the $\mathit{Softmax}$ normalization if $f_{1}\left(x\right)=e^{x}$.
The normalization method we propose here is
$\mathit{L2}\left(f_{2}\left(x\right)\right)$ with
$f_{2}\left(x\right)=\sqrt{e^{x}}$. We use $S\left(x\right)$ and
$L\left(x\right)$ to denote $\mathit{Softmax}$ and the proposed normalization
method, respectively. According to above discussion, we formulate the final
form as,

$S\left(x_{i}\right)=\mathit{L1}\left(e^{x_{i}}\right)=\frac{e^{x_{i}}}{\sum_{k}e^{x_{k}}},$
$L\left(x_{i}\right)=\mathit{L2}\left(\sqrt{e^{x_{i}}}\right)=\sqrt{\frac{e^{x_{i}}}{\sum_{k}e^{x_{k}}}}.$

The corresponding partial derivatives become,

(3.4) $\frac{\partial S\left(x_{i}\right)}{\partial
x_{j}}=\frac{1}{\sum_{k}e^{x_{k}}}\left(-\frac{e^{x_{i}}}{\sum_{k}e^{x_{k}}}\right)e^{x_{j}},$
(3.5) $\frac{\partial L\left(x_{i}\right)}{\partial
x_{j}}=\frac{1}{\sqrt{\sum_{k}e^{x_{k}}}}\left(-\frac{\sqrt{e^{x_{i}}}\sqrt{e^{x_{j}}}}{\sum_{k}e^{x_{k}}}\right)\frac{1}{2}\sqrt{e^{x_{j}}}.$

Since our goal is to enhance the convergence, we must find a condition that
satisfies the proposition about the partial derivatives, $\frac{\partial
L\left(x_{i}\right)}{\partial x_{j}}\geq\frac{\partial
S\left(x_{i}\right)}{\partial x_{j}}$, i.e.,

$\frac{1}{\sqrt{\sum_{k}e^{x_{k}}}}\left(-\frac{\sqrt{e^{x_{i}}}\sqrt{e^{x_{j}}}}{\sum_{k}e^{x_{k}}}\right)\frac{1}{2}\sqrt{e^{x_{j}}}\geq\frac{1}{\sum_{k}e^{x_{k}}}\left(-\frac{e^{x_{i}}}{\sum_{k}e^{x_{k}}}\right)e^{x_{j}}.$

We simplify this and obtain,

(3.6) $\sum_{k}e^{x_{k}}\leq 4e^{x_{i}}.$

It can be found that Equation 3.6 leads to the necessary condition
$N<4e^{x_{i}}$, which can be rewritten as,

$x_{i}>\ln\frac{N}{4},$

when we assume all $x_{i}>0$, because there is a $\mathit{Softplus}$ function
at the end of the feature extraction. In summary, by using our normalization
method with necessary condition $x_{1},x_{2},\cdots,x_{k}>\ln\frac{N}{4}$, the
enhancement of convergence becomes more and more obvious with the increase of
accuracy. Based on the theory, the proposed classifiers must have the lower
bound $\ln\frac{N}{4}$. In fact, the $\mathit{Softplus}$ function always
satisfies with $x>0\geq\ln\frac{N}{4}$ for $N\geq 4$.

Figure 3. The complete internal design of the proposed classifier. $h_{t}$ is
the $t$-th output feature produced from $\mathit{Set}_{t}$, $h_{-1}$ is the
last output feature, which is the same as in Figure 2, $h.\mathit{ch}$ and
$h.\mathit{size}$ denote the number of $\mathit{channels}$ and the
$\mathit{size}$ of feature $h$, respectively, $N$ denotes the number of
categories.

As shown in Figure 3, we detail the structure of all the stacked layers in the
classifier. There are three parts of the procedures for each classifier.
First, $\mathit{Downsample}$ performs the feature extraction. Next,
$\mathit{FC}$ projects the extracted information into each category as a
reference. Finally, $\mathit{Score}$ normalizes these references into the
output vector $\vec{c}_{t}$ for the final decision making with other
classifiers.

###  3.3. Network Architectures

Following the last stage of the classification neural network, we must finally
provide a probability vector to calculate the cross entropy loss through the
$\mathit{Softmax}$ function. Considering the variable length of future
concatenate-designed network, the number of vectors $\vec{c}_{t}$ produced by
our classifier that are used in other networks may also be different.
Therefore, we want to maintain its scalability and use the sum of the vectors
in Equation 3.3 as the final output of the multiple classifier strategy. The
overall design of the proposed method can inherit the backbone architecture of
any concatenate-designed neural network, making it easy to implement and apply
to other tasks. This can be achieved simply by adding a classifier following
each $\mathit{Set}_{t}$ to the existing classification networks. Under a well
optimized deep learning platform, each classifier requires only a fixed amount
of computational cost and memory consumption, making the deployment very
efficient.

Figure 4. The complete structure of proposed classification method. $\mathit{Set}_{t}$ is contributed by the original network design (like Figure 1). The proposed $\mathit{Classifier}_{t}$ collects output feature $h_{t}$, and then produces a vector $\vec{c}_{t}$ as a reference for final decision, instead of using a classifier at the end. Table 1. The architecture and complexity of our re-implemented concatenate-designed neural networks. We present their required training parameters and computational cost using the FLOPS with input size of one $3\times 32\times 32$ image. Stage | VGG16 | ResNet18 | DLA34 | DenseNet121
---|---|---|---|---
$\mathit{Set}_{1}$ | $\begin{bmatrix}3\times 3,64\end{bmatrix}\times 2$ | $\begin{bmatrix}3\times 3,64\end{bmatrix}\times 1$ | $\begin{bmatrix}3\times 3,16\end{bmatrix}\times 1$ | $\begin{bmatrix}3\times 3,64\end{bmatrix}\times 1$
$\mathit{Set}_{2}$ | $\begin{bmatrix}3\times 3,128\end{bmatrix}\times 2$ | $\begin{bmatrix}3\times 3,64\\\ 3\times 3,64\end{bmatrix}\times 2$ | $\begin{bmatrix}3\times 3,32\end{bmatrix}\times 1$ | $\begin{matrix}\begin{bmatrix}1\times 1,128\\\ 3\times 3,32\end{bmatrix}\times 6\\\ \hline\cr\begin{bmatrix}1\times 1,128\end{bmatrix}\times 1\end{matrix}$
$\mathit{Set}_{3}$ | $\begin{bmatrix}3\times 3,256\end{bmatrix}\times 3$ | $\begin{bmatrix}3\times 3,128\\\ 3\times 3,128\end{bmatrix}\times 2$ | $\begin{matrix}\begin{bmatrix}3\times 3,64\\\ 3\times 3,64\end{bmatrix}\times 2\\\ \hline\cr\begin{bmatrix}1\times 1,64\end{bmatrix}\times 1\end{matrix}$ | $\begin{matrix}\begin{bmatrix}1\times 1,128\\\ 3\times 3,32\end{bmatrix}\times 12\\\ \hline\cr\begin{bmatrix}1\times 1,256\end{bmatrix}\times 1\end{matrix}$
$\mathit{Set}_{4}$ | $\begin{bmatrix}3\times 3,512\end{bmatrix}\times 3$ | $\begin{bmatrix}3\times 3,256\\\ 3\times 3,256\end{bmatrix}\times 2$ | $\begin{matrix}\begin{array}[]{c|c}\begin{matrix}\begin{bmatrix}3\times 3,128\\\ 3\times 3,128\end{bmatrix}\times 2\\\ \hline\cr\begin{bmatrix}1\times 1,128\end{bmatrix}\times 1\end{matrix}&\times 2\end{array}\\\ \hline\cr\begin{bmatrix}1\times 1,128\end{bmatrix}\times 1\end{matrix}$ | $\begin{matrix}\begin{bmatrix}1\times 1,128\\\ 3\times 3,32\end{bmatrix}\times 24\\\ \hline\cr\begin{bmatrix}1\times 1,512\end{bmatrix}\times 1\end{matrix}$
$\mathit{Set}_{5}$ | $\begin{bmatrix}3\times 3,512\end{bmatrix}\times 3$ | $\begin{bmatrix}3\times 3,512\\\ 3\times 3,512\end{bmatrix}\times 2$ | $\begin{matrix}\begin{array}[]{c|c}\begin{matrix}\begin{bmatrix}3\times 3,256\\\ 3\times 3,256\end{bmatrix}\times 2\\\ \hline\cr\begin{bmatrix}1\times 1,256\end{bmatrix}\times 1\end{matrix}&\times 2\end{array}\\\ \hline\cr\begin{bmatrix}1\times 1,256\end{bmatrix}\times 1\end{matrix}$ | $\begin{bmatrix}1\times 1,128\\\ 3\times 3,32\end{bmatrix}\times 16$
$\mathit{Set}_{6}$ |  |  | $\begin{matrix}\begin{bmatrix}3\times 3,512\\\ 3\times 3,512\end{bmatrix}\times 2\\\ \hline\cr\begin{bmatrix}1\times 1,512\end{bmatrix}\times 1\end{matrix}$ |
$\mathit{Classifier}$ | $\begin{bmatrix}512\times 4096\\\ 4096\times 4096\\\ 4096\times N\end{bmatrix}$ | $\begin{bmatrix}512\times N\end{bmatrix}$ | $\begin{bmatrix}512\times N\end{bmatrix}$ | $\begin{bmatrix}1024\times N\end{bmatrix}$
Training parameters | $15.7\times 10^{6}$ | $10.2\times 10^{6}$ | $15.1\times 10^{6}$ | $18.0\times 10^{6}$
FLOPS | $0.3\times 10^{9}$ | $0.9\times 10^{9}$ | $0.3\times 10^{9}$ | $1.1\times 10^{9}$
Table 2. The architecture and complexity of our re-implemented concatenate-designed neural networks with the proposed multiple classifier strategy. We present their required training parameters and computational cost using the FLOPs with input size of one $3\times 32\times 32$ image. Stage | VGG16 | ResNet18 | DLA34 | DenseNet121
---|---|---|---|---
$\mathit{Set}_{1}$ | $\begin{bmatrix}3\times 3,64\end{bmatrix}\times 2$ | $\begin{bmatrix}3\times 3,64\end{bmatrix}\times 1$ | $\begin{bmatrix}3\times 3,16\end{bmatrix}\times 1$ | $\begin{bmatrix}3\times 3,64\end{bmatrix}\times 1$
$\mathit{Set}_{2}$ | $\begin{bmatrix}3\times 3,128\end{bmatrix}\times 2$ | $\begin{bmatrix}3\times 3,64\\\ 3\times 3,64\end{bmatrix}\times 2$ | $\begin{bmatrix}3\times 3,32\end{bmatrix}\times 1$ | $\begin{matrix}\begin{bmatrix}1\times 1,128\\\ 3\times 3,32\end{bmatrix}\times 6\\\ \hline\cr\begin{bmatrix}1\times 1,128\end{bmatrix}\times 1\end{matrix}$
$\mathit{Set}_{3}$ | $\begin{bmatrix}3\times 3,256\end{bmatrix}\times 3$ | $\begin{bmatrix}3\times 3,128\\\ 3\times 3,128\end{bmatrix}\times 2$ | $\begin{matrix}\begin{bmatrix}3\times 3,64\\\ 3\times 3,64\end{bmatrix}\times 2\\\ \hline\cr\begin{bmatrix}1\times 1,64\end{bmatrix}\times 1\end{matrix}$ | $\begin{matrix}\begin{bmatrix}1\times 1,128\\\ 3\times 3,32\end{bmatrix}\times 12\\\ \hline\cr\begin{bmatrix}1\times 1,256\end{bmatrix}\times 1\end{matrix}$
$\mathit{Set}_{4}$ | $\begin{bmatrix}3\times 3,512\end{bmatrix}\times 3$ | $\begin{bmatrix}3\times 3,256\\\ 3\times 3,256\end{bmatrix}\times 2$ | $\begin{matrix}\begin{array}[]{c|c}\begin{matrix}\begin{bmatrix}3\times 3,128\\\ 3\times 3,128\end{bmatrix}\times 2\\\ \hline\cr\begin{bmatrix}1\times 1,128\end{bmatrix}\times 1\end{matrix}&\times 2\end{array}\\\ \hline\cr\begin{bmatrix}1\times 1,128\end{bmatrix}\times 1\end{matrix}$ | $\begin{matrix}\begin{bmatrix}1\times 1,128\\\ 3\times 3,32\end{bmatrix}\times 24\\\ \hline\cr\begin{bmatrix}1\times 1,512\end{bmatrix}\times 1\end{matrix}$
$\mathit{Set}_{5}$ | $\begin{bmatrix}3\times 3,512\end{bmatrix}\times 3$ | $\begin{bmatrix}3\times 3,512\\\ 3\times 3,512\end{bmatrix}\times 2$ | $\begin{matrix}\begin{array}[]{c|c}\begin{matrix}\begin{bmatrix}3\times 3,256\\\ 3\times 3,256\end{bmatrix}\times 2\\\ \hline\cr\begin{bmatrix}1\times 1,256\end{bmatrix}\times 1\end{matrix}&\times 2\end{array}\\\ \hline\cr\begin{bmatrix}1\times 1,256\end{bmatrix}\times 1\end{matrix}$ | $\begin{bmatrix}1\times 1,128\\\ 3\times 3,32\end{bmatrix}\times 16$
$\mathit{Set}_{6}$ |  |  | $\begin{matrix}\begin{bmatrix}3\times 3,512\\\ 3\times 3,512\end{bmatrix}\times 2\\\ \hline\cr\begin{bmatrix}1\times 1,512\end{bmatrix}\times 1\end{matrix}$ |
$\mathit{Proposed\,Classifier}$ | $\begin{bmatrix}3\times 3,512\\\ 512\times N\end{bmatrix}\times 5$ | $\begin{bmatrix}3\times 3,512\\\ 512\times N\end{bmatrix}\times 5$ | $\begin{bmatrix}3\times 3,512\\\ 512\times N\end{bmatrix}\times 6$ | $\begin{bmatrix}3\times 3,1024\\\ 1024\times N\end{bmatrix}\times 5$
Training parameters | $21.5\times 10^{6}$ | $15.9\times 10^{6}$ | $20.0\times 10^{6}$ | $26.3\times 10^{6}$
FLOPS | $0.5\times 10^{9}$ | $1.4\times 10^{9}$ | $0.6\times 10^{9}$ | $2.0\times 10^{9}$

As listed in Table 1 and Table 2, we measure the model complexity by counting
the total number of training parameters within each neural network, and
measure the computational cost of each deeper model using the floating-point
operations per second (FLOPS). As found in the results, the required
parameters of the neural network when using the proposed multiple classifiers
are about 50.0% more than that of the original network, and the FLOPS reaches
up to 200.0% in a training epoch.

##  4\. Experiments Result and Discussion

In order to evaluate the proposed method on a variety of state-of-art
classification models, we applied our approach to the CIFAR-10 and CIFAR-100
datasets [19], which are labeled by 10 and 100 classes color images,
respectively, with 50k for training and 10k for testing. All our experiments
are conducted on an NVIDIA GeForce RTX 2080 Ti with 11.0GB of video memory. In
order to compare the proposed strategy with the original networks, we re-
implement the VGG16 [14], ResNet18 [18], DLA34 [43] and DenseNet121 [40] with
and without our multiple classifiers in PyTorch [44], respectively, and used
the advanced gradient-related optimizer, the Adam [45] method with a learning
rate of 0.001. All experiments use the same dataset in each test with a batch
size of 100 per iteration set, and with the same configuration and the same
number of neural nodes, as shown in Table 1 and Table 2. For a justified
comparison, we also train the original and the proposed models with the same
training procedure. We use random cropping and horizontal flipping with color
normalization, followed by the random erasure data augmentation [15]. We
target to process 300 epochs to compare the accuracy and convergence of the
various models. In addition, there is a scheduler for adjusting the learning
rate. It reduces the learning rate when the loss becomes stagnant.

Figure 5. The accuracy of training data after 300 epoch in the original and
proposed VGG16. Figure 6. The accuracy of training data after 300 epoch in the
original and proposed ResNet18. Figure 7. The accuracy of training data after
300 epoch in the original and proposed DenseNet181. Figure 8. The accuracy of
training data after 300 epoch in the original and proposed DenseNet181.

For more credibility, each model has been tested eight times. However, we only
show the results with the best accuracy in the figures from Figure 5 to Figure
8, with the purpose to visualize the convergence during the training period.
As expected in subsection 3.2, we see that the convergence contributed by the
proposed method is significantly improved. They converge faster with
increasing accuracy. All the experiments can be well trained within 150
epochs. Note that in addition to the (best) results shown in these figures,
the convergences of all the experiments using the proposed method are better
than those of the original when the accuracy is greater than 0.5. On the other
hand, Table 1 and Table 2 show that the proposed classifier requires more
training parameters, and the FLOPS also increases, but the increase of the
FLOPS can almost overcome the time increase caused by the increased
parameters, so this is only a little more than the original time.

Table 3. The average accuracy of CIFAR-10 test data with the error range. Method | VGG16 | ResNet18 | DLA34 | DenseNet121
---|---|---|---|---
Original | $0.9258\pm 0.0031$ | $0.9334\pm 0.0042$ | $0.9205\pm 0.0032$ | $0.9359\pm 0.0037$
Proposed | $0.9336\pm 0.0005$ | $0.9505\pm 0.0013$ | $0.9337\pm 0.0021$ | $0.9528\pm 0.0029$
Table 4. The average accuracy of CIFAR-100 test data with the error range. Method | VGG16 | ResNet18 | DLA34 | DenseNet121
---|---|---|---|---
Original | $0.6626\pm 0.0033$ | $0.6933\pm 0.0063$ | $0.6056\pm 0.0124$ | $0.6815\pm 0.0069$
Proposed | $0.7174\pm 0.0036$ | $0.7450\pm 0.0061$ | $0.7185\pm 0.0027$ | $0.7669\pm 0.0052$

Meanwhile, Table 3 and Table 4 list the average accuracy of the test data with
the error range, with the purpose to illustrate the improvement by our method.
It can be seen that the accuracy and the stability in our method have been
significantly improved. The accuracy by the proposed method is found to
improve the overall classification performance in multiple runs. The most
obvious improvement is the CIFAR-100 classification, by a gap of 5.0%.
Commonly, with the discovery of VGG16, ResNet18, DLA34 and DensNet121, the
performance improves, and our method can further improve these functions
without a lot of work. For more complex and deeper networks, the gain of our
method will be even higher. In addition to these network backbones, our
experiment also requires more pre-processing and post-processing, the reason
for that is less relevant and we are not going into the detail. Please find
the complete source code and experimental results in the supplemental files.

##  5\. Conclusion

We present a multiple classifier method that can improve the performance from
a new perspective, and as the number of modules multiplying their connectivity
get greater, the method is more effective. By adjusting the concatenated
architectures used for classification tasks, we identify the need for multiple
classifiers to participate, and make the final decision according to their
results. We further discover the condition to enhance the convergence, and
embed it into the proposed classifier. Compared with the original model, our
method is more accurate, and can make use of parameters and computations more
efficiently. Experiments show that the dominant architectures can all be
improved by using the multiple classifiers. The gap of accuracy improvement is
obvious.

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# Precision Measurement of the Decay $\Sigma^{+}\rightarrow p\gamma$ in the
Process $J/\psi\rightarrow\Sigma^{+}\bar{\Sigma}^{-}$

M. Ablikim1, M. N. Achasov13,b, P. Adlarson73, R. Aliberti34, A.
Amoroso72A,72C, M. R. An38, Q. An69,56, Y. Bai55, O. Bakina35, I.
Balossino29A, Y. Ban45,g, V. Batozskaya1,43, K. Begzsuren31, N. Berger34, M.
Bertani28A, D. Bettoni29A, F. Bianchi72A,72C, E. Bianco72A,72C, J. Bloms66, A.
Bortone72A,72C, I. Boyko35, R. A. Briere5, A. Brueggemann66, H. Cai74, X.
Cai1,56, A. Calcaterra28A, G. F. Cao1,61, N. Cao1,61, S. A. Cetin60A, J. F.
Chang1,56, T. T. Chang75, W. L. Chang1,61, G. R. Che42, G. Chelkov35,a, C.
Chen42, Chao Chen53, G. Chen1, H. S. Chen1,61, M. L. Chen1,56,61, S. J.
Chen41, S. M. Chen59, T. Chen1,61, X. R. Chen30,61, X. T. Chen1,61, Y. B.
Chen1,56, Y. Q. Chen33, Z. J. Chen25,h, W. S. Cheng72C, S. K. Choi10A, X.
Chu42, G. Cibinetto29A, S. C. Coen4, F. Cossio72C, J. J. Cui48, H. L. Dai1,56,
J. P. Dai77, A. Dbeyssi19, R. E. de Boer4, D. Dedovich35, Z. Y. Deng1, A.
Denig34, I. Denysenko35, M. Destefanis72A,72C, F. De Mori72A,72C, B. Ding64,1,
X. X. Ding45,g, Y. Ding39, Y. Ding33, J. Dong1,56, L. Y. Dong1,61, M. Y.
Dong1,56,61, X. Dong74, S. X. Du79, Z. H. Duan41, P. Egorov35,a, Y. L. Fan74,
J. Fang1,56, S. S. Fang1,61, W. X. Fang1, Y. Fang1, R. Farinelli29A, L.
Fava72B,72C, F. Feldbauer4, G. Felici28A, C. Q. Feng69,56, J. H. Feng57, K
Fischer67, M. Fritsch4, C. Fritzsch66, C. D. Fu1, Y. W. Fu1, H. Gao61, Y. N.
Gao45,g, Yang Gao69,56, S. Garbolino72C, I. Garzia29A,29B, P. T. Ge74, Z. W.
Ge41, C. Geng57, E. M. Gersabeck65, A Gilman67, K. Goetzen14, L. Gong39, W. X.
Gong1,56, W. Gradl34, S. Gramigna29A,29B, M. Greco72A,72C, M. H. Gu1,56, Y. T.
Gu16, C. Y Guan1,61, Z. L. Guan22, A. Q. Guo30,61, L. B. Guo40, R. P. Guo47,
Y. P. Guo12,f, A. Guskov35,a, X. T. H.1,61, W. Y. Han38, X. Q. Hao20, F. A.
Harris63, K. K. He53, K. L. He1,61, F. H. Heinsius4, C. H. Heinz34, Y. K.
Heng1,56,61, C. Herold58, T. Holtmann4, P. C. Hong12,f, G. Y. Hou1,61, Y. R.
Hou61, Z. L. Hou1, H. M. Hu1,61, J. F. Hu54,i, T. Hu1,56,61, Y. Hu1, G. S.
Huang69,56, K. X. Huang57, L. Q. Huang30,61, X. T. Huang48, Y. P. Huang1, T.
Hussain71, N Hüsken27,34, W. Imoehl27, M. Irshad69,56, J. Jackson27, S.
Jaeger4, S. Janchiv31, J. H. Jeong10A, Q. Ji1, Q. P. Ji20, X. B. Ji1,61, X. L.
Ji1,56, Y. Y. Ji48, Z. K. Jia69,56, P. C. Jiang45,g, S. S. Jiang38, T. J.
Jiang17, X. S. Jiang1,56,61, Y. Jiang61, J. B. Jiao48, Z. Jiao23, S. Jin41, Y.
Jin64, M. Q. Jing1,61, T. Johansson73, X. K.1, S. Kabana32, N. Kalantar-
Nayestanaki62, X. L. Kang9, X. S. Kang39, R. Kappert62, M. Kavatsyuk62, B. C.
Ke79, A. Khoukaz66, R. Kiuchi1, R. Kliemt14, L. Koch36, O. B. Kolcu60A, B.
Kopf4, M. Kuessner4, A. Kupsc43,73, W. Kühn36, J. J. Lane65, J. S. Lange36, P.
Larin19, A. Lavania26, L. Lavezzi72A,72C, T. T. Lei69,k, Z. H. Lei69,56, H.
Leithoff34, M. Lellmann34, T. Lenz34, C. Li42, C. Li46, C. H. Li38, Cheng
Li69,56, D. M. Li79, F. Li1,56, G. Li1, H. Li69,56, H. B. Li1,61, H. J. Li20,
H. N. Li54,i, Hui Li42, J. R. Li59, J. S. Li57, J. W. Li48, Ke Li1, L. J
Li1,61, L. K. Li1, Lei Li3, M. H. Li42, P. R. Li37,j,k, S. X. Li12, T. Li48,
W. D. Li1,61, W. G. Li1, X. H. Li69,56, X. L. Li48, Xiaoyu Li1,61, Y. G.
Li45,g, Z. J. Li57, Z. X. Li16, Z. Y. Li57, C. Liang41, H. Liang69,56, H.
Liang1,61, H. Liang33, Y. F. Liang52, Y. T. Liang30,61, G. R. Liao15, L. Z.
Liao48, J. Libby26, A. Limphirat58, D. X. Lin30,61, T. Lin1, B. J. Liu1, B. X.
Liu74, C. Liu33, C. X. Liu1, D. Liu19,69, F. H. Liu51, Fang Liu1, Feng Liu6,
G. M. Liu54,i, H. Liu37,j,k, H. B. Liu16, H. M. Liu1,61, Huanhuan Liu1, Huihui
Liu21, J. B. Liu69,56, J. L. Liu70, J. Y. Liu1,61, K. Liu1, K. Y. Liu39, Ke
Liu22, L. Liu69,56, L. C. Liu42, Lu Liu42, M. H. Liu12,f, P. L. Liu1, Q.
Liu61, S. B. Liu69,56, T. Liu12,f, W. K. Liu42, W. M. Liu69,56, X. Liu37,j,k,
Y. Liu37,j,k, Y. B. Liu42, Z. A. Liu1,56,61, Z. Q. Liu48, X. C. Lou1,56,61, F.
X. Lu57, H. J. Lu23, J. G. Lu1,56, X. L. Lu1, Y. Lu7, Y. P. Lu1,56, Z. H.
Lu1,61, C. L. Luo40, M. X. Luo78, T. Luo12,f, X. L. Luo1,56, X. R. Lyu61, Y.
F. Lyu42, F. C. Ma39, H. L. Ma1, J. L. Ma1,61, L. L. Ma48, M. M. Ma1,61, Q. M.
Ma1, R. Q. Ma1,61, R. T. Ma61, X. Y. Ma1,56, Y. Ma45,g, F. E. Maas19, M.
Maggiora72A,72C, S. Maldaner4, S. Malde67, A. Mangoni28B, Y. J. Mao45,g, Z. P.
Mao1, S. Marcello72A,72C, Z. X. Meng64, J. G. Messchendorp14,62, G.
Mezzadri29A, H. Miao1,61, T. J. Min41, R. E. Mitchell27, X. H. Mo1,56,61, N.
Yu. Muchnoi13,b, Y. Nefedov35, F. Nerling19,d, I. B. Nikolaev13,b, Z.
Ning1,56, S. Nisar11,l, Y. Niu 48, S. L. Olsen61, Q. Ouyang1,56,61, S.
Pacetti28B,28C, X. Pan53, Y. Pan55, A. Pathak33, Y. P. Pei69,56, M.
Pelizaeus4, H. P. Peng69,56, K. Peters14,d, J. L. Ping40, R. G. Ping1,61, S.
Plura34, S. Pogodin35, V. Prasad32, F. Z. Qi1, H. Qi69,56, H. R. Qi59, M.
Qi41, T. Y. Qi12,f, S. Qian1,56, W. B. Qian61, C. F. Qiao61, J. J. Qin70, L.
Q. Qin15, X. P. Qin12,f, X. S. Qin48, Z. H. Qin1,56, J. F. Qiu1, S. Q. Qu59,
C. F. Redmer34, K. J. Ren38, A. Rivetti72C, V. Rodin62, M. Rolo72C, G.
Rong1,61, Ch. Rosner19, S. N. Ruan42, N. Salone43, A. Sarantsev35,c, Y.
Schelhaas34, K. Schoenning73, M. Scodeggio29A,29B, K. Y. Shan12,f, W. Shan24,
X. Y. Shan69,56, J. F. Shangguan53, L. G. Shao1,61, M. Shao69,56, C. P.
Shen12,f, H. F. Shen1,61, W. H. Shen61, X. Y. Shen1,61, B. A. Shi61, H. C.
Shi69,56, J. Y. Shi1, Q. Q. Shi53, R. S. Shi1,61, X. Shi1,56, J. J. Song20, T.
Z. Song57, W. M. Song33,1, Y. X. Song45,g, S. Sosio72A,72C, S. Spataro72A,72C,
F. Stieler34, Y. J. Su61, G. B. Sun74, G. X. Sun1, H. Sun61, H. K. Sun1, J. F.
Sun20, K. Sun59, L. Sun74, S. S. Sun1,61, T. Sun1,61, W. Y. Sun33, Y. Sun9, Y.
J. Sun69,56, Y. Z. Sun1, Z. T. Sun48, Y. X. Tan69,56, C. J. Tang52, G. Y.
Tang1, J. Tang57, Y. A. Tang74, L. Y Tao70, Q. T. Tao25,h, M. Tat67, J. X.
Teng69,56, V. Thoren73, W. H. Tian50, W. H. Tian57, Y. Tian30,61, Z. F.
Tian74, I. Uman60B, B. Wang1, B. L. Wang61, Bo Wang69,56, C. W. Wang41, D. Y.
Wang45,g, F. Wang70, H. J. Wang37,j,k, H. P. Wang1,61, K. Wang1,56, L. L.
Wang1, M. Wang48, Meng Wang1,61, S. Wang37,j,k, S. Wang12,f, T. Wang12,f, T.
J. Wang42, W. Wang70, W. Wang57, W. H. Wang74, W. P. Wang69,56, X. Wang45,g,
X. F. Wang37,j,k, X. J. Wang38, X. L. Wang12,f, Y. Wang59, Y. D. Wang44, Y. F.
Wang1,56,61, Y. H. Wang46, Y. N. Wang44, Y. Q. Wang1, Yaqian Wang18,1, Yi
Wang59, Z. Wang1,56, Z. L. Wang70, Z. Y. Wang1,61, Ziyi Wang61, D. Wei68, D.
H. Wei15, F. Weidner66, S. P. Wen1, C. W. Wenzel4, U. Wiedner4, G.
Wilkinson67, M. Wolke73, L. Wollenberg4, C. Wu38, J. F. Wu1,61, L. H. Wu1, L.
J. Wu1,61, X. Wu12,f, X. H. Wu33, Y. Wu69, Y. J Wu30, Z. Wu1,56, L. Xia69,56,
X. M. Xian38, T. Xiang45,g, D. Xiao37,j,k, G. Y. Xiao41, H. Xiao12,f, S. Y.
Xiao1, Y. L. Xiao12,f, Z. J. Xiao40, C. Xie41, X. H. Xie45,g, Y. Xie48, Y. G.
Xie1,56, Y. H. Xie6, Z. P. Xie69,56, T. Y. Xing1,61, C. F. Xu1,61, C. J. Xu57,
G. F. Xu1, H. Y. Xu64, Q. J. Xu17, W. L. Xu64, X. P. Xu53, Y. C. Xu76, Z. P.
Xu41, F. Yan12,f, L. Yan12,f, W. B. Yan69,56, W. C. Yan79, X. Q Yan1, H. J.
Yang49,e, H. L. Yang33, H. X. Yang1, Tao Yang1, Y. Yang12,f, Y. F. Yang42, Y.
X. Yang1,61, Yifan Yang1,61, Z. W. Yang37,j,k, M. Ye1,56, M. H. Ye8, J. H.
Yin1, Z. Y. You57, B. X. Yu1,56,61, C. X. Yu42, G. Yu1,61, T. Yu70, X. D.
Yu45,g, C. Z. Yuan1,61, L. Yuan2, S. C. Yuan1, X. Q. Yuan1, Y. Yuan1,61, Z. Y.
Yuan57, C. X. Yue38, A. A. Zafar71, F. R. Zeng48, X. Zeng12,f, Y. Zeng25,h, Y.
J. Zeng1,61, X. Y. Zhai33, Y. H. Zhan57, A. Q. Zhang1,61, B. L. Zhang1,61, B.
X. Zhang1, D. H. Zhang42, G. Y. Zhang20, H. Zhang69, H. H. Zhang33, H. H.
Zhang57, H. Q. Zhang1,56,61, H. Y. Zhang1,56, J. J. Zhang50, J. L. Zhang75, J.
Q. Zhang40, J. W. Zhang1,56,61, J. X. Zhang37,j,k, J. Y. Zhang1, J. Z.
Zhang1,61, Jiawei Zhang1,61, L. M. Zhang59, L. Q. Zhang57, Lei Zhang41, P.
Zhang1, Q. Y. Zhang38,79, Shuihan Zhang1,61, Shulei Zhang25,h, X. D. Zhang44,
X. M. Zhang1, X. Y. Zhang53, X. Y. Zhang48, Y. Zhang67, Y. T. Zhang79, Y. H.
Zhang1,56, Yan Zhang69,56, Yao Zhang1, Z. H. Zhang1, Z. L. Zhang33, Z. Y.
Zhang42, Z. Y. Zhang74, G. Zhao1, J. Zhao38, J. Y. Zhao1,61, J. Z. Zhao1,56,
Lei Zhao69,56, Ling Zhao1, M. G. Zhao42, S. J. Zhao79, Y. B. Zhao1,56, Y. X.
Zhao30,61, Z. G. Zhao69,56, A. Zhemchugov35,a, B. Zheng70, J. P. Zheng1,56, W.
J. Zheng1,61, Y. H. Zheng61, B. Zhong40, X. Zhong57, H. Zhou48, L. P.
Zhou1,61, X. Zhou74, X. K. Zhou6, X. R. Zhou69,56, X. Y. Zhou38, Y. Z.
Zhou12,f, J. Zhu42, K. Zhu1, K. J. Zhu1,56,61, L. Zhu33, L. X. Zhu61, S. H.
Zhu68, S. Q. Zhu41, T. J. Zhu12,f, W. J. Zhu12,f, Y. C. Zhu69,56, Z. A.
Zhu1,61, J. H. Zou1, J. Zu69,56
(BESIII Collaboration)
1 Institute of High Energy Physics, Beijing 100049, People’s Republic of China
2 Beihang University, Beijing 100191, People’s Republic of China
3 Beijing Institute of Petrochemical Technology, Beijing 102617, People’s
Republic of China
4 Bochum Ruhr-University, D-44780 Bochum, Germany
5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
6 Central China Normal University, Wuhan 430079, People’s Republic of China
7 Central South University, Changsha 410083, People’s Republic of China
8 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
9 China University of Geosciences, Wuhan 430074, People’s Republic of China
10 Chung-Ang University, Seoul, 06974, Republic of Korea
11 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind
Road, 54000 Lahore, Pakistan
12 Fudan University, Shanghai 200433, People’s Republic of China
13 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
14 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
15 Guangxi Normal University, Guilin 541004, People’s Republic of China
16 Guangxi University, Nanning 530004, People’s Republic of China
17 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
18 Hebei University, Baoding 071002, People’s Republic of China
19 Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
20 Henan Normal University, Xinxiang 453007, People’s Republic of China
21 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
22 Henan University of Technology, Zhengzhou 450001, People’s Republic of
China
23 Huangshan College, Huangshan 245000, People’s Republic of China
24 Hunan Normal University, Changsha 410081, People’s Republic of China
25 Hunan University, Changsha 410082, People’s Republic of China
26 Indian Institute of Technology Madras, Chennai 600036, India
27 Indiana University, Bloomington, Indiana 47405, USA
28 INFN Laboratori Nazionali di Frascati , (A)INFN Laboratori Nazionali di
Frascati, I-00044, Frascati, Italy; (B)INFN Sezione di Perugia, I-06100,
Perugia, Italy; (C)University of Perugia, I-06100, Perugia, Italy
29 INFN Sezione di Ferrara, (A)INFN Sezione di Ferrara, I-44122, Ferrara,
Italy; (B)University of Ferrara, I-44122, Ferrara, Italy
30 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
31 Institute of Physics and Technology, Peace Avenue 54B, Ulaanbaatar 13330,
Mongolia
32 Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D,
Arica, Chile
33 Jilin University, Changchun 130012, People’s Republic of China
34 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
35 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
36 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
37 Lanzhou University, Lanzhou 730000, People’s Republic of China
38 Liaoning Normal University, Dalian 116029, People’s Republic of China
39 Liaoning University, Shenyang 110036, People’s Republic of China
40 Nanjing Normal University, Nanjing 210023, People’s Republic of China
41 Nanjing University, Nanjing 210093, People’s Republic of China
42 Nankai University, Tianjin 300071, People’s Republic of China
43 National Centre for Nuclear Research, Warsaw 02-093, Poland
44 North China Electric Power University, Beijing 102206, People’s Republic of
China
45 Peking University, Beijing 100871, People’s Republic of China
46 Qufu Normal University, Qufu 273165, People’s Republic of China
47 Shandong Normal University, Jinan 250014, People’s Republic of China
48 Shandong University, Jinan 250100, People’s Republic of China
49 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
50 Shanxi Normal University, Linfen 041004, People’s Republic of China
51 Shanxi University, Taiyuan 030006, People’s Republic of China
52 Sichuan University, Chengdu 610064, People’s Republic of China
53 Soochow University, Suzhou 215006, People’s Republic of China
54 South China Normal University, Guangzhou 510006, People’s Republic of China
55 Southeast University, Nanjing 211100, People’s Republic of China
56 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
57 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
58 Suranaree University of Technology, University Avenue 111, Nakhon
Ratchasima 30000, Thailand
59 Tsinghua University, Beijing 100084, People’s Republic of China
60 Turkish Accelerator Center Particle Factory Group, (A)Istinye University,
34010, Istanbul, Turkey; (B)Near East University, Nicosia, North Cyprus,
99138, Mersin 10, Turkey
61 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
62 University of Groningen, NL-9747 AA Groningen, The Netherlands
63 University of Hawaii, Honolulu, Hawaii 96822, USA
64 University of Jinan, Jinan 250022, People’s Republic of China
65 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
66 University of Muenster, Wilhelm-Klemm-Strasse 9, 48149 Muenster, Germany
67 University of Oxford, Keble Road, Oxford OX13RH, United Kingdom
68 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
69 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
70 University of South China, Hengyang 421001, People’s Republic of China
71 University of the Punjab, Lahore-54590, Pakistan
72 University of Turin and INFN, (A)University of Turin, I-10125, Turin,
Italy; (B)University of Eastern Piedmont, I-15121, Alessandria, Italy;
(C)INFN, I-10125, Turin, Italy
73 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
74 Wuhan University, Wuhan 430072, People’s Republic of China
75 Xinyang Normal University, Xinyang 464000, People’s Republic of China
76 Yantai University, Yantai 264005, People’s Republic of China
77 Yunnan University, Kunming 650500, People’s Republic of China
78 Zhejiang University, Hangzhou 310027, People’s Republic of China
79 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia
b Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
c Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia
d Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
e Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology,
Ministry of Education; Shanghai Key Laboratory for Particle Physics and
Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240,
People’s Republic of China
f Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and
Institute of Modern Physics, Fudan University, Shanghai 200443, People’s
Republic of China
g Also at State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, People’s Republic of China
h Also at School of Physics and Electronics, Hunan University, Changsha
410082, China
i Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, China
j Also at Frontiers Science Center for Rare Isotopes, Lanzhou University,
Lanzhou 730000, People’s Republic of China
k Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou
730000, People’s Republic of China
l Also at the Department of Mathematical Sciences, IBA, Karachi, Pakistan

###### Abstract

Using $(10\,087\pm 44)\text{\times}{10}^{6}$ $J/\psi$ events collected with
the BESIII detector, the radiative hyperon decay $\Sigma^{+}\to p\gamma$ is
studied at an electron-positron collider experiment for the first time. The
absolute branching fraction is measured to be $\left($0.996\pm 0.021_{\rm
stat.}\pm 0.018_{\rm syst.}$\right)\times 10^{-3}$, which is lower than its
world average value by $4.2$ standard deviations. Its decay asymmetry
parameter is determined to be $-0.652\pm 0.056_{\rm stat.}\pm 0.020_{\rm
syst.}$. The branching fraction and decay asymmetry parameter are the most
precise to date, and the accuracies are improved by $78\text{\,}\mathrm{\char
37\relax}$ and $34\text{\,}\mathrm{\char 37\relax}$, respectively.

Radiative hyperon decays provide valuable insight into the nature of
nonleptonic weak interactions [1]. In general, the radiative decays of a
spin-$\frac{1}{2}$ hyperon are described by a parity conserving (P-wave) and a
parity violating (S-wave) amplitude. The non-vanishing parity violating
amplitude produces an asymmetric angular distribution of the daughter baryon
in the hyperon rest frame, as
$\mathop{}\\!\mathrm{d}N/\mathop{}\\!\mathrm{d}\Omega=(N/4\pi)(1+\alpha_{\gamma}\mathbf{P}_{i}\cdot\hat{\mathrm{p}})$.
Here $\mathbf{P}_{i}$ is the polarization of the decaying hyperon,
$\hat{\mathrm{p}}$ is the unit vector of the daughter baryon’s momenta in the
hyperon rest frame, and $\alpha_{\gamma}$ is the decay asymmetry parameter
characterizing the mixing of S- and P-waves. In the limit of unitary symmetry,
the parity violating amplitude of radiative hyperon decays is predicted to be
small, resulting in $\alpha_{\gamma}=0$ [2]. However, the $\alpha_{\gamma}$
values measured in experiments are large [3, 4, *PhysRevLett.64.843,
*PhysRevLett.72.808, *PhysRevLett.86.3239, 8]. Of all radiative hyperon
decays, the $\Sigma^{+}\to p\gamma$ decay was observed first and stimulated
continued controversy over several decades. It was first observed in bubble
chamber experiments [9, *PhysRev.188.2077, *MANZ1980217, *Ang:1969hg], and was
studied later at modern particle physics spectrometers [3, 13, *Bristol-
Geneva-Heidelberg-Lausanne-QueenMaryColl-Rutherford:1985ksh]. The current
world average value [15] is dominated by precise measurements using a
polarized charged hyperon beam at Fermilab [8, 16], where both branching
fraction (BF) and $\alpha_{\gamma}$ are obtained as ratios to those of the
$\Sigma^{+}\to p\pi^{0}$ decay.

Various phenomenological models have been proposed to explain the experimental
results of radiative hyperon decays [17, 18, 19, 20, *chang_challenges_2000,
*dubovik_weak_2008, *Zenczykowski:2020hmg], but none of them gives a unified
picture in describing all radiative hyperon decays. The theoretical
predictions given by Ref. [24] in the framework of chiral perturbation theory
(ChPT) are consistent with the experimental results of almost all radiative
hyperon decays except for $\Sigma^{+}\to p\gamma$. Therefore, more precise
measurements on $\Sigma^{+}\to p\gamma$ are crucial to test theoretical
approaches such as ChPT. In addition, it has long been argued that
$\Sigma^{+}\rightarrow pl^{+}l^{-}~{}(l=e,\mu)$ decays are good probes for
physics beyond the standard model (SM) [25, 26]. More precise BF and
$\alpha_{\gamma}$ measurements of $\Sigma^{+}\to p\gamma$ offer critical
information on the form factor of $\Sigma^{+}\rightarrow pl^{+}l^{-}$ [26],
and thus provide better constraints on the SM predictions for these decays.

The unique properties of $J/\psi\to\Sigma^{+}\bar{\Sigma}^{-}$ events produced
in electron-positron collisions, _e.g._ the spin correlation and transverse
polarization of hyperon and anti-hyperon [27, 28], and the well defined
kinematics, provide ideal conditions to study the decay $\Sigma^{+}\to
p\gamma$. A double-tagged technique is applied to determine the absolute BF
[29]. This approach was applied in a recent work on the $\Lambda\rightarrow
n\gamma$ decay investigated in the $J/\psi\rightarrow\Lambda\bar{\Lambda}$
process by BESIII [30], which achieved a better precision than fixed target
experiments. Moreover, the decay $\Sigma^{+}\to p\gamma$ and the corresponding
charge conjugate one provide a good opportunity to search for charge-parity
($CP$) violation [31, *Tandean:2002vy], whose experimental information is
absent for $\Sigma^{+}\to p\gamma$ [33, *PhysRevD.51.2271] currently.

In this Letter, using $J/\psi\to\Sigma^{+}\bar{\Sigma}^{-}$ decays from
$(10\,087\pm 44)\text{\times}{10}^{6}$ $J/\psi$ events [35] collected at
BESIII, we report the measurements of the absolute BF and decay asymmetry
parameter of $\Sigma^{+}\to p\gamma$, as well as a test of $CP$ violation in
hyperon decays. Throughout this Letter, charge conjugation is always implied
unless noted otherwise.

A detailed description of the design and performance of the BESIII detector
can be found in Ref. [36, *HUANG2022142], while the simulation and analysis
software framework for BESIII are described in Refs. [38, *Asner:2009zza,
*zhang2018low]. First, events of $J/\psi\to\Sigma^{+}\bar{\Sigma}^{-}$ are
selected with a single-tag (ST) approach, _i.e._ reconstructing a
$\bar{\Sigma}^{-}$ candidate in one of its dominant decay modes
$\bar{\Sigma}^{-}\to\bar{p}\pi^{0}$ ($\mathrm{BF}=$51.57\pm
0.30\text{\,}\mathrm{\char 37\relax}$$) [15]. Then, the double-tag (DT) decay
$\Sigma^{+}\to p\gamma$ is searched for in the system recoiling against the ST
$\bar{\Sigma}^{-}$ hyperon. The corresponding absolute BF is calculated by

$\mathrm{BF}(\Sigma^{+}\to
p\gamma)=\frac{N^{\mathrm{obs}}_{\mathrm{DT}}~{}\varepsilon_{\mathrm{ST}}}{N^{\mathrm{obs}}_{\mathrm{ST}}~{}\varepsilon_{\mathrm{DT}}},$
(1)

where $N^{\mathrm{obs}}_{\mathrm{ST(DT)}}$ and $\varepsilon_{\mathrm{ST(DT)}}$
are the ST (DT) yields and the corresponding detection efficiencies,
respectively.

The helicity decay amplitudes of the processes [41, *PhysRevD.99.056008],
which will be used for the measurement of $\alpha_{\gamma}$, are functions of
the five observables
$\xi=(\theta_{\Sigma^{+}},\theta_{p},\phi_{p},\theta_{\bar{p}},\phi_{\bar{p}})$.
Here, $\theta_{\Sigma^{+}}$ is the angle between the $\Sigma^{+}$ hyperon and
the electron beam in the $J/\psi$ rest frame, $\theta_{p}$
($\theta_{\bar{p}}$) and $\phi_{p}$ ($\phi_{\bar{p}}$) are the polar and
azimuthal angles of the proton (anti-proton) with respect to the $\Sigma^{+}$
helicity frame, respectively. The differential cross-section is given as
$\mathop{}\\!\mathrm{d}\sigma\propto\mathcal{W}(\xi)\mathop{}\\!\mathrm{d}\xi$,
and $\mathcal{W}(\xi)$ follows

$\displaystyle\mathcal{W}(\mathcal{\xi})=$
$\displaystyle\mathcal{F}_{0}(\xi)+\alpha_{\psi}\mathcal{F}_{5}(\xi)+\alpha_{\gamma}\bar{\alpha}_{0}$
(2)
$\displaystyle\times\left(\mathcal{F}_{1}(\xi)+\sqrt{1-\alpha^{2}_{\psi}}\cos(\Delta\Phi)\mathcal{F}_{2}(\xi)+\alpha_{\psi}\mathcal{F}_{6}(\xi)\right)$
$\displaystyle+\sqrt{1-\alpha_{\psi}^{2}}\sin(\Delta\Phi)(\alpha_{\gamma}\mathcal{F}_{3}(\xi)+\bar{\alpha}_{0}\mathcal{F}_{4}(\xi)),$

where $\alpha_{\psi}$ and $\Delta\Phi$ are the hyperon production parameters
of the process $J/\psi\to\Sigma^{+}\bar{\Sigma}^{-}$, $\bar{\alpha}_{0}$ is
the decay asymmetry parameter for the $\bar{\Sigma}^{-}\to\bar{p}\pi^{0}$
decays, and $\mathcal{F}_{i}(\xi)$ ($i=0,1,...,6$) are the angular functions
as described in detail in Ref. [41]. The parameters used here are fixed to the
values in Ref. [43], except for $\alpha_{\gamma}$ to be determined in this
analysis.

A sample of Monte Carlo simulated events (MC) of generic $J/\psi$ decays
corresponding to the luminosity of data is used to study possible background
reactions. On the ST side, the signal MC sample of
$J/\psi\rightarrow\Sigma^{+}\left(\rightarrow\mathrm{anything}\right)\bar{\Sigma}^{-}\left(\rightarrow\bar{p}\pi^{0}\right)$
is generated with its helicity decay amplitude. A phase space (PHSP) MC sample
of
$J/\psi\rightarrow\Delta(1232)^{+}(\rightarrow{\mathrm{anything}})\bar{\Delta}(1232)^{-}(\rightarrow\bar{p}\pi^{0})$
is generated to study ST background. On the DT side, MC samples for the
$J/\psi\to\Sigma^{+}\left(\rightarrow
p\gamma\right)\bar{\Sigma}^{-}\left(\rightarrow\bar{p}\pi^{0}\right)$ signal
and the dominant background $J/\psi\to\Sigma^{+}\left(\rightarrow
p\pi^{0}\right)\bar{\Sigma}^{-}\left(\rightarrow\bar{p}\pi^{0}\right)$ are
generated according to their helicity decay amplitudes.

Charged tracks and photon candidates are detected in the main drift chamber
and electromagnetic calorimeter (EMC), respectively. Requirements on the polar
angle, distance of closest approach to the interaction point, particle
identification probability of charged tracks, the polar angle, energy, EMC
time and distance to the nearest charged track of photon candidates are
applied as in Ref. [43]. The $\pi^{0}$ candidates are reconstructed from pairs
of photon candidates with an invariant mass $M_{\gamma\gamma}$ satisfying
$$116\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}^{2}$<M_{\gamma\gamma}<$148\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}^{2}$$.
A kinematic fit constraining their invariant mass to the $\pi^{0}$ mass [15]
is performed on these photon pairs, and the updated four-momentum is used in
further analysis.

The candidate ST $\bar{\Sigma}^{-}$ events are required to have at least one
$\bar{p}$ and one $\pi^{0}$, and the $\bar{p}\pi^{0}$ combination is required
to have an invariant mass of
$\left|M_{\bar{p}\pi^{0}}-M_{\bar{\Sigma}^{-}}\right|<$13.5\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}^{2}$$,
where $M_{\bar{\Sigma}^{-}}$ is the nominal $\bar{\Sigma}^{-}$ mass [15]. If
there is more than one combination satisfying this requirement, the one with
the minimum $\left|M_{\bar{p}\pi^{0}}-M_{\bar{\Sigma}^{-}}\right|$ is kept for
further analysis. Finally, the ST yield is obtained from the mass distribution
recoiling against the reconstructed $\bar{\Sigma}^{-}$ candidate, which is
defined as

$M_{\mathrm{rec}}=\sqrt{\left(E_{\mathrm{cms}}-E_{\bar{p}}-E_{\pi^{0}}\right)^{2}/c^{4}-\left(\mathbf{P}_{\bar{p}}+\mathbf{P}_{\pi^{0}}\right)^{2}/c^{2}}.$
(3)

Here, $E_{\mathrm{cms}}$ is the center-of-mass energy, $E_{\bar{p}}$
($\mathbf{P}_{\bar{p}}$) and $E_{\pi^{0}}$ ($\mathbf{P}_{\pi^{0}}$) are the
energies (momenta) of the anti-proton and $\pi^{0}$ in the $J/\psi$ rest
frame, respectively.

With the above selection criteria, the distribution of $M_{\rm{rec}}$ of the
ST candidates is shown in Fig. 1, where there is a clear peak representing the
signal of $J/\psi\to\Sigma^{+}\bar{\Sigma}^{-}$. For ST candidates, the
background events are dominated by mis-combinations and
$J/\psi\rightarrow\Delta(1232)^{+}(\rightarrow{\mathrm{anything}})\bar{\Delta}(1232)^{-}(\rightarrow\bar{p}\pi^{0})$,
which contributes as a broad peak. An unbinned maximum likelihood fit is
performed on the $M_{\rm{rec}}$ distribution to determine the ST yield, with
the fit result shown in Fig. 1. In the fit, the signal and
$J/\psi\rightarrow\Delta(1232)^{+}(\rightarrow{\mathrm{anything}})\bar{\Delta}(1232)^{-}(\rightarrow\bar{p}\pi^{0})$
background are described by their MC simulated shapes convolved with a
Gaussian function, which represents the resolution difference between data and
MC simulation. Other non-peaking background is described by a third order
Chebyshev polynomial function. The ST yield and the detection efficiency
evaluated using the signal MC sample are summarized in Table 1.

Figure 1: Fit to the $M_{\rm{rec}}$ distribution of ST candidates in data.
Points with error bars represent data. The blue solid curve is the total fit
result. The red dashed curve, magenta dotted curve and blue dash-dotted curve
show the shapes of signal,
$J/\psi\rightarrow\Delta(1232)^{+}(\rightarrow{\mathrm{anything}})\bar{\Delta}(1232)^{-}(\rightarrow\bar{p}\pi^{0})$
background and other non-peaking background components, respectively.

The signal process $\Sigma^{+}\to p\gamma$ is searched for in the remaining
charged tracks and photons recoiling against the selected ST
$\bar{\Sigma}^{-}$ candidates. One remaining $p$ and at least one remaining
photon candidate are required for DT candidate events. A five-constraint (5C)
kinematic fit under the hypothesis $J/\psi\rightarrow p\bar{p}\pi^{0}\gamma$
is performed. The fit constrains energy-momentum conservation and the
invariant mass of the photon pair on the ST side to the $\pi^{0}$ nominal mass
[15]. The $\chi^{2}$ of the 5C kinematic fit ($\chi^{2}_{\mathrm{5C}}$) is
required to be less than $30$. In case of multiple photon candidates on the DT
side, the one with the minimum $\chi^{2}_{\mathrm{5C}}$ is kept.

MC studies with a generic event type analysis tool [44] indicate that the
dominant background events on the DT side are from the processes
$J/\psi\rightarrow\Sigma^{+}(\rightarrow
p\pi^{0})\bar{\Sigma}^{-}(\rightarrow\bar{p}\pi^{0})$ and
$J/\psi\rightarrow\Delta(1232)^{+}(\rightarrow
p\pi^{0})\bar{\Delta}(1232)^{-}(\rightarrow\bar{p}\pi^{0})$, denoted as
$\Sigma^{+}\rightarrow p\pi^{0}$ and $\Delta(1232)^{+}\rightarrow p\pi^{0}$
background. To remove the $\Delta(1232)^{+}\rightarrow p\pi^{0}$ background,
we make use of the life time difference between $\Sigma^{+}$ and
$\Delta(1232)^{+}$. A secondary vertex fit [45] is performed for the $p$ and
$\bar{p}$ combination, and the length ($L$) from the intersection point to the
interaction point is required to satisfy $L/\sigma_{L}>1.5$, where
$\sigma_{L}$ is the decay length resolution. The remaining
$\Sigma^{+}\rightarrow p\pi^{0}$ background usually includes a low energy
photon from an asymmetric $\pi^{0}$ decay. To suppress this background, a 5C
kinematic fit under the $J/\psi\to p\bar{p}\pi^{0}\gamma\gamma$ hypothesis is
performed for all photon pair combinations. A candidate is eliminated if any
$\chi^{2}$ of the kinematic fit under the $J/\psi\to
p\bar{p}\pi^{0}\gamma\gamma$ hypothesis is less than $\chi^{2}_{5C}$ under the
signal hypothesis.

After applying all the above selection criteria, the distribution of the
proton momentum in the rest frame of $\Sigma^{+}$ ($P_{p}$) is shown in Fig.
2. The prominent peak at 0.204 GeV/$c$ is the major background from
$\Sigma^{+}\to p\pi^{0}$, and the peaking structure around 0.223 GeV/$c$ is
from the $\Sigma^{+}\to p\gamma$ signal. The contribution from the
$\Delta(1232)^{+}\rightarrow p\pi^{0}$ background is negligible.

To obtain the DT signal yield, an unbinned maximum likelihood fit is performed
on the $P_{p}$ distribution. In the fit, the $\Sigma^{+}\to p\gamma$ signal
and the $\Sigma^{+}\to p\pi^{0}$ background are described by MC simulated
shapes convolved with a Gaussian function. A second order Chebyshev polynomial
function is included to represent the residual background such as
$\Delta(1232)^{+}\rightarrow p\pi^{0}$. The fit results are shown in Fig. 2,
and the DT signal yields are summarized in Table 1. To account for the small
difference in the selection efficiencies of charged tracks and photons between
data and MC simulation, an efficiency correction has been performed on the DT
side [43]. The resultant DT efficiencies and the BFs are also summarized in
Table 1. The BFs for $\Sigma^{+}\to p\gamma$ and
$\bar{\Sigma}^{-}\to\bar{p}\gamma$ are consistent with each other within their
uncertainties. Therefore, a simultaneous fit assuming the same BF between the
charge conjugate channels is performed. The resultant BF is also shown in
Table 1.

Figure 2: Fits to the $P_{p}$ distributions for (a) $\Sigma^{+}\rightarrow
p\gamma$ and (b) $\bar{\Sigma}^{-}\rightarrow\bar{p}\gamma$ candidate events.
Points with error bars are data. The red solid curves are the total fit
results. The blue dashed, orange dotted and green dash-dotted curves show the
shapes of signal, $\Sigma^{+}\to p\pi^{0}$ background and other background,
respectively. The insets show the details of the fit in the signal region.

The decay asymmetry parameter $\alpha_{\gamma}$ is determined by a joint
angular analysis according to Eq. (2). To further improve the purity of the
data sample used for the $\alpha_{\gamma}$ measurements, additional kinematic
fits under the hypotheses of $J/\psi\to p\bar{p}\pi^{0}+\pi^{0}$ and
$J/\psi\to p\bar{p}\pi^{0}+\gamma$ are performed for the surviving DT
candidates, where the momenta information of the $\pi^{0}$ and photon are
missing in the fit. The DT candidates are removed if
$\chi^{2}_{\pi^{0}}<\chi^{2}_{\gamma}$, where $\chi^{2}_{\pi^{0}}$ and
$\chi^{2}_{\gamma}$ are the $\chi^{2}$ from the two kinematic fits. Finally,
$2345$ events (including $243$ background events) for the two charge conjugate
modes in the signal region
$0.215<P_{p}<$0.235\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}$$
are used in the following analysis.

$\alpha_{\gamma}$ is obtained from an unbinned maximum likelihood fit with the
likelihood function defined as

$\mathcal{L}\left(H\right)=\prod^{N}_{i=1}\frac{\mathcal{W}\left(\xi_{i},H\right)\varepsilon\left(\xi_{i}\right)}{C\left(H\right)},$
(4)

where
$H=\left(\alpha_{\psi},\Delta\Phi,\alpha_{\gamma},\bar{\alpha}_{0}\right)$
represents the decay parameters, $N$ is the total number of DT candidate
events, $i$ is the corresponding event index,
$\mathcal{W}\left(\xi_{i},H\right)$ is the decay amplitude as defined in Eq.
(2), $\varepsilon\left(\xi_{i}\right)$ is the detection efficiency and
$C\left(H\right)$ is the normalization factor calculated with the PHSP MC
sample [27]. Due to the small number of events, the parameters
$\alpha_{\psi}$, $\Delta\Phi$ and $\bar{\alpha}_{0}$ are fixed to those in
Ref. [43] in the fit.

The value of $\alpha_{\gamma}$ is obtained by minimizing the likelihood
function
$S=-\left(\ln\mathcal{L}_{\rm{data}}-\ln\mathcal{L}_{\rm{bkg}}\right)$, where
$\mathcal{L}_{\rm{data}}$ is the likelihood value from the data sample, and
$\mathcal{L}_{\rm{bkg}}$ represents the background contributions including
$\Sigma^{+}\to p\pi^{0}$ and other background components. The former
background contribution is estimated with a MC simulated event sample that is
five times the size of data and is generated according to its decay amplitude.
The latter is estimated with data events in the side-band region defined as
$0.11\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}$
$<P_{p}<$
$0.16\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}$ and
$0.24\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}$
$<P_{p}<$$0.29\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}$.
$\mathcal{L}_{\rm{bkg}}$ is estimated with the above samples and normalized to
data. The fits are performed for the $\Sigma^{+}\to p\gamma$ and
$\bar{\Sigma}^{-}\to\bar{p}\gamma$ decays individually. The $\alpha_{\gamma}$
results are consistent within their uncertainties and summarized in Table 1.
Further, a simultaneous fit, assuming the same magnitude but opposite sign for
the decay asymmetry parameters of the charge conjugate channels, is also
performed, and the result is also shown in Table 1.

Table 1: The values of $N_{\rm{ST}}^{\rm{obs}}$, $\varepsilon_{\rm{ST}}$, $N_{\rm{DT}}^{\rm{obs}}$, and $\varepsilon_{\rm{DT}}$ for the decays $\Sigma^{+}\to p\gamma$ and $\bar{\Sigma}^{-}\to\bar{p}\gamma$. The BF and $\alpha_{\gamma}$ are obtained from both individual and simultaneous fits. The first uncertainties are statistical and the second systematic. Mode | $\Sigma^{+}\rightarrow p\gamma$ | $\bar{\Sigma}^{-}\rightarrow\bar{p}\gamma$
---|---|---
$N^{\mathrm{obs}}_{\mathrm{ST}}$ | $$2\,177\,771$\pm$2285$$ | $$2\,509\,380$\pm$2301$$
$\varepsilon_{\mathrm{ST}}$ ($\mathrm{\char 37\relax}$) | $$39.00$\pm$0.04$$ | $$44.31$\pm$0.04$$
$N^{\mathrm{obs}}_{\mathrm{DT}}$ | $$1189$\pm$38$$ | $$1306$\pm$39$$
$\varepsilon_{\mathrm{DT}}$ ($\mathrm{\char 37\relax}$) | $$21.16$\pm$0.03$$ | $$23.20$\pm$0.03$$
Individual BF (${10}^{-3}$) | $$1.005$\pm$0.032$$ | $$0.993$\pm$0.030$$
Simultaneous BF (${10}^{-3}$) | $0.996\pm 0.021\pm 0.018$
Individual $\alpha_{\gamma}$ | $$-0.587$\pm$0.082$$ | $$0.710$\pm$0.076$$
Simultaneous $\alpha_{\gamma}$ | $-0.651\pm 0.056\pm 0.020$

To visualize the effect of the decay asymmetry, two moments are calculated for
$m=8$ intervals in $\cos\theta_{\Sigma^{+}}$:

$\displaystyle
M_{1}(\cos\theta_{\Sigma^{+}})=\frac{m}{N}\sum^{N_{k}}_{i=1}\cos\theta_{\bar{p}}^{i}\cos\theta_{p}^{i},$
(5) $\displaystyle
M_{2}(\cos\theta_{\Sigma^{+}})=\frac{m}{N}\sum^{N_{k}}_{i=1}\sin\theta_{p}^{i}\sin\phi_{p}^{i},$

where $N$ is the total number of events and $N_{k}$ is the number of events in
the kth $\cos\theta_{\Sigma^{+}}$ interval. Fig. 3 shows the comparison of
moments between data and MC projection based on the fit. Good data-MC
consistencies are observed.

Figure 3: Moment distributions for (a)(b)
$\bar{\Sigma}^{-}\rightarrow\bar{p}\gamma$ and (c)(d) $\Sigma^{+}\to p\gamma$.
The black points with error bars are data. The cyan dashed curves are the
phase space MC simulation. The red solid curves show the MC simulation based
on the fit results. For the comparison between charge conjugate channels,
plots (c)(d) adopt the polar angle of $\bar{\Sigma}^{-}$ hyperon as the
abscissa variable.

The systematic uncertainties due to proton tracking and particle
identification ($0.4\text{\,}\mathrm{\char 37\relax}$), as well as photon
detection ($0.3\text{\,}\mathrm{\char 37\relax}$), are studied with
$J/\psi\rightarrow p\bar{p}\pi^{+}\pi^{-}$ and
$J/\psi\rightarrow\gamma\mu^{+}\mu^{-}$ control samples. The uncertainties of
$\chi^{2}$ requirements on the kinematic fits ($0.9\text{\,}\mathrm{\char
37\relax}$) and decay length requirement ($0.4\text{\,}\mathrm{\char
37\relax}$) are obtained by applying corresponding requirements on a
$J/\psi\to\Sigma^{+}\left(\rightarrow
p\pi^{0}\right)\bar{\Sigma}^{-}\left(\rightarrow\bar{p}\pi^{0}\right)$ control
sample. The uncertainties of ST ($0.4\text{\,}\mathrm{\char 37\relax}$) and DT
($1.2\text{\,}\mathrm{\char 37\relax}$) yields are estimated by changing the
fit parameters. The decay parameters used for signal MC generation are varied
by $\pm 1\sigma$ [43] to study the MC model related uncertainty
($0.6\text{\,}\mathrm{\char 37\relax}$). Details can be found in the
supplemental material [46]. By assuming all the sources to be independent, the
total systematic uncertainty in the BF measurement is
$1.8\text{\,}\mathrm{\char 37\relax}$ by summing up all above values in
quadrature.

The systematic uncertainties in the determination of the decay asymmetry
parameter $\alpha_{\gamma}$ are separated into two categories: the fit-related
uncertainties and event selection uncertainties. The fit-related uncertainties
are estimated with alternative fits by shifting the sideband region ($0.004$),
changing the signal region in the fit ($0.014$), varying the number of
background events by $\pm 1\sigma$ ($0.002$) and varying the fixed decay
parameters by $\pm 1\sigma$ [43] ($0.011$), individually.

For the event selection induced uncertainties, only the angular dependent
event selection criteria are considered, since other effects are negligible.
For each requirement, its dependence on the angular distribution, and effect
on the systematic uncertainty are detailed in the supplemental material [46].
The systematic uncertainty due to the track detection efficency ($0.001$) is
estimated by performing an efficiency correction on the PHSP MC. The decay
length requirement ($0.005$) and $\chi^{2}_{\gamma}<\chi^{2}_{\pi^{0}}$
requirement ($0.006$) uncertainties are studied with a
$J/\psi\to\Sigma^{+}\left(\rightarrow
p\pi^{0}\right)\bar{\Sigma}^{-}\left(\rightarrow\bar{p}\pi^{0}\right)$ control
sample. The total systematic uncertainty is assigned to be $0.020$ by adding
all individual uncertainties quadratically.

Based on the above results, the $CP$ asymmetry is calculated using the BFs and
decay asymmetry parameters between the charge conjugate channels:

$\displaystyle\Delta_{CP}=\frac{\mathcal{B}_{+}-\mathcal{B}_{-}}{\mathcal{B}_{+}+\mathcal{B}_{-}}=$0.006\pm
0.011_{\rm{stat.}}\pm 0.004_{\rm{syst.}}$,$ $\displaystyle
A_{CP}=\frac{\alpha_{-}+\alpha_{+}}{\alpha_{-}-\alpha_{+}}=$0.095\pm
0.087_{\rm{stat.}}\pm 0.018_{\rm{syst.}}$.$

Here, $\mathcal{B}_{+}$ ($\alpha_{+}$) denotes the BF ($\alpha_{\gamma}$) of
$\Sigma^{+}\to p\gamma$ and $\mathcal{B}_{-}$ ($\alpha_{-}$) is that of
$\bar{\Sigma}^{-}\to\bar{p}\gamma$. The systematic uncertainties of
$\Delta_{CP}$ and $A_{CP}$ only consider the uncorrelated uncertainties of
$B_{+/-}$ and $\alpha_{+/-}$. This represents the first search for $CP$
violation in $\Sigma^{+}\to p\gamma$. No obvious $CP$ violation is observed.

In summary, using $(10\,087\pm 44)\text{\times}{10}^{6}$ $J/\psi$ events
collected with the BESIII detector, the radiative hyperon decay $\Sigma^{+}\to
p\gamma$ is studied at an electron-positron collider for the first time. The
absolute BF of this decay is determined to be $($0.996\pm 0.021_{\rm stat.}\pm
0.018_{\rm syst.}$)\times 10^{-3}$, with a decay asymmetry parameter of
$-0.651\pm 0.056_{\rm stat.}\pm 0.020_{\rm syst.}$. Two independent
observables between the two charge conjugate channels, $A_{CP}$ and
$\Delta_{CP}$, are used to search for $CP$ violation, and no evidence of $CP$
violation is found.

Figure 4: Distribution of $\alpha_{\gamma}$ versus BF of the $\Sigma^{+}\to
p\gamma$ decay. The black star denotes the results measured by this work and
the orange contours correspond to the $$68\text{\,}\mathrm{\char
37\relax}$/$95\text{\,}\mathrm{\char 37\relax}$$ confidence-level of the
results. The red diamond represents the PDG values [15] of the BF and
$\alpha_{\gamma}$. Other symbols in blue stand for the results predicted by
the vector meson dominance model (VMD) [47], pole model (PM I refers to Ref.
[17] and PM II refers to Ref. [18]), nonrelativistic constituent quark model
(NRCQM) [19], broken SU(3) model (BSU(3)) [48] and ChPT [49].

The accuracies of the BF and $\alpha_{\gamma}$ are improved by
$78\text{\,}\mathrm{\char 37\relax}$ and $34\text{\,}\mathrm{\char 37\relax}$,
respectively. The comparison between the measurement results with the PDG
values and theoretical predictions are shown in Fig. 4. The measured BF is
lower than the PDG value [15] by $4.2\sigma$, where all previous measurement
results were obtained as relative ratios to $\Sigma^{+}\to p\pi^{0}$. The
decay asymmetry parameter is consistent with the world average value within
$1.1\sigma$. A similar BF difference is also observed in another radiative
hyperon decay $\Lambda\to n\gamma$ [30] at BESIII. With the updated BF and
$\alpha_{\gamma}$ results of $\Sigma^{+}\to p\gamma$, further theoretical
efforts are needed to clarify the physics of radiative hyperon decays. The SM
prediction of $\Sigma^{+}\rightarrow pl^{+}l^{-}$ decays can also be further
improved [26] to search for new physics beyond the SM.

###### Acknowledgements.

The authors thank Prof. L. S. Geng and Prof. X. G. He for helpful discussion.
The BESIII Collaboration thanks the staff of BEPCII and the IHEP computing
center and the supercomputing center of USTC for their strong support. This
work is supported in part by National Key R&D Program of China under Contracts
Nos. 2020YFA0406300, 2020YFA0406400; National Natural Science Foundation of
China (NSFC) under Contracts Nos. 11335008, 11625523, 11635010, 11705192,
11735014, 11835012, 11935015, 11935016, 11935018, 11950410506, 11961141012,
12022510, 12025502, 12035009, 12035013, 12061131003, 12105276, 12122509,
12192260, 12192261, 12192262, 12192263, 12192264, 12192265; the Chinese
Academy of Sciences (CAS) Large-Scale Scientific Facility Program; the CAS
Center for Excellence in Particle Physics (CCEPP); Joint Large-Scale
Scientific Facility Funds of the NSFC and CAS under Contract No. U1732263,
U1832103, U1832207, U2032111; CAS Key Research Program of Frontier Sciences
under Contracts Nos. QYZDJ-SSW-SLH003, QYZDJ-SSW-SLH040; 100 Talents Program
of CAS; The Institute of Nuclear and Particle Physics (INPAC) and Shanghai Key
Laboratory for Particle Physics and Cosmology; ERC under Contract No. 758462;
European Union’s Horizon 2020 research and innovation programme under Marie
Sklodowska-Curie grant agreement under Contract No. 894790; German Research
Foundation DFG under Contracts Nos. 443159800, 455635585, Collaborative
Research Center CRC 1044, FOR5327, GRK 2149; Istituto Nazionale di Fisica
Nucleare, Italy; Ministry of Development of Turkey under Contract No.
DPT2006K-120470; National Research Foundation of Korea under Contract No.
NRF-2022R1A2C1092335; National Science and Technology fund; National Science
Research and Innovation Fund (NSRF) via the Program Management Unit for Human
Resources & Institutional Development, Research and Innovation under Contract
No. B16F640076; Polish National Science Centre under Contract No.
2019/35/O/ST2/02907; Suranaree University of Technology (SUT), Thailand
Science Research and Innovation (TSRI), and National Science Research and
Innovation Fund (NSRF) under Contract No. 160355; The Royal Society, UK under
Contract No. DH160214; The Swedish Research Council; U. S. Department of
Energy under Contract No. DE-FG02-05ER41374.

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