[1]Ashley Jisue Hong

[1] Punahou School, 1601 Punahou St., Honolulu, 96813, HI, USA

2]Psychology, Tufts University, 490 Boston Ave., Medford, 02155, MA, USA

3]Data Science, Georgia Institute of Technology, 225 North Ave., Atlanta,
30332, GA, USA

# Can CNNs Accurately Classify Human Emotions? A Deep-Learning Facial
Expression Recognition Study

<EMAIL_ADDRESS>David DiStefano<EMAIL_ADDRESS>Sejal Dua
<EMAIL_ADDRESS>*  [  [

###### Abstract

Emotional Artificial Intelligences are currently one of the most anticipated
developments of AI. If successful, these AIs will be classified as one of the
most complex, intelligent nonhuman entities as they will possess sentience,
the primary factor that distinguishes living humans and mechanical machines.
For AIs to be classified as “emotional,” they should be able to empathize with
others and classify their emotions because without such abilities they cannot
normally interact with humans. This study investigates the CNN model’s ability
to recognize and classify human facial expressions (positive, neutral,
negative). The CNN model made for this study is programmed in Python and
trained with preprocessed data from the Chicago Face Database. The model is
intentionally designed with less complexity to further investigate its
ability. We hypothesized that the model will perform better than chance
(33.3%) in classifying each emotion class of input data. The model accuracy
was tested with novel images. Accuracy was summarized in a percentage report,
comparative plot, and confusion matrix. Results of this study supported the
hypothesis as the model had 75% accuracy over 10,000 images (data),
highlighting the possibility of AIs that accurately analyze human emotions and
the prospect of viable Emotional AIs.

###### keywords:

Robotics and Intelligent Machines, Machine Learning, Convolutional Neural
Network (CNN), Human Emotion

##  1 Introduction

Artificial Intelligence systems, most commonly referred to as AI systems, are
computer-based systems or machines that perform tasks that have traditionally
been thought to require human intelligence or higher order thinking. AI
systems are most commonly used to help solve human problems and better
procedures [1]. Because AI systems are developed to imitate human cognition,
scientists constantly compare the limits and abilities of an AI system to
those of a human. Currently, studies still find that humans are better than AI
systems in solving broader problems under various circumstances, especially in
social interactions [2]. However, unlike simple machines that can only perform
one task without further development, AI systems can constantly evolve and
learn by themselves to expand their limits by developing their own decision-
making process based on the given information. The subset of processes by
which an AI develops its own way of thinking is called machine learning.
According to the National Library of Medicine, machine learning or deep
learning refers to the various algorithms that generate educated predictions
based on a large dataset [3].

Artificial neural networks are a popular type of machine learning algorithm.
According to the National Library of Medicine, artificial neural networks are
developed based on the biological concept of human neurons [4]. Within each
artificial neural network are multiple nodes, which each act like a human
neuron. Each node is connected to corresponding layers that help pass on and
assess the initial message. A typical artificial neural network contains an
input layer, various nodes, a couple of hidden layers, and an output layer.
When an input is given, each node connected to the input layer uses its
individually assigned weight and threshold to assess the input’s average
numeric values, which represents how close the input is to the stored data for
classification. When the output (input’s average numeric value) of any
singular node is above the threshold value, it activates the node to pass on
its output to the next layer, and the process continues until it reaches the
final output node and layer. If the input of a node is below the threshold, it
will not continue any process and halt.

Among the various kinds of neural networks are convolutional neural networks,
also known as CNNs. Convolutional neural networks are the most popular neural
network for computer vision and image analysis [5]. While they can be used in
other applications like natural language processing as well, CNNs are mostly
used for classification of images. Although looking at an object and being
able to classify it is a basic human trait in vision, computers cannot
differentiate simple objects before they go through complex deep learning like
CNNs. With multiple hidden (convolutional) layers, pooling layers, and fully
connected layers, CNNs allow AI to assess each image data with more accuracy
and store more data. The reason why CNNs are most often used for image
analysis is because traditional neural networks, also called multilayer
perceptrons (MLP), are quite inefficient in their way of processing images.
When MLP tries to process large images, they run out of storage space and lose
crucial information about the input images. While MLPs struggle just to store
an image in its network to process, CNNs quickly find patterns in an image and
assess the classification based on the features it detects by itself through
their convolutional layers that each assess a different feature of the image
and altogether compile the closeness of an image to an existing class in the
neural network’s realm of information. While the input data for MLPs are
vectors, CNNs accept matrices as input and compile them into tensors,
containers of numeric data, expanding the number of hidden layers and
increasing the range of patterns it can detect. By having multiple layers, the
CNN will have a larger compile of hidden layers (larger tensor), which means
that it is “deeper” in terms of tensor dimensions. Because CNNs have a more
complex structure with more advanced mathematical components, they are able to
process images with higher accuracy.

Emotional AI systems are machines that can recognize and classify human
emotions. While current AI systems have already exhibited the potential to
surpass the limits of human intelligence, they are still not the equivalent to
humans because AIs lack the ability to feel and understand human emotions. If
successful, emotional AIs will be regarded the most similar to human minds and
be considered the most intelligent form of AI. As AIs become more and more
sophisticated and intelligent compared to real human beings, a key question
that determines the relationship between human beings and AI systems emerges:
can AIs ever be considered sentient? In 2022, an engineer working for Google,
Blake Lemoine, was placed on paid administrative leave after arguing that
Google’s AI chatbot LaMDA is sentient, showing signs of being “alive.” One of
the key examples Lemoine used to support his argument was how LaMDA said it
feels its own emotions, such as “pleasure, joy, love, sadness, depression,
contentment, anger, and many others.” From this it can be inferred that LaMDA
could be considered sentient if it is validated as a functioning emotional AI.

In order for Lemoine’s theory to be accepted as true, LaMDA not only has to
show the ability to generate its own emotions but also prove that it has
emotional intelligence, commonly described as empathy or the ability to
understand others’ emotions. According to a journal-published study by
neuroscientist Aron K. Barbey et. al., in 2012, scientists proved that
“emotional intelligence and cognitive intelligence share many neural systems
for integrating cognitive, social, and affective processes,” so “there are
interdependencies between emotional intelligence and general intelligence”
[6]. An important factor of emotional intelligence is empathy, a feeling that
unites every well human being. According to the European Journal of
Personality, psychopaths are people who have shallow emotions and lack
empathy. In relation to these facts, according to the European Journal of
Personality, psychopaths scored significantly lower on intelligence tests when
tested against normal humans without any mental illnesses [7]. Oftentimes,
psychopaths are distinguished from others because their lack of empathy and
understanding in human emotions cause them to only think for themselves,
preventing them from normally interacting with others. It is therefore crucial
for an emotional AI to have emotional intelligence since they will be
classified more similar to socially isolated psychopaths than normal humans
without it.

One possible way for an AI to gain emotional intelligence and show empathy is
to be able to assess and classify others’ emotions. To assess the possibility
of emotional AIs and learn how well AIs can recognize emotions, this study
investigates the accuracy of a CNN model in classifying human facial
expressions. This study investigates the classification of human emotions
through facial expression exhibited within the Chicago Face Database where
pictures of individuals may display a positive, neutral, or negative
expression [8]. The model CNN is programmed with Python and PyTorch. Based on
other similar research, we hypothesize that this model will perform better
than chance (33.3%) in classifying each class of given emotions based on the
input data. This study’s results will not only be helpful in determining the
possibility of developing an emotional AI but will also be useful in
developing technology that simply requires AIs to read facial expressions.
Example uses of this study’s results are managing workplace and employee
satisfaction, student stress monitoring, remote therapy and guidance, etc [9].

Over the last few years, studies on AIs and machine learning models that
recognize facial expressions have increased. Though there are some previous
studies on the same subject, this study differs with them for various reasons.
Unlike how this study aims to investigate the possibility of an Emotional AI
and the accessibility of CNN-based Facial Expression Recognition Systems
(FER), the study published in 2017 by V. Tümen aimed to diversify computer
vision applications through building a CNN based FER [10]. Tümen’s model was
not trained with CFD but was trained with the popular FER2013 database, and
had a 57.1% success rate [11]. The study published in 2019 by I. Larsi took
this subject at another angle and elaborated how CNNs can be used to analyze
students’ emotions, but like Tümen’s study they too used the FER2013 dataset
[11]. Some other differences between this study and Larsi’s was that they kept
the RGB color factor of the pictures for their model to process, had a smaller
pixel size for the input compared to this study, had more emotion classes than
the three this study has, and thus had a different model architecture [11].
One major difference between the studies of Tümen and Larsi was their success
rate. While they used the same dataset, their results varied by a lot in
percentage, with Tümen’s model producing a 57.1% accuracy while Larsi’s
producing a 70.1% accuracy [10][11]. To clarify and investigate if CNN-based
FERs are actually viable, this study uses CFD’s Expression Morph dataset
besides the popular FER2013 dataset commonly used for CNN-based FERs. To
mitigate racial bias during training, this study uses grayscale to normalize
all images (data), and uses a simpler model architecture to match the smaller
size of the dataset.

##  2 Methods

###  2.1 Dataset

The dataset used in this study was provided by the Chicago Face Database
developed by Debbie S. Ma, Joshua Correll, and Bernd Wittenbrink of the
University of Chicago [8]. The specific dataset used in study is the
“Expression Morphs” set based on the original CFD dataset developed by
Vikhanova, Mareschal, and Tibber Queen Mary University of London [12]. The
data contained in the datasets were 320 headshots of multiple human
individuals with varying facial expressions per individual. The individuals in
the data varied in sex (male and female), race (Black and White), and age
(unspecified), but all headshots were taken with the same white background and
all individuals wore the same gray shirt.

###  2.2 Data Processing

Data was preprocessed before it was used for training the CNN model. Figure 1
shows the procedure of data processing with examples of the data. Firstly, all
headshots were manually labeled as displaying “positive,” “negative,” or
“neutral” expressions before they were used to train the model. All data
manually sorted from the CFD datasets with labeled emotions were uploaded to
the Google Drive folder connected to the model and each emotion label was set
as classes. Then the data was processed to all have the same pixel size
because the layers of the model require a fixed input size to produce
comparable outputs. During resizing the headshots were also centered in order
to focus on the facial features that display human expression. Finally, the
data was all processed in grayscale to mitigate possible racial bias and
reduce another factor for the model to practice so that the model runs with
better efficiency.

Figure 1: Image processing procedure including resizing and scaling.

###  2.3 Experimental Design

The model CNN is designed with various architectures for the best performance.
Figure 2 shows the model architecture and each layer’s dimensions. The model
is built with four convolutional layers and three fully connected layers to
amplify its accuracy. The images are separated into two data groups, train
data and test data. The dataset went through full randomization when it was
split into train data and test data. Because it was fully randomized, the
train and test data have a chance of having data (headshot) of the same
individual. However, since the two groups of data were completely randomized
the expression that the individual displays in train and test data is
different and there is no headshot that exists in both the train data group
and test data group. The train data is 75% (240 pictures), and the test data
is 25% (80 pictures) of the total data.

Figure 2: Model architecture diagram.

After the model was trained with the train data over 100 epochs, the model
went through two tests to assess its accuracy. First it iterated over the
entire train data again. Second, the model was tested with novel images
(a.k.a. the test data) to investigate if it is intelligent enough to assess
new data. The model’s performance was evaluated using three methods:
classification accuracy (%), confusion matrix, and graphing test accuracy over
100 epochs.

##  3 Results

When commanded to calculate the model’s accuracy, the code printed that the
accuracy of the network on the 10,000 test images is 75%. Figure 3 displays
how the model’s accuracy reaches for both the train and test data were similar
to chance (33.3%) for the first 35 epochs, but after the 35th epoch the
training accuracy reaches 100% and the validation (testing) accuracy reaches
75%.

Figure 3: Graph of the model’s accuracy (%) history over each epoch run.
Figure 4: Confusion matrix illustrating types of misclassified observations.

Figure 4 shows the model’s testing accuracy on predicting each emotion class
through a confusion matrix. On Figure 4, the vertical column represents true
labels and the horizontal row represents predicted labels. Each centered
number on the boxes of the matrix represents the percentage of data in each
true emotion class to be classified as a certain predicted emotion class by
the model. As shown in the rightmost color legend, the darker the color of the
box the higher the probability is for that box. For example, the upper
leftmost box with the true label “Negative” and predicted label “Negative”
shows that about 80% of data with the true class “Negative” is classified
“Negative” by the model. Figure 4 shows that the model had about 80% accuracy
in classifying “Negative” data, 39% accuracy in classifying “Neutral” data,
and 91% accuracy in classifying “Positive” data. In conclusion, the results
supported the hypothesis.

##  4 Discussion

The results supported the hypothesis, with the model producing an overall
accuracy rate of 75% even with a limited dataset and less layers. There are
many factors which lead up to this achievement of the model. Starting with the
positive factors, the model’s performance met the expectations, and was higher
than Tümen’s study with the conventional FER2013 dataset.10 As shown in Figure
3, the model had varying accuracy between the three emotion classes. The model
showed significantly high accuracy in classifying data with classes “Negative”
and “Positive,” each with about 80% accuracy and 91% accuracy. In contrast,
the model did not show high accuracy in classifying “Neutral” data, with about
39% accuracy that is slightly higher than chance (33.3%). A possible
explanation of this result could be that while facial features displaying
negative and positive expressions are more distinguishable and similar,
neutral expressions of individuals can easily vary depending on their original
facial features such as existing wrinkles.

One factor that might have allowed the model to perform better is the data
normalization of colors. According to a study in 2012 by Klare et al., FERs
often have a negative racial bias towards certain races, especially towards
people of color [13]. Klare’s study highlights that their models showed the
most bias towards Black females between the ages of 18 and 30 [13]. Though it
may be important for the model to differentiate race for identification AIs,
this study focused on assessing the model’s ability to recognize human
expressions and emotions, so it was decided that the RGB factor would be
removed from all data through normalization before training the model, thus
having all data in grayscale. Besides eliminating possible racial bias,
removing colors from the pictures has also allowed the model to run faster
since there was one less component for it to process. One concern regarding
racial bias is that CFD’s Expression Morph dataset did not have any data from
races besides Black and White [12]. This may cause the model to show racial
bias against races other than Black and White if it encounters other novel
data because even though all training data is normalized with grayscale,
people from other races have physically different facial features (i.e. eye
shape, mouth shape, etc.) that the model was not trained to recognize. In
order to avoid such possible racial bias, more data of other races should be
added to the training and testing data for future models.

Another factor that could have influenced the model’s performance is the small
dataset and minimal layers in its architecture. The model’s architecture was
purposefully designed minimally with only four convolutional layers and three
fully connected layers to investigate if such CNN FER models can be simple
enough to implement accessibly for various purposes. The architecture seemed
to work well with the inputted dataset, and it ran through well over the 100
epochs. This showed that the minimalistic model architecture can be applicable
for classification of other datasets in use cases with acceptable speed and
accuracy. The size of the dataset had both pros and cons. While the small
dataset with fewer classes allowed the model to run and train faster, it might
have also caused the model to overfit over the training data. Evidence of this
can be seen in Figure 2, where the epoch accuracy for the train data remains
near 100% starting from about 45 epochs run while the epoch accuracy for the
test data stabilizes around 75% starting from about 70 epochs run. Adding more
data and classes for the model or reducing the number of epochs in training
could possibly enhance the model’s accuracy by avoiding overfitting.

The last factor to consider for future references is the amount of information
the model has to process when it trains and tests through the inputted data.
It is a possible concern that CFD and its Expression Morph dataset might have
too many unnecessary components for the model to process, including hair,
clothes, ears, and other components of the picture that do not reflect a
person’s emotion through facial expression. It is also possible that those
components irrelevant to the person’s facial expressions not only slow down
the model but also create biases that tie those components with emotion
labels. An example could be that a person’s ear shape may be considered by the
model in classifying her emotions. To make the model more efficient and
accurate, it may be beneficial to process the data even more before training
so that the model only focuses on facial components (i.e. eyes, wrinkles,
eyebrows, mouth, etc.) that actually display the emotions.

##  5 Conclusion

In conclusion, the CNN model meets the expectations that provide the
possibility of emotional AIs and how well they can recognize emotions. There
are definitely some areas that can be better supported with more data within
this study, but the model has shown potential in accuracy and efficiency with
minimal architecture and an unconventional dataset, adding support to the
conclusion that CNN FERs can be pretty accurate. If more improvements are
implemented, it is possible that the model can show greater accuracy that
introduces more feasibility in creating an Emotional AI in the future.

## Acknowledgments

I thank David DiStefano for his project mentoring, Sejal Dua for her
showcasing mentoring, and the CFD Database for providing the data for this
project.

## References

  *   * [1] Tai, M. C.-T.  The impact of artificial intelligence on human society and bioethics.  _Tzu Chi Med. J._ 32, 339–343 (2020).
  * [2] Korteling, J. E. H., van de Boer-Visschedijk, G. C., Blankendaal, R. A. M., Boonekamp, R. C. & Eikelboom, A. R.  Human- versus artificial intelligence.  _Front. Artif. Intell._ 4, 622364 (2021).
  * [3] Nichols, J. A., Herbert Chan, H. W. & Baker, M. A. B.  Machine learning: applications of artificial intelligence to imaging and diagnosis.  _Biophys. Rev._ 11, 111–118 (2019).
  * [4] Han, S.-H., Kim, K. W., Kim, S. & Youn, Y. C.  Artificial neural network: Understanding the basic concepts without mathematics.  _Dement. Neurocognitive Disord._ 17, 83–89 (2018).
  * [5] Yamashita, R., Nishio, M., Do, R. K. G. & Togashi, K.  Convolutional neural networks: an overview and application in radiology.  _Insights Imaging_ 9, 611–629 (2018).
  * [6] Barbey, A. K., Colom, R. & Grafman, J.  Distributed neural system for emotional intelligence revealed by lesion mapping.  _Soc. Cogn. Affect. Neurosci._ 9, 265–272 (2014).
  * [7] Sánchez de Ribera, O., Kavish, N., Katz, I. M. & Boutwell, B. B.  Untangling intelligence, psychopathy, antisocial personality disorder, and conduct problems: A meta–analytic review.  _Eur. J. Pers._ 33, 529–564 (2019).
  * [8] Ma, D. S., Correll, J. & Wittenbrink, B.  The chicago face database: A free stimulus set of faces and norming data.  _Behav. Res. Methods_ 47, 1122–1135 (2015).
  * [9] Mantello, P. & Ho, M.-T.  Emotional ai and the future of wellbeing in the post-pandemic workplace.  _AI & SOCIETY_ (2023).
  * [10] Tumen, V., Soylemez, O. F. & Ergen, B.  _Facial emotion recognition on a dataset using convolutional neural network_ (IEEE, 2017).
  * [11] Lasri, I., Solh, A. R. & Belkacemi, M. E.  _Facial emotion recognition of students using convolutional neural network_ (IEEE, 2019).
  * [12] Vikhanova, A., Mareschal, I. & Tibber, M.  Emotion recognition bias depends on stimulus morphing strategy.  _Atten. Percept. Psychophys._ 84, 2051–2059 (2022).
  * [13] Klare, B. F., Burge, M. J., Klontz, J. C., Vorder Bruegge, R. W. & Jain, A. K.  Face recognition performance: Role of demographic information.  _IEEE Transactions on Information Forensics and Security_ 7, 1789–1801 (2012).


# Congestion-Aware Routing in Dynamic IoT Networks: A Reinforcement Learning
Approach

Hossam Farag and Čedomir Stefanović
Department of Electronic Systems, Aalborg University, Denmark
<EMAIL_ADDRESS>
###### Abstract

The innovative services empowered by the Internet of Things (IoT) require a
seamless and reliable wireless infrastructure that enables communications
within heterogeneous and dynamic low-power and lossy networks (LLNs). The
Routing Protocol for LLNs (RPL) was designed to meet the communication
requirements of a wide range of IoT application domains. However, a load
balancing problem exists in RPL under heavy traffic-load scenarios, degrading
the network performance in terms of delay and packet delivery. In this paper,
we tackle the problem of load-balancing in RPL networks using a reinforcement-
learning framework. The proposed method adopts Q-learning at each node to
learn an optimal parent selection policy based on the dynamic network
conditions. Each node maintains the routing information of its neighbours as
Q-values that represent a composite routing cost as a function of the
congestion level, the link-quality and the hop-distance. The Q-values are
updated continuously exploiting the existing RPL signalling mechanism. The
performance of the proposed approach is evaluated through extensive
simulations and compared with the existing work to demonstrate its
effectiveness. The results show that the proposed method substantially
improves network performance in terms of packet delivery and average delay
with a marginal increase in the signalling frequency.

##  I Introduction

Internet of Things (IoT) represents a pervasive ecosystem of innovative
services within different application areas [1]. Low-power and Lossy Networks
(LLNs) constitute the wireless infrastructure for a variety of IoT
applications. The LLN is a set of interconnected resource-constrained devices,
featuring limitations on power, processing and storage capabilities [2].

The IPv6 Routing Protocol for LLNs (RPL) [3] was standardized as the de-facto
routing protocol to meet the requirements of LLN applications. RPL organizes
an IoT network as a Destination-Oriented Acyclic Graph (DODAG) rooted at the
sink node, namely the DODAG root. The DODAG root represents the final
destination for the traffic within the network level, bridging the topology
with other IPv6 domains such as the Internet. Uplink routes are constructed
and maintained using the periodic DODAG Information Object (DIO) messages [3].
RPL is a single path routing protocol where each node transmits its traffic
directly to a preferred parent, which is exclusively selected based on the
adopted Objective Function (OF) [4]. Although RPL is originally designed for
low traffic scenarios, performance issues arise under heavy traffic loads due
to congestion, such as load imbalance, high packet loss and fast energy
depletion of bottleneck nodes.

Most of the proposed approaches to improve RPL against congestion are based on
fixed strategies that incur expensive overhead [5]; as a result, the network
does not work properly for dynamic IoT environments. In this respect, using
machine learning frameworks could enable self-organizing operation of LLN
devices in dynamic IoT environments [6]. Reinforcement learning (RL) is a
machine learning technique capable of training an agent (IoT device) to
interact intelligently and autonomously with an environment [7]. Ultimately,
adopting learning-based protocols to improve network performance can provide
support for advanced IoT applications.

In this paper, we introduce an RL-based routing strategy to tackle the
congestion problem in RPL networks. The proposed method is based on Q-learning
where each node maintains a Q-table that represents the routing information of
all its neighbours. The Q-values at each node are updated according to the
received feedback function which is formulated as a composite routing metric
that reflects the congestion level and link quality of each neighbour. Based
on the received feedback, the node selects the neighbour with the minimum
Q-value as its preferred parent. In order to comply with the standard RPL and
keep the overhead to a minimum, the congestion information is embedded in the
periodic DIO control messages. The period of DIO message exchanges is governed
by the RPL Trickle timer [8], whose reset strategy is modified to achieve a
balance between overhead and fast dissemination of the feedback function. To
the best of our knowledge, this is the first work that adopts machine learning
to tackle the congestion problem in RPL-based networks. Performance
evaluations and comparative analysis with respect to solutions available in
the literature are carried out, showing that the proposed method achieves load
balancing with improved performance in terms of packet delivery and average
delay with a marginal increase in the signalling overhead.

The remainder of the text is organized as follows. Section II presents the
background and related work. Section III introduces the load-balancing problem
in RPL and the proposed Q-learning method. The performance evaluation is given
in Section IV, followed by the conclusions in Section V.

##  II Background and Related Work

Figure 1: RPL network model.

In RPL standard, the OF is used to describe the set of rules and policies that
governs the process of route selection and optimization in a way that meets
the different requirements of various applications. RPL decouples the route
selection and optimization mechanisms from the core protocol specifications to
enable its users to define routing strategies and local policies to fulfill
the conflicting requirements of different LLN applications [3]. Currently, two
OFs have been standardized for RPL, namely, the Objective Function Zero (OF0)
[9] and the Minimum Rank with Hysteresis Objective Function (MRHOF) [10]. The
two OFs rely solely on a single metric as the routing decision metric, hop-
count in OF0 and Expected Transmission Count (ETX) in MRHOF. In heavy-traffic
scenarios, LLN nodes are triggered to forward high amount of traffic which may
cause congestion at particular parent nodes when the offered load exceeds the
available queuing capacity [11]. Consequently, the network suffers from
consecutive packet delivery failures due to output queue overflows, also
denoted as node congestion (henceforth, also simply referred to as
congestion). Node congestion occurs even in the case of low-rate traffic
because nodes near to the DODAG root have to relay high amount of traffic.
Therefore, node congestion and imbalanced routing tree topology have a
profound impact on the network performance in terms of packet loss, delay, and
energy consumption. In RPL specifications [3], there is no explicit mechanism
to detect and react to node congestion. Instead, all traffic will be forwarded
through the selected preferred parent as long as it is reachable, without any
attempt to perform load balancing, which ultimately leads to poor quality of
service (QoS). Therefore, incorporating load-balancing mechanism in RPL
routing is essential to maintain efficient network performance in terms of
delay and reliability.

Several rule-based approaches have been proposed in the literature to tackle
the congestion problem in RPL networks. The authors in [12] developed an OF
using fuzzy logic to improve packet delivery and energy consumption. The
method utilizes the hop-count, ETX and residual energy to select the best
parent, however, it does not consider the congestion level in the selection
process. In [13], the authors proposed a non-cooperative game theory-based
solution to mitigate node congestion in RPL, which attempts to find the
optimal sending rate of each node based on the information of buffer loss and
channel loss. Besides the increased overhead, adjusting the sending rate may
violate the potential requirement of IoT applications where a fixed refresh
rate is required. A context-aware OF was proposed in [14] to mitigate node
congestion and improve network lifetime under heavy traffic. The method
considers the residual energy and queue utilization when selecting the
preferred parent, however, it suffers from increased overhead when
distributing the routing information among neighbour nodes. A load-balancing
parent selection was proposed in [15], where all feasible parents compete for
packet forwarding when a node transmits a packet. The method incurs a
significant energy waste when all the feasible parents wake up their radio,
and also does not guarantee that the winner is the one with best link quality.

A number of machine learning-based protocols was introduced to improve routing
in next-generation networks [16, 17, 18], however, these works are not
compliant with the RPL specification. The authors in [19] introduced iCPLA, a
cross-layer optimization method using reinforcement learning to improve RPL
performance under heterogeneous traffic. The proposed method utilizes the
collision probability to select the optimal parent. However, the method
considers only the channel congestion due to collisions and neglects node
congestion which has a crucial impact on the network performance, as shown in
the next section.

Figure 2: Packet delivery performance under varying traffic load.

##  III The Proposed Method

First, we emphasize the load balancing problem in the standard RPL. Then, we
introduce the proposed Q-learning method to mitigate node congestion and
achieve load balancing.

###  III-A Load-Balancing Problem in RPL

We consider a typical LLN model in an IoT application, e.g., industrial
automation as depicted in Fig. 1. In this model, a set of sensor nodes are
distributed to form the LLN that is rooted at the DODAG root (border router).
The DODAG root connects the LLN to the public Internet or a private IP-based
network. The nodes utilize IEEE 802.15.4 links to communicate with each other,
and use RPL to construct routes towards the DODAG root. We first investigate
the packet delivery performance towards the DODAG root of an LLN consisting of
30 nodes via MATLAB simulations. The LLN structure shown in Fig. 1 depicts a
characteristic snapshot of the RPL topology using MRHOF at the end of a
simulation. Fig. 2 shows the percentage of packets lost at the link layer and
also at the output queue under different traffic rates with each node having a
buffer size of 10 packets; note that the assumed traffic-load range covers
values that can be expected in practice. The results shown in Fig. 2
represents the average over all nodes in the network. The figure demonstrates
that, as the traffic rate increases, the losses due to the node congestion
tend to increasingly dominate over the channel losses, quickly becoming the
main reason of packet delivery degradation.

Figure 3: Queue losses at each node at 90 ppm/node rate.

Fig. 3 shows the Queue Loss Ratio (QLR) of each node according to its hop-
distance from the DODAG root. The QLR is calculated as the ratio between the
packets dropped due to buffer overflow and the total transmitted packets.
Initially, it might be intuitive that nodes that are one-hop distance from the
DODAG root experience the highest queue loss levels. However, the results
presented in Fig. 3 (corresponding to the RPL topology in Fig. 1) reveal that,
under heavy load, there is an imbalance among queue losses among nodes at the
same hop-distance from the root. For instance, node $A$ has the highest QLR
among five nodes that are 2 hops from the root. Similarly, node $E$ has the
highest QLR among four nodes that are a single hop from the root. This is due
to the inefficient parent selection mechanism in RPL, which leads to
imbalanced sub-tree sizes among nodes at the same hop-distance. Since LLN
nodes have small queue sizes, typically 4 to 10 packets, their queues start to
overflow before the congestion level is heavy enough to be detected through
the ETX parameter. Therefore, children nodes are not aware of the backlog
status of their parents, continuing to transmit packets even if the parents
suffer from consecutive queue losses. In summary, there is a need for an
efficient load-balancing strategy in RPL.

###  III-B The Proposed Q-learning for Load-balancing in RPL

Q-learning is a model-free RL technique, where the agent learns to select the
best action according to the maintained Q-table [20]. In our approach, the
Q-table corresponds to the routing table at each node (agent) and the action
represents the selection of the preferred parent. The learning goal is to find
the optimal parent selection policy according to the network dynamics, e.g
congestion level and link quality.

Each node $x$ maintains a candidate neighbour set $N(x)$ which includes all
nodes that are within the communication range of $x$. Denote by $Q_{x}(y)$ a
Q-value that is the preference of node $x$ to select node $y$ as its parent,
where $y\in N(x)$. The Q-values are periodically updated in the following way

$Q_{x}^{\text{new}}(y)=Q_{x}^{\text{old}}(y)+\alpha\left[R(y)-Q_{x}^{\text{old}}(y)\right]$
(1)

where $Q_{x}^{\text{new}}$ and $Q_{x}^{\text{old}}(y)$ are the Q-values for
the current and the previous intervals, respectively; the interval duration is
determined by the Trickle timer, as will be elaborated later. $\alpha$ is the
learning rate, and $R(y)$ is the estimate of the feedback function, i.e., the
reinforcement signal. Note that the Q-values are initialized to zeros for all
nodes upon network deployment. The value $Q_{x}(y)$ represents the routing
metric of the path between $x$ and $y$. The main idea of the proposed method
is to enable the nodes to learn the congestion levels at their neighbours
through the feedback $R(y)$, and utilize this information to select the best
parent in order to achieve load-balancing. We represent the congestion level
at node $y$ through the Backlog Factor $\mathrm{BF}(y)$, which denotes the
ratio between the current queue length and the total queue size. An
exponentially weighted moving average filter is applied for calculation of
$\mathrm{BF}(y)$. The congestion status $\mathrm{BF}(y)$ is distributed to all
nodes in the set $N(y)$, as illustrated later. To obtain the optimal network
performance, in heavy traffic mode, the node learns to select the parent that
is less congested, while in light traffic mode, i.e., no congestion, the node
learns to select the parent with the best link quality and the shortest hop-
distance. Thus, we formulate $R(y)$ as a composite routing metric as follows

$R(y)=\lambda(y)\mathrm{BF}(y)+\mathrm{ETX}(x,y)+H(y)$ (2)

where $H(y)$ denotes the hop-count of node $y$ towards the DODAG root, and
$\mathrm{ETX}(x,y)$ is the ETX measurement between $x$ and $y$.111More complex
formulations for $R(y)$ can be devised. However, the simple metric in (2)
fosters an excellent performance, as shown in Section IV. In RPL standard,
$\mathrm{ETX}(x,y)$ is exchanged periodically, and is calculated as [21]

$\mathrm{ETX}(x,y)=\frac{\\#\,\textrm{of total transmissions
from\,}x\,\textrm{to}\,\,y}{\\#\,\textrm{of successful transmissions
from}\,x\,\textrm{to}\,\,y}.$ (3)

The parameter $\lambda(y)$ is used to control the weight of $\mathrm{BF}(y)$
in (2), such that it reflects congestion level of the node $y$. Since it holds
that $0\leq\mathrm{BF}(y)\leq 1$, we define $\lambda(y)$ as

$\lambda(y)=\max\left(\frac{\mathrm{BF}(y)}{\mathrm{BF}_{\text{th}}},1-\frac{\mathrm{BF}(y)}{\mathrm{BF}_{\text{th}}}\right)$
(4)

where $\mathrm{BF}_{\text{th}}$ is a design parameter that represents the
threshold after which the parent node is assumed to be congested. That way, in
congestion situations, i.e., when $\mathrm{BF}(y)>\mathrm{BF}_{\text{th}}$,
$\mathrm{BF}(y)$ will have a noticeable effect in (2) compared to
$\mathrm{ETX}(x,y)$ and $H(y)$. Conversely, in light traffic scenarios, when
$\mathrm{BF}(y)<\mathrm{BF}_{\text{th}}$, the feedback function is influenced
more by $\mathrm{ETX}(x,y)$ and $H(y)$ compared to $\mathrm{BF}(y)$. Based on
the received feedback $R(y)$, the node updates the value of $Q_{x}(y)$
according to (1).

Figure 4: Illustration of the thundering herd problem.

A straightforward approach would be to use the greedy policy and select the
neighboring node with the minimum Q-value as its preferred parent. However,
the approach in which all nodes utilize only the exploitation phase that
follows the greedy policy may lead to the thundering herd problem [22],
illustrated in Fig. 4. The four leaf nodes in Fig. 4 may select node $A$ as
their preferred parent as it has the minimum Q-value, which may incur
congestion. Upon detecting congestion and selecting a new preferred parent,
the four nodes may change simultaneously to the same parent node $B$ causing
congestion again; such cycle may be repeated indefinitely without achieving
load balancing. To overcome this problem, the nodes have to explore other
alternatives using a certain probability. Specifically, we refer to $P_{x}(y)$
as the probability of node $x$ selecting node $y$ as the preferred parent,
given by

$P_{x}(y)=1-\frac{e^{Q_{x}(y)/\theta}}{\sum_{k\in N(y)}e^{Q_{x}(k)/\theta}}$
(5)

where $\theta$ is a parameter that determines the amount of exploration. The
probabilistic parent selection in (5) represents a modified version of the
softmax action selection strategy. In other words, it implies that a neighbour
with low Q-value will be most likely to be selected as a preferred parent,
while other neighbours with higher Q-values will be selected with smaller
probabilities. Accordingly, the nodes will obtain an efficient experience and
avoid thundering herd problem as well.

The congestion level $\mathrm{BF}(y)$ needs to be distributed to the set
$N(y)$ in order to make the right action according to the network status. In
RPL standard, DIO messages are periodically exchanged between neighbour nodes
and carry the RANK of the transmitted node, i.e, $H(y)$. In our proposed
approach, we implicitly embed $\mathrm{BF}(y)$ into the RANK value in the
transmitted DIO message as follows

$\mathrm{RANK}^{\text{new}}(y)=\eta\left(H(y)+1\right)+(\eta-1)\mathrm{BF}(y),$
(6)

where $\eta$ is a positive integer that enables decoding two values
($\mathrm{BF}(y)$ and $H(y)$) from a single numeric field
($\mathrm{RANK}^{\text{new}}$). The value of $\eta$ should be selected in a
way to keep $\mathrm{RANK}^{\text{new}}(y)$ within its 16 bit boundary [3].
Specifically, a neighbor node that receives the DIO message from $y$ extracts
the two values $\mathrm{BF}(y)$ and $H(y)$ from
$\mathrm{RANK}^{\text{new}}(y)$ as

$\begin{split}\mathrm{BF}(y)&=\frac{\text{mod}\left(\mathrm{RANK}^{\text{new}}(y),\eta\right)}{\eta-1}\\\
H(y)&=\left\lfloor\frac{\mathrm{RANK}^{\text{new}}(y)}{\eta}\right\rfloor-1\end{split}$
(7)

where $\text{mod}()$ is the modulo operation. Therefore, in the proposed
method, the congestion information are distributed among neighbour nodes
without changing the DIO message format nor adding any additional control
message overhead.

Figure 5: The proposed Trickle timer reset strategy.

The transmission interval of the DIO messages is controlled by the Trickle
timer algorithm [8]. According to the standard, the Trickle timer is reset to
its minimum interval $I_{\min}$ only when changes occur in the network
topology. As long as the network is consistent, the DIO message interval is
doubled up to a certain maximum value. With such strategy, the nodes may have
inaccurate and outdated congestion information from neighbor nodes. On the
other hand, frequent resetting of the Trickle timer may increase the routing
overhead. To tackle this conflict, we introduce a modified reset strategy to
the Trickle timer. The proposed strategy is shown in Fig. 5 and is described
as follows. A node resets its Trickle timer to $I_{\min}$ when it experiences
a certain number $\phi$ of consecutive queue losses. Specifically, the small
queues of LLN nodes my fill up temporarily even when there is no congestion,
that often results in false positive for congestion if it is declared too
early, which in turn incurs unnecessary DIO overhead. For this reason, we use
consecutive queue losses to reset the Trickle timer. After resetting the timer
to $I_{\min}$, the value of $\phi$ is increased by a fixed value $\phi_{0}$ to
limit the reset frequency and the DIO overhead. If no queue losses are
detected within a timer $X$, the algorithm values are reinitialized as shown
in Fig. 5.

##  IV Evaluation

TABLE I: Evaluation parameters. Parameter | Value
---|---
Network size |  $30$ nodes
Packet length | $100\text{\,}\mathrm{B}\text{/}$
Traffic load per node |  $30$, $60$, $90$, $120$ ppm
Propagation model | Shadowing (Log-normal)
Standard deviation | $14\text{\,}\mathrm{dB}\text{/}$
PHY and MAC protocol | IEEE 802.15.4 with CSMA/CA
Slotframe length |  $500$ slots
Time slot duration | $10\text{\,}\mathrm{ms}\text{/}$
No. of retransmissions | $3$
$\alpha$ | $0.3$
$\mathrm{BF}_{\text{th}}$ | $0.5$
$\eta$ | $100$
$\phi_{0}$ | $2$
Timer $X$ | $100\text{\,}\mathrm{ms}\text{/}$
$I_{min}$ | $3\text{\,}\mathrm{s}\text{/}$

We consider a typical IoT network in a monitoring scenario, where a set of 30
randomly placed nodes are reporting to a single DODAG root. The nodes
communicate using IEEE 802.15.4 PHY layer and CSMA/CA for channel access. Each
node generates a 100-byte packet following a Poisson arrival model and
forwards it to its preferred parent; the traffic arrival intensity per node
(the traffic load) is a parameter whose values are given in Table I. We
consider a log-normal shadowing propagation model with the standard deviation
specified in Table I [23]. The table also lists the values of other relevant
parameters.222The values of the parameters $\mathrm{BF_{th}}$, $\phi_{0}$ and
$X$ were found via optimization and are independent of the traffic load. Our
investigations also showed that these values are robust to the change in the
number of network nodes. To verify the effectiveness of the proposed method,
we compare its performance with iCPLA [19] and RPL using MRHOF (RPL-MRHOF)
[10]. The evaluation was performed using Monte Carlo simulations in MATLAB;
the presented results are the averages of 10 simulation runs for each value of
the traffic load, where each run lasted for a duration of 1000 consecutive
slotframes.

Fig. 6 shows a sample of the routing topology of both RPL-MRHOF and the
proposed method. Obviously, the proposed method achieves load balancing
between intermediate nodes, as the nodes learn to avoid congested nodes when
selecting their parents. Quantitatively, the method reduces the standard
deviation of the number of children per node from 1.3 to 0.75.

Figure 6: Routing topology of: (a) RPL-MRHOF; (b) Proposed method.

Fig. 7 presents a comparison of the average QLR with varying traffic load:
While iCPLA fares better than RPL-MRHOF, the proposed Q-learning-based method
achieves the lowest QLR due to a more balanced load distribution For instance,
the proposed method reduces the QLR of RPL-MRHOF by 71% at 120 ppm/node, while
iCPLA achieves only 5% reduction at the same traffic load. Specifically, the
parent selection strategy in iCPLA considers only the link-level information,
i.e, collision probability, which is insufficient to solve node congestion
under heavy load.

Figure 7: QLR comparison with varying traffic load.  Figure 8: Effect of
buffer size on QLR.

As already mentioned, LLN devices have small queue sizes that start to
overflow quickly as the traffic load increases. In this respect, a solution
could be to increase the Buffer Size (BS) to avoid node congestion. However,
as shown by Fig. 8, increasing the BS has a marginal effect to mitigate queue
losses, especially under heavy load scenarios. Specifically, increasing the BS
of RPL-MRHOF from 20 to 40 packets reduces the QLR by 12% and 3% at traffic
loads of 90 ppm/node and 120 ppm/node, respectively. The figure also shows
that, in order for RPL-MRHOF to match QLR performance of the proposed method
when the traffic load is 90 ppm/node, the BS should be 90 packets, which is
infeasible in practice for the resource-constrained LLN nodes. In summary,
achieving a balanced routing topology has a more significant impact on node
congestion mitigation than increasing the BS.

Figure 9: PDR comparison with varying traffic load.

As noted in Section III-A, queue losses are the main reason for the packet
delivery degradation in the network. Therefore, the reduction in QLR is
directly translated to improved packet delivery performance, as shown in Fig.
9. The figure depicts the Packet Delivery Ratio (PDR), which is the number of
packets successfully delivered to the DODAG root divided by the total
generated packets. At 120 ppm/node, the proposed method enhances the PDR
performance of RPL-MRHOF by 160% compared to a 60% improvement achieved by
iCPLA.

Figure 10: Average delay comparison with varying traffic load.

Fig. 10 compares the average delay of successfully received packets by the
DODAG root under varying traffic load. As the traffic load increases, the
average delay in all three methods increases as more packets are backlogged
and a packet has to spend more time in the output queue before transmission.
However, the proposed method guides the nodes to a fair queue utilization
strategy that helps to reduce the average backlogged packets per node.
Accordingly, the queuing time is decreased which reduces the average delay in
turn. For instance, at 90 ppm/node, the proposed method reduces the average
delay of RPL-MRHOF and iCPLA by 27% and 12%, respectively. In other words, the
proposed method outperforms the competitors with a significantly higher packet
delivery ratio, and a lower delay of the delivered packets.

Fig. 11 depicts the average number of transmitted DIO messages of each node
under different traffic loads for the proposed method and RPL-MRHOF. As the
figure demonstrates, the proposed method incurs higher DIO overhead compared
to RPL-MRHOF, especially under heavy load. This is because our proposed method
resets the Trickle timer more frequently to fast distribute the feedback
function when a node suffers from congestion. However, under heavy load
scenarios, the increased overhead is still insignificant compared to the total
traffic in the network. For instance, at 90 ppm/node, the DIO overhead
represents less than 0.5% of the total traffic. Moreover, the increased DIO
overhead could be a reasonable cost to the obtained performance improvements
in the PDR and the average delay.

##  V Conclusion

In this paper, we have proposed a RL-based routing approach to mitigate node
congestion and improve routing performance of RPL networks. Each node adopts
Q-learning to select it preferred parent based on a composite feedback
function. The congestion level at each node is embedded in the RANK metric and
distributed using a modified Trickle timer strategy. Performance evaluations
have been carried out and proved the effectiveness of our proposed method to
achieve load balancing and improve the network performance in terms of packet
delivery and average delay.

Figure 11: Average DIO overhead with varying traffic load.

## References

  * [1] H. Tran-Dang, N. Krommenacker, P. Charpentier and D. Ki, “Toward the Internet of Things for Physical Internet: Perspectives and Challenges,” IEEE Internet of Things J., vol. 7, no. 6, pp. 4711-4736, Jun. 2020.
  * [2] B. Ghaleb, et al., “A survey of limitations and enhancements of the ipv6 routing protocol for low-power and lossy networks: A focus on core operations,” IEEE Commun. Surv. Tut., vol. 21, no. 2, pp. 1607-1635, Secondquarter 2019.
  * [3] T. Winter et al., RPL: IPv6 routing protocol for low-power and lossy networks, IETF RFC 6550, Mar. 2012.
  * [4] B. Ghaleb, et al., “A survey of limitations and enhancements of the ipv6 routing protocol for low-power and lossy networks: A focus on core operations,” IEEE Commun. Surv. Tut., vol. 21, no. 2, pp. 1607-1635, Second Quarter 2019.
  * [5] J. V. V. Sobral, et al., “Routing protocols for low power and lossy networks in internet of things applications,” Sensors, vol. 19, no. 9, pp. 1-40, May 2019.
  * [6] S. K. Sharma and X. Wang, “Toward massive machine type communications in ultra-dense cellular iot networks: Current issues and machine learning-assisted solutions,” IEEE Commun. Surv. Tut., vol. 22, no. 1, pp. 426-471, Firstquarter 2020.
  * [7] D. Praveen Kumar, T. Amgoth and C. S. R. Annavarapu, “Machine learning algorithms for wireless sensor networks: A survey,” Information Fusion, vol. 49, pp. 1-25, Sept. 2019.
  * [8] P. Levis et al., The Trickle Algorithm, [Online]. Available: https://tools.ietf.org/html/rfc6206.
  * [9] P. Thubert, Objective function zero for the routing protocol for low-power and lossy networks (rpl), IETF RFC 6552, Mar. 2012.
  * [10] O. Gnawali and P. Levis, The minimum rank with hysteresis objective function, IETF RFC 6719, Sept. 2012.
  * [11] H. -S. Kim, H. Kim, J. Paek and S. Bahk, “Load balancing under heavy traffic in rpl routing protocol for low power and lossy networks,” IEEE Trans. Mobile Comput., vol. 16, no. 4, pp. 964-979, Apr. 2017.
  * [12] P. Fabian, A. Rachedi, C. Gueguen and S. Lohier, “Fuzzy-based objective function for routing protocol in the internet of things,” in Proc. IEEE GLOBECOM 2018, Dec. 2018, pp. 1-6.
  * [13] Chowdhury, A. Benslimane and C. Giri, “Non-cooperative gaming for energy-efficient congestion control in 6Lowpan,” IEEE Internet of Things J., vol. 7, no. 6, pp. 4777-4788, Jun. 2020.
  * [14] S. Taghizadeh, H. Bobarshad and H. Elbiaze, “CLRPL: Context-aware and load balancing rpl for iot networks under heavy and highly dynamic load,” IEEE Access, vol. 6, pp. 23277-23291, Apr. 2018.
  * [15] S. L. Sampayo, J. Montavont and T. Noël, “LoBaPS: Load balancing parent selection for rpl using wake-up radios,” in Proc. IEEE ISCC 2019, Jul. 2019, pp. 1-6.
  * [16] H. Yao, X. Yuan, P. Zhang, J. Wang, C. Jiang and M. Guizani, “Machine learning aided load balance routing scheme considering queue utilization,” IEEE Trans Veh. Technol., vol. 68, no. 8, pp. 7987-7999, Aug. 2019.
  * [17] B.-S. Roh, M.-H. Han, J.-H. Ham and K-I. Kim, “Q-LBR: Q-learning based load balancing routing for uav-assisted vanet,” Sensors, vol. 20, no. 19, pp. 1-18, Oct. 2020.
  * [18] H. Yao, X. Yuan, P. Zhang, J. Wang, C. Jiang and M. Guizani, “A machine learning approach of load balance routing to support next-generation wireless networks,” in Proc. IEEE IWCMC 2019, Jun. 2019, pp.1317-1322.
  * [19] A. Musaddiq,et al., “Reinforcement learning-enabled cross-layer optimization for low-power and lossy networks under heterogeneous traffic patterns, ” Sensors, vol. 20, no. 15, pp. 1-26, Jul. 2020.
  * [20] M. L. Littman, “Reinforcement learning improves behavior from evaluative feedback,” Nature, vol. 521, pp. 445-451, May. 2015.
  * [21] JP. Vasseur et al., Routing metrics used for path calculation in low-power and lossy networks, IETF RFC 6551, 2012.
  * [22] J. Hou, J. Rahul and Z. Luo, Optimization of parent-node selection in rpl-based networks, IETF RFC 6921, Mar. 2017.
  * [23] J. Miranda et al., “Path loss exponent analysis in Wireless Sensor Networks: Experimental evaluation,” in Proc. IEEE INDIN 2013, Jul. 2013, pp. 54-58.


# Locating Quark-Antiquark String Breaking in QCD through Chiral Symmetry
Restoration and Hawking-Unruh Effect

Wei Kou<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  School of Nuclear Science and Technology,
University of Chinese Academy of Sciences, Beijing 100049, China  Xurong Chen
<EMAIL_ADDRESS>(Corresponding Author) Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou 730000, China  School of Nuclear Science and
Technology, University of Chinese Academy of Sciences, Beijing 100049, China
Southern Center for Nuclear Science Theory (SCNT), Institute of Modern
Physics, Chinese Academy of Sciences, Huizhou 516000, Guangdong Province,
China

###### Abstract

The relationship between QCD and the string model offers a valuable
perspective for exploring the interaction potential between quarks. In this
study, we investigate the restoration of chiral symmetry in connection with
the Unruh effect experienced by accelerating observers. Utilizing the
Schwinger model, we analyze the critical point at which the string or
chromoelectric flux tube between quark-antiquarks breaks with increasing
separation between quarks. In this study, the critical distance for quark-
antiquark chromoelectric flux tube or string breaking is determined to be
$r_{c}=1.294\pm 0.040$ fm. The acceleration and Unruh temperature
corresponding to this critical point signify the transition of the system’s
chiral symmetry from a broken to a restored state. Our estimates for the
critical acceleration ($a_{c}=1.14\times 10^{34}$ cm/s2) and Unruh temperature
($T_{c}=0.038$ GeV) align with previous studies. This analysis illuminates the
interplay between chiral symmetry restoration, the Unruh effect, and the
breaking of the string or chromoelectric flux tube within the context of quark
interactions.

##  I Introduction

The absence of experimental observations of asymptotic quarks and gluons is
commonly attributed to the confinement property of Quantum Chromodynamics
(QCD) [1]. This phenomenon can be elucidated by the confinement mechanism,
which manifests as the existence of a potential $V(r)$ confining static quark-
antiquark pairs within a colorless system embedded in the QCD vacuum. This
energy is dependent on the distance $r$ between the color sources, exhibiting
a linear increase in asymptotic behavior as the sources are separated further
apart. The well-known string tension $\sigma$ represents the long-range linear
asymptotic growth rate of potential energy. When examining the dynamics of the
QCD vacuum in the formation and dissolution of light quark pairs, it implies
that the potential energy between the initial quark pairs reaches a critical
distance at which new light quark pairs can emerge. This phenomenon is
commonly referred to as “string breaking” in QCD phenomenology. Previous
lattice simulations [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] have
extensively explored the topic of string breaking, with recent studies
explicitly pinpointing the location of string breaking in quark-antiquark
pairs [14, 15], suggesting that string breaking typically occurs at a distance
of approximately $r_{c}\sim 1.2-1.3$ fm between the quark and antiquark.

The fluctuation of fermions or quark pairs within the QCD vacuum introduces
quark dynamics, breaking the chiral symmetry [16, 17] of the original theory.
The breaking of chiral symmetry is intricately linked to string breaking
between static quark pairs (excluding complex glueball scenarios for now).
During the occurrence of string breaking, new quark-antiquark pairs emerge
from the vacuum at the “breakpoints” of the two strings, merging with the
initial quark pair and resulting in an elevation of chiral fermion condensate.
This hypothesis has been incorporated into recent quantum simulation studies
[18, 19] utilizing the Schwinger model [20, 21].

Another concept related to symmetry breaking and restoration is the Hawking-
Unruh effect (referred to as the Unruh effect) [22, 23, 24, 25, 26, 27], which
posits that an observer accelerated in the QCD vacuum will experience a
thermal bath corresponding to the Unruh temperature $T_{U}=a/2\pi$ ($a$ is the
acceleration). The existence of the Unruh effect in interacting field theories
is demonstrated by the presence of thermal radiation, a phenomenon that has
been explored within low-energy effective field theories such as the instanton
model [28, 29] and the Nambu-Jona-Lasinio (NJL) models [16, 17, 30].
Specifically, the relationship between dynamical chiral symmetry breaking and
the Unruh effect has been investigated in the NJL model, leading to the
determination of the critical Unruh temperature $T_{c}=a_{c}/2\pi$ associated
with chiral symmetry restoration. It is worth mentioning here that in Ref.
[31], the authors propose that the phase transition from color glass
condensate to quark-gluon plasma can be observed in relativistic heavy-ion
collisions through the Hawking-Unruh thermalization mechanism.

When considering the elementary particles as thermometers to measure
spontaneous symmetry breaking, the Unruh effect introduces the mechanism of a
vacuum thermal bath induced by acceleration. Through this perspective, it is
realized that the phase transition points of spontaneous symmetry breaking and
restoration can be traced by the temperature and acceleration corresponding to
the thermal effects [32]. This concept has inspired inducing transitions
between symmetry breaking and restoration in interacting systems through
acceleration - akin to the dissolution of condensates. Such discussions serve
as a key exploration, shedding light on the intricate relationship between
quantum field theory and general relativity [33]. The related studies are all
based on the pioneering work in Ref. [27], followed by many research efforts
that offer their own perspectives on this issue [34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45].

In this study, we investigate the impact of chromoelectric flux tubes
generated by quark-antiquark pairs at different distances on chiral symmetry.
As the distance between the quark-antiquark pairs increases, the structure of
the flux tube changes, leading to varying degrees of dissolution in the chiral
condensate influenced by the chromoelectric flux tube. The complete
disappearance of the condensate signifies the full restoration of chiral
symmetry, indicating that the transition critical point between chiral
symmetry breaking and restoration should correspond to the critical distance
of the chromoelectric flux tube when the quark-antiquark pairs are at a
critical separation. At this point, the strength of the flux tube and the
assumed acceleration and Unruh temperature correspond to the critical
acceleration and temperature for chiral symmetry restoration. We propose that
this may represent the limit distance for chromoelectric flux tubes or string
breaking in QCD, corresponding to the transition point of chiral symmetry.

##  II 1+1 Schwinger model and chiral symmetry restoration

We begin by establishing a model of chromoelectric flux tubes - the 1+1
dimensional Schwinger model [20, 21]. The reason for using this model is that
it shares the property of spontaneous chiral symmetry breaking with QCD and
has been the subject of several previous studies [46, 47, 48, 19, 18, 49]. The
Lagrangian of the Abelian gauge theory in (1+1) dimensions for the case of
massless fermions is given by

$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-g\gamma^{\mu}A_{\mu})\psi,$
(1)

where $g$ is the coupling constant with the dimension of mass, $F_{\mu\nu}$ is
gauge strength, $\psi$ denotes the fermion field. This theory is exactly
solvable and processes properties such as confinement, chiral symmetry
breaking, axial anomaly, and periodic $\theta$-vacuum [21, 50, 51].
Additionally, it is used to calculate the chiral condensate depended on the
electric field [52]. The presence of a background electric field suppresses
the magnitude of the chiral condensate [46, 49], with its periodicity
depending on the $\theta$-vacuum of the theoretical model.

The Schwinger model could be known as bosonization. In the Abelian gauge case,
the correspondence is as follows [53, 54]

$\displaystyle\bar{\psi}i\gamma^{\mu}\partial_{\mu}\psi\to\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi,$
(2)
$\displaystyle\bar{\psi}\gamma^{\mu}\psi\to-\frac{1}{\sqrt{\pi}}\epsilon^{\mu\nu}\partial_{\nu}\phi,$
$\displaystyle\bar{\psi}\psi\to-c\frac{g}{\sqrt{\pi}}\cos(2\sqrt{\pi}\phi),$

where $c=e^{\gamma_{E}}/2\pi$ with the Euler constant $\gamma_{E}\simeq
0.5772$. Note that the scalar field has mass dimension $[\phi]=0$. Thus the
chiral condensate can be expressed in terms of the scalar field
$\bar{\psi}\psi=-\frac{ge^{\gamma_{E}}}{2\pi^{3/2}}\cos(2\sqrt{\pi}\phi)$.
Based on the Feynman-Hellman theorem the chiral condensate with the
chromoelectric field $E_{1+1}$ in the chiral limit $m\to 0$ can be written as
[52]

$\langle\bar{\psi}\psi\rangle_{E_{1+1}}=-\frac{ge^{\gamma_{E}}}{2\pi^{3/2}}\cos\theta,$
(3)

with $\theta=2\pi E_{1+1}/g\leq\pi/4$. It is evident that when the field
$E_{1+1}$ vanished, the chiral condensate must degenerate to the vaccum case
$\langle\bar{\psi}\psi\rangle_{0}=-\frac{ge^{\gamma_{E}}}{2\pi^{3/2}}$, which
depends on the coupling $g$. The chiral condensate exhibits strong sensitivity
to the field $E_{1+1}$. If we consider the complete disappearance of the
chiral condensate, the cosine function in Eq. (3) must be equal to 0. It is
important to emphasize that, based on physical considerations, we only
consider the first quarter of the period, during which the cosine function is
positive.

##  III 3+1 dimensional physical chromoelectric field and field strength

In this section, we need to provide some explanations regarding the selection
of the strength of the chromoelectric flux tube. This study is an extension of
previous work [49], and the model used to describe the flux tube structure is
consistent. The description of (3+1)-dimensional non-Abelian gauge fields is
complex. In order to simplify the framework, a quasi-Abelian picture based on
the dual Meissner effect has been proposed, providing support for many lattice
QCD studies [55, 56, 57, 58, 59, 60]. This is also referred to as the “maximum
Abelian projection.” These arguments are summarized and reviewed in Ref. [46],
where the relationship between confinement and vacuum magnetic monopole
condense is discussed. In a simple string model, confined quarks are connected
by Abelian chromoelectric flux tubes. These flux tubes are dual to Abrikosov-
Nielsen-Olesen (ANO) vortices [61, 62, 60] in type-II superconductors and
their dynamical behavior can be described by (1+1) dimensional effective
theory [63].

In lattice simulations, the strength of the 3+1 dimensional chromoelectric
flux tube is measured, exhibiting transverse distribution in the $x_{t}$
direction, while the longitudinal aspect can be described by $E_{1+1}$. It is
commonly assumed that the two descriptions of the flux tube tension are
consistent, leading to the following relationship [46, 49]

$\sigma_{E}\simeq\frac{1}{2}\int
d^{2}x_{t}E_{3+1}^{2}(x_{t})=\frac{1}{2}(E_{1+1})^{2}.$ (4)

Clearly, the information on the 3+1 dimensional flux tubes can be obtained
through lattice simulations, similar to the previous work [49], where we
extract flux tube data from different static color sources in Ref. [64]. The
measured chromoelectric field as a function of the transverse coordinate can
be well described by the following parameterization [65]

$E_{3+1}(x_{t})=\frac{\varphi}{2\pi}\frac{\mu^{2}}{\alpha}\frac{K_{0}[(\mu^{2}x_{t}^{2}+\alpha^{2})^{1/2}]}{K_{1}[\alpha]},$
(5)

where $\varphi$, $\alpha$ and $\mu$ are fitting parameters. Moreover,
$K_{n}(x)$ is the modified Bessel function of the second kind. The parameter
$\varphi$ represents the external flux, and $\mu$ is the inverse of the London
penetration depth $\lambda$. When the London penetration depth is much larger
than the coherence length of the magnetic monopole condensate, i.e., when the
Ginzburg-Landau parameter is much greater than 1, the chromomagnetic field is
dual to the type II superconductor (see [60, 65]). Additionally, $\alpha$
represents the ratio of the vortex core radius to the London penetration
depth, characterizing the scale of the vortex core. Eq. (5) is a simple
analytical expression derived from Ampère’s law and the Ginzburg-Landau
equation. In particular, it can be simplified to the London model outside the
vortex core in the presence of a transverse magnetic field distribution. It
reflects the compatibility of color flux tubes with dual superconductor
theories.

This study presents the distribution of 3+1 dimensional flux tubes from
lattice simulations at different color source distances [64]. The lattice data
can be well described by Eq. (5), allowing for the quick extraction of
parameter values. The specific parameters are presented in Table 1, and we
compare the data points with the parameterized form using the Set 3 parameter
group as an example, as shown in Figure. Noted that $\beta=2N_{c}/g^{2}$ is
the lattice coupling constant with color number $N_{c}$ and $r$ represents the
quark-antiquark’s separation. It is evident that the transverse spatial
distribution of the chromoelectric flux tubes resembles a Gaussian
distribution.

Table 1: The Clem parameters describing the non-perturbative filed transverse
section going through the midpoint between the quark and antiquark positions
[64].

Sets | $\beta$ |  $r$ (fm) | $\varphi$ |  $\mu$ (fm-1) | $\alpha$
---|---|---|---|---|---
1 | 6.475 | 0.37 | 3.474(4) | 4.999(9) | 1.192(4)
2 | 6.333 | 0.45 | 3.83(3) | 5.30(6) | 1.55(3)
3 | 6.240 | 0.51 | 4.028(11) | 6.039(26) | 2.141(20)
4 | 6.500 | 0.54 | 4.370(15) | 5.71(4) | 2.02(3)
5 | 6.539 | 0.69 | 4.50(7) | 6.25(20) | 2.47(16)
6 | 6.370 | 0.85 | 5.40(25) | 6.7(9) | 4.0(1.1)
7 | 6.299 | 0.94 | 5.2(4) | 7.8(1.9) | 5.5(2.8)
8 | 6.240 | 1.02 | 8.0(7) | 4.4(8) | 2.4(9)
9 | 6.218 | 1.06 | 6.6(7) | 6.0(1.8) | 4.0(2.4)

Figure 1: Comparison of lattice simulation (red circle) [64] with the
parameterization (5). The parameters are chosen as Set 3 in Table 1.

The transverse distribution of chromoelectric flux tubes is utilized to
explain the partial restoration of chiral condensate, a topic that can be
directly addressed by lattice simulations, as discussed in Refs. [66, 67, 68,
69, 70, 71, 46, 49]. On the other hand, if we do not focus on the information
of the transverse distribution, we need to integrate over the complete $x_{t}$
space to obtain a quantity akin to a field strength, denoted as $E_{1+1}$. We
reserve the discussion of this aspect for the subsequent sections of this
paper.

##  IV Chiral symmetry restoration by acceleration

We have mentioned some interesting works in the preceding sections that
utilize the NJL model [36, 37], one-loop corrections in $\lambda\phi^{4}$
theory [42], and linear or nonlinear sigma models (L$\sigma$M or NL$\sigma$M)
[72, 73, 74] to demonstrate a correspondence between the restoration of
spontaneous breaking of global symmetry and accelerated observers. It is worth
noting that all studies are conducted in Rindler spacetime. Similarly, we
introduce the quantity $R$ to describe the restoration of chiral condensate,
representing the ratio of chiral condensate dependent on the flux tube to that
in the vacuum:

$R\equiv\frac{\langle\bar{\psi}\psi\rangle_{E_{1+1}}}{\langle\bar{\psi}\psi\rangle_{0}},$
(6)

where the numerator and denominator represent the results of chiral condensate
in the $E_{1+1}$ field and the vacuum chiral condensate, respectively. It is
important to note that only the influence of the chromoelectric flux tube
intensity is considered here, without introducing the transverse information
of the flux tube, which has been extensively discussed in Refs. [46, 49].

The Unruh effect describes how the symmetry breaking in interacting field
theories appears as symmetry restoration in the eyes of accelerated observers.
Taking chiral condensate as an example, an accelerated observer implies that
the system of chiral condensate experiences an effective gravitational force
with an acceleration $a$, which affects the condensed system causing it to
either “collapse or dissolve.” Clearly, chiral condensate is sensitive to
acceleration, and it can be anticipated that when the acceleration provided by
the external field increases to a certain critical value, the chiral
condensate completely disappears, indicating the restoration of the broken
chiral symmetry. Our goal is to identify this critical acceleration $a_{c}$,
the corresponding Unruh temperature $T_{c}=a_{c}/2\pi$, and the underlying
source that induces them – the chromoelectric field $E_{1+1,c}$.

The NL$\sigma$M provides the result of $R$ in the chiral and large-N limits
which is written as [72, 74]

$R\equiv\frac{\langle\bar{\psi}\psi\rangle_{a}}{\langle\bar{\psi}\psi\rangle_{0}}=\sqrt{1-\frac{a^{2}}{a_{c}^{2}}},$
(7)

where $a_{c}$ is the critical acceleration. This result is consistent in
Rindler spacetime and flat Minkowski spacetime [74], and according to the
definition of the Unruh temperature, $a/a_{c}$ also plays the role of
$T_{U}/T_{c}$. It can be observed that Eq. (7) corresponds to the typical
second-order Landau-Ginzburg phase transition.

Building on the above discussion, we provide the results of chiral condensate
restoration in an accelerated version of the Schwinger model. Considering an
observer with a color charge $q$ and mass $M$ (for simplicity, assume
$q/M=1$), it experiences an acceleration $a=qE_{1+1}/M\sim E_{1+1}$ in the
color field, i.e., the chromoelectric flux tube. This can be viewed as a semi-
classical Newtonian mechanics approximation, upon which our work is based.
Returning to the $1+1$-dimensional Schwinger model, the ratio of chiral
condensate restoration in the Schwinger model can be obtained based on the
Eqs. (3, 6):

$R_{\mathrm{Sch}}\equiv\frac{\langle\bar{\psi}\psi\rangle_{E_{1+1}}}{\langle\bar{\psi}\psi\rangle_{0}}=\cos\left(\frac{2\pi
E_{1+1}}{g}\right).$ (8)

The above formula exhibits a simple mathematical expression, where
$R_{\mathrm{Sch}}$ is suppressed with the increase of $E_{1+1}$ within the
first quarter period of the cosine function. When $E_{1+1}/g=E_{1+1,c}/g=1/4$,
$R_{\mathrm{Sch}}$ vanishes, indicating the complete restoration of chiral
condensate.

To contrast with the results of the NL$\sigma$ model, we provide Figure 2 for
readers’ understanding. The horizontal axis in the figure is scaled to the
dimensionless physical quantity $a/a_{c}$ due to our adoption of the
assumption $E_{1+1}\sim a$.

Figure 2: Variation of chiral quark condensate with acceleration ratio
$a/a_{c}$. The blue solid line corresponds to the results from Ref. [74],
while the red dashed-dotted line is generated based on Eq. (8).

Both results in Figure 2 exhibit a suppression behavior with the increase in
the acceleration ratio $a/a_{c}$, which shares the same trend as the chiral
perturbation theory calculations mentioned in earlier work [75] on chiral
phase transitions. The transition from broken to restored chiral symmetry
depends on the acceleration or the corresponding gravitational-like field,
which is also applicable in the chromoelectric field. A straightforward
inference can be drawn from the above analysis: the critical acceleration,
critical Unruh temperature, and critical chromoelectric field have equivalent
effects on chiral condensate. Once the critical acceleration $a_{c}$ is
determined, the other two quantities are correspondingly fixed, and we reserve
these discussions for the next section.

##  V Determining the critical distance for string breaking between quarks-
antiquarks

We have previously introduced and discussed the relationship between string
breaking and chiral symmetry breaking, considering the mechanism of chiral
symmetry restoration triggered by the variation of the field in the
chromoelectric flux tube. The next step is to determine the critical value of
this chromoelectric field $E_{1+1,c}\sim a_{c}$ and its critical color source
distance associated with string or flux tube breaking. At the core of the
preceding work [49], we aim to determine the information of the chromoelectric
flux tube corresponding to different color source distances. Specifically, the
objective is to ascertain the chromoelectric field strength $E_{1+1}$
corresponding to various color source distances $r$, which can be obtained
from the Eqs. (4, 5) and parameters presented in Table 1. The specific results
are shown in Table 2, where, for dimensionless representation, we directly
provide the outcomes of $E_{1+1}/g$, with $g$ obtainable from the lattice
configurations in Ref. [64].

Table 2: The $E_{1+1}/g$ outcomes corresponding to the chromoelectric flux
tube distribution $E_{3+1}(x_{t})$ at different color source distances $r$,
obtained from lattice simulation results. Here, $\beta=2N_{c}/g^{2}$
represents the lattice coupling dependent on the color number $N_{c}$.

$\beta$ | 6.475 | 6.333 | 6.240 | 6.500 | 6.539 | 6.370 | 6.299 | 6.240 | 6.218
---|---|---|---|---|---|---|---|---|---
$r$ [fm] | 0.37 | 0.45 | 0.51 | 0.54 | 0.69 | 0.85 | 0.94 | 1.02 | 1.06
$E_{1+1}/g$ | 0.1503(2) | 0.1625(16) | 0.1751(7) | 0.1832(11) | 0.192(5) | 0.206(25) | 0.202(46) | 0.243(39) | 0.225(60)

The chromoelectric field strength $E_{1+1}$ is correlated with the string
tension of the flux tube, which increases concomitantly with the color source
distance $r$. Moreover, the quantity $\sqrt{w^{2}}$ representing the average
radius of the flux tube also enlarges with increasing $r$, which is discussed
under the assumption that the flux tube width can infinitely logarithmically
increase without breaking [76, 77].

Although we need to consider string breaking, before the breaking occurs, the
string tension and the corresponding chromoelectric field strength $E_{1+1}$
are still assumed to exhibit logarithmic growth behavior with the color source
distance [76, 77], i.e.

$\frac{E_{1+1}}{g}(r)=A+B\log\left(\frac{r}{r_{0}}\right),$ (9)

where $A$, $B$ and $r_{0}$ are parameters to be determined by fitting the data
from Table 2. Figure 3 depicts the relationship between the ratio of the
computed field strength to the coupling $g$, $E_{1+1}/g$, and the color source
distance $r$. We have conducted fitting using a parameterized Eq. (9). The
fitting parameters are determined as $A=0.125\pm 0.003$, $B=0.080\pm 0.002$,
$r_{0}=0.269\pm 0.009$ fm. By determining the intersection point of the fitted
curve and $E_{1+1}/g=1/4$, which corresponds to the critical external field
value for chiral restoration, the critical color source distance $r_{c}$ can
be identified. Through analysis, we obtain $r_{c}=1.294\pm 0.040$ fm.

Figure 3: Reduced flux tube strength $E_{1+1}/g$ with the logarithmic
$Ansatz$. The red data points are numerical calculations of $E_{1+1}/g$ and
the blue solid line with uncertainties corresponds to the fit of the growth of
$E_{1+1}/g$.

The utilization of logarithmic parameterization for fitting the flux tube
intensity is intended to correspond to the classical Luscher $Ansatz$ [76]. It
is worth noting that another study employed a linear function for fitting and
discussion. In summary, we have obtained a color source distance $r_{c}$
corresponding to the complete collapse of chiral condensate in the
chromoelectric field. Interestingly, our finding is in good agreement with
those in Refs. [78, 79, 80, 81, 82, 14, 15, 83, 84], despite the different
methods or assumptions employed in this work. We will compare the results of
several works that provide specific values of $r_{c}$ [14, 79, 15, 84, 81] (in
chronological order of publication dates) with our finding, as illustrated in
Figure 4.

Figure 4: Comparision of the critical distances of string breaking. The blue
marked points come from Refs. [14, 79, 15, 84, 81] (in chronological order of
publication dates), and our result is listed at the end.

By fitting the relationship between the chromoelectric flux tube field
strength and the distance from the color source generating the flux tube, we
have determined the critical length $r_{c}$ for the flux tube or string
breaking at the critical point where the flux tube leads to the restoration of
chiral symmetry. We note that our results are in the same order of magnitude
as the findings in [79] for the separation of heavy quarks and lattice
simulations of meson spectrum [14, 15] and Yang-Mills fields [84, 81],
although some works did not explicitly provide the sources of uncertainties.
Our results also have uncertainties, primarily stemming from the parameter
sets (see Table 1) used due to the parameterization (5) and the transverse
distribution of the flux tube calculated on the lattice [64] to determine
$E_{1+1}/g$. Furthermore, the determination of $r_{c}$ is obtained through the
fitting (see Figure 3), hence its uncertainty arises from the fitting
uncertainties of the three parameters in (9).

Finally, let us draw a connection between our results and the Unruh effect. In
Section IV, we outlined how observers within the chromoelectric flux tube
experience an acceleration $a\sim E_{1+1}$ and can detect thermal radiation
with an Unruh temperature of $T_{U}=a/2\pi$. Building on the hypothesis that
acceleration leads to symmetry restoration in the Unruh effect, we can also
determine the critical acceleration $a_{c}$ and critical temperature
$T_{c}=a_{c}/2\pi$ for the chiral symmetry phase transition within the
chromoelectric flux tube. In this section, we have identified that the
chromoelectric field strength corresponding to the complete restoration of
chiral symmetry in the flux tube is $E_{1+1,c}=g/4$, indicative of flux tube
or string breaking. One can naturally estimate the critical acceleration as
$a_{c}\sim E_{1+1,c}=g/4\simeq 1.14\times 10^{34}$ cm$/$s2 and the critical
temperature $T_{c}=0.038$ GeV $\sim 4.6\times 10^{11}$ K. Conversion factors
are utilized here: $T_{U}=a/(2.5\times 10^{22}$ (cm$/$s2)) K and 1 GeV
$=1.2\times 10^{13}$ K. This is consistent in order of magnitude with the
conclusions of Refs [36, 37], although there are some computational
differences. Additionally, we observe that the critical temperature value for
the chiral phase transition considered in [37] is higher than the results of
[36], but the critical temperature for the color symmetry phase transition
($T_{c}=0.04$ GeV) is closer to ours.

##  VI Conclusion and outlook

In this work, we consider the description of chiral symmetry restoration in
the 1+1-dimensional Schwinger model and utilize lattice simulation results of
chromoelectric flux tubes to provide the flux tube field strength excited by
different static quark-antiquark distances, based on the input of the chiral
quark condensate expression in the Schwinger model. Subsequently, we establish
a correspondence between the Hawking-Unruh effect and symmetry restoration by
determining the flux tube strength corresponding to the complete disappearance
of chiral condensate in the chromoelectric field, i.e., the strength
corresponding to the complete restoration of chiral symmetry, to identify the
critical distance $r_{c}$ for QCD flux tube or string breaking and compare it
with other relevant studies. Finally, we revisit the correspondence between
the Unruh effect and chiral symmetry restoration, providing the critical
temperature and acceleration for the chiral phase transition corresponding to
string breaking, and engage in discussions on this matter.

Based on the conjecture of static quark flux tubes, this study can, in
principle, be extended to similar processes involving the Schwinger mechanism
generating quark-antiquark pairs, as mentioned in the relevant studies [18,
19]. Furthermore, since the free energy of the quark-antiquark system
increases as their separation grows, ultimately approaching the sum of the
free energies of two free quarks [85, 82], this indirectly indicates that the
correlation between two correlated quarks weakens as they separate, leading to
the eventual disappearance of correlation. This suggests the possibility of
investigating the correlation or entanglement between quark-antiquark pairs, a
topic that has been partially discussed in [47, 48]. This is an area of
research that we need to pay attention to in the future.

## Acknowledgments

This work has been supported by the Strategic Priority Research Program of
Chinese Academy of Sciences (Grant NO. XDB34030301) and Guangdong Major
Project of Basic and Applied Basic Research (Grant NO. 2020B0301030008).

## References

  * Wilson [1974] K. G. Wilson, Confinement of Quarks, Phys. Rev. D 10, 2445 (1974).
  * Bali  _et al._ [1998] G. S. Bali _et al._ (TXL, TkL), Glueballs and string breaking from full QCD, Nucl. Phys. B Proc. Suppl. 63, 209 (1998), arXiv:hep-lat/9710012 .
  * Paladini and Sexton [1999] B. Paladini and J. C. Sexton, Asymptotic expansion of the lattice scalar propagator in coordinate space, Phys. Lett. B 448, 76 (1999), arXiv:hep-lat/9805021 .
  * Philipsen and Wittig [1998] O. Philipsen and H. Wittig, String breaking in nonAbelian gauge theories with fundamental matter fields, Phys. Rev. Lett. 81, 4056 (1998), [Erratum: Phys.Rev.Lett. 83, 2684 (1999)], arXiv:hep-lat/9807020 .
  * Knechtli and Sommer [1998] F. Knechtli and R. Sommer (ALPHA), String breaking in SU(2) gauge theory with scalar matter fields, Phys. Lett. B 440, 345 (1998), [Erratum: Phys.Lett.B 454, 399–399 (1999)], arXiv:hep-lat/9807022 .
  * Detar  _et al._ [1999] C. E. Detar, O. Kaczmarek, F. Karsch, and E. Laermann, String breaking in lattice quantum chromodynamics, Phys. Rev. D 59, 031501 (1999), arXiv:hep-lat/9808028 .
  * Drummond and Horgan [1999] I. T. Drummond and R. R. Horgan, Lattice string breaking and heavy meson decays, Phys. Lett. B 447, 298 (1999), arXiv:hep-lat/9811016 .
  * Drummond [1998] I. T. Drummond, Mixing scenarios for lattice string breaking, Phys. Lett. B 442, 279 (1998), arXiv:hep-lat/9808014 .
  * Bonnet  _et al._ [1999] F. D. R. Bonnet, D. B. Leinweber, A. G. Williams, and J. M. Zanotti, Towards string breaking in the static quark potential, (1999), arXiv:hep-lat/9912044 .
  * Duncan  _et al._ [2001] A. Duncan, E. Eichten, and H. Thacker, String breaking in four-dimensional lattice QCD, Phys. Rev. D 63, 111501 (2001), arXiv:hep-lat/0011076 .
  * Bolder  _et al._ [2001] B. Bolder, T. Struckmann, G. S. Bali, N. Eicker, T. Lippert, B. Orth, K. Schilling, and P. Ueberholz, A High precision study of the Q anti-Q potential from Wilson loops in the regime of string breaking, Phys. Rev. D 63, 074504 (2001), arXiv:hep-lat/0005018 .
  * Knechtli and Sommer [2000] F. Knechtli and R. Sommer (ALPHA), String breaking as a mixing phenomenon in the SU(2) Higgs model, Nucl. Phys. B 590, 309 (2000), arXiv:hep-lat/0005021 .
  * Bernard  _et al._ [2001] C. W. Bernard, T. A. DeGrand, C. E. Detar, P. Lacock, S. A. Gottlieb, U. M. Heller, J. Hetrick, K. Orginos, D. Toussaint, and R. L. Sugar, Zero temperature string breaking in lattice quantum chromodynamics, Phys. Rev. D 64, 074509 (2001), arXiv:hep-lat/0103012 .
  * Bali  _et al._ [2005] G. S. Bali, H. Neff, T. Duessel, T. Lippert, and K. Schilling (SESAM), Observation of string breaking in QCD, Phys. Rev. D 71, 114513 (2005), arXiv:hep-lat/0505012 .
  * Bulava  _et al._ [2019] J. Bulava, B. Hörz, F. Knechtli, V. Koch, G. Moir, C. Morningstar, and M. Peardon, String breaking by light and strange quarks in QCD, Phys. Lett. B 793, 493 (2019), arXiv:1902.04006 [hep-lat] .
  * Nambu and Jona-Lasinio [1961a] Y. Nambu and G. Jona-Lasinio, Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. 1., Phys. Rev. 122, 345 (1961a).
  * Nambu and Jona-Lasinio [1961b] Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. II., Phys. Rev. 124, 246 (1961b).
  * Florio  _et al._ [2023] A. Florio, D. Frenklakh, K. Ikeda, D. Kharzeev, V. Korepin, S. Shi, and K. Yu, Real-Time Nonperturbative Dynamics of Jet Production in Schwinger Model: Quantum Entanglement and Vacuum Modification, Phys. Rev. Lett. 131, 021902 (2023), arXiv:2301.11991 [hep-ph] .
  * Ikeda  _et al._ [2023] K. Ikeda, D. E. Kharzeev, R. Meyer, and S. Shi, Detecting the critical point through entanglement in the Schwinger model, Phys. Rev. D 108, L091501 (2023), arXiv:2305.00996 [hep-ph] .
  * Schwinger [1962a] J. S. Schwinger, Gauge Invariance and Mass, Phys. Rev. 125, 397 (1962a).
  * Schwinger [1962b] J. S. Schwinger, Gauge Invariance and Mass. 2., Phys. Rev. 128, 2425 (1962b).
  * Hawking [1975] S. W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43, 199 (1975), [Erratum: Commun.Math.Phys. 46, 206 (1976)].
  * Fulling [1973] S. A. Fulling, Nonuniqueness of canonical field quantization in Riemannian space-time, Phys. Rev. D 7, 2850 (1973).
  * Davies [1975] P. C. W. Davies, Scalar particle production in Schwarzschild and Rindler metrics, J. Phys. A 8, 609 (1975).
  * Unruh [1976] W. G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14, 870 (1976).
  * Takagi [1986] S. Takagi, Vacuum Noise and Stress Induced by Uniform Acceleration: Hawking-Unruh Effect in Rindler Manifold of Arbitrary Dimension, Prog. Theor. Phys. Suppl. 88, 1 (1986).
  * Unruh and Weiss [1984] W. G. Unruh and N. Weiss, Acceleration Radiation in Interacting Field Theories, Phys. Rev. D 29, 1656 (1984).
  * Rapp  _et al._ [1998] R. Rapp, T. Schäfer, E. V. Shuryak, and M. Velkovsky, Diquark Bose condensates in high density matter and instantons, Phys. Rev. Lett. 81, 53 (1998), arXiv:hep-ph/9711396 .
  * Rapp  _et al._ [2000] R. Rapp, T. Schäfer, E. V. Shuryak, and M. Velkovsky, High density QCD and instantons, Annals Phys. 280, 35 (2000), arXiv:hep-ph/9904353 .
  * Vaks and Larkin [1961] V. Vaks and A. Larkin, Application of superconductivity theory methods to the problem of the masses of elementary particles, Zhur. Eksptl’. i Teoret. Fiz. 40, 282 (1961).
  * Kharzeev and Tuchin [2005] D. Kharzeev and K. Tuchin, From color glass condensate to quark gluon plasma through the event horizon, Nucl. Phys. A 753, 316 (2005), arXiv:hep-ph/0501234 .
  * Dolan and Jackiw [1974] L. Dolan and R. Jackiw, Symmetry Behavior at Finite Temperature, Phys. Rev. D 9, 3320 (1974).
  * Salluce  _et al._ [2024] D. G. Salluce, M. Pasini, A. Flachi, A. Pittelli, and S. Ansoldi, Symmetry Restoration and Uniformly Accelerated Observers in Minkowski Spacetime, (2024), arXiv:2401.16483 [hep-th] .
  * Hill [1985] C. T. Hill, Can the Hawking Effect Thaw a Broken Symmetry?, Phys. Lett. B 155, 343 (1985).
  * Hill [1986] C. T. Hill, One Loop Operator Matrix Elements in the Unruh Vacuum, Nucl. Phys. B 277, 547 (1986).
  * Ohsaku [2004] T. Ohsaku, Dynamical chiral symmetry breaking and its restoration for an accelerated observer, Phys. Lett. B 599, 102 (2004), arXiv:hep-th/0407067 .
  * Ebert and Zhukovsky [2007] D. Ebert and V. C. Zhukovsky, Restoration of Dynamically Broken Chiral and Color Symmetries for an Accelerated Observer, Phys. Lett. B 645, 267 (2007), arXiv:hep-th/0612009 .
  * Peeters and Zamaklar [2008] K. Peeters and M. Zamaklar, Dissociation by acceleration, JHEP 01, 038, arXiv:0711.3446 [hep-th] .
  * Paredes  _et al._ [2009] A. Paredes, K. Peeters, and M. Zamaklar, Temperature versus acceleration: The Unruh effect for holographic models, JHEP 04, 015, arXiv:0812.0981 [hep-th] .
  * Ebert  _et al._ [2008] D. Ebert, V. C. Zhukovsky, and A. V. Tyukov, Dynamical breaking and restoration of chiral and color symmetries for an accelerated observer and in the static Einstein universe, J. Phys. A 41, 164064 (2008).
  * Lenz  _et al._ [2011] F. Lenz, K. Ohta, and K. Yazaki, Static interactions and stability of matter in Rindler space, Phys. Rev. D 83, 064037 (2011), arXiv:1012.3283 [hep-th] .
  * Castorina and Finocchiaro [2012] P. Castorina and M. Finocchiaro, Symmetry Restoration By Acceleration, J. Mod. Phys. 3, 1703 (2012), arXiv:1207.3677 [hep-th] .
  * Takeuchi [2015a] S. Takeuchi, Unruh effect in a real scalar field with the Higgs potential on a dynamically variable background space-time, Eur. Phys. J. C 75, 404 (2015a), arXiv:1412.0338 [hep-th] .
  * Takeuchi [2015b] S. Takeuchi, Bose–Einstein condensation in the Rindler space, Phys. Lett. B 750, 209 (2015b), arXiv:1501.07471 [hep-th] .
  * Benic and Fukushima [2015] S. Benic and K. Fukushima, Unruh effect and condensate in and out of an accelerated vacuum, (2015), arXiv:1503.05790 [hep-th] .
  * Kharzeev and Loshaj [2014] D. E. Kharzeev and F. Loshaj, Partial restoration of chiral symmetry in a confining string, Phys. Rev. D 90, 037501 (2014), arXiv:1404.7746 [hep-ph] .
  * Grieninger  _et al._ [2023a] S. Grieninger, D. E. Kharzeev, and I. Zahed, Entanglement in a holographic Schwinger pair with confinement, Phys. Rev. D 108, 086030 (2023a), arXiv:2305.07121 [hep-th] .
  * Grieninger  _et al._ [2023b] S. Grieninger, D. E. Kharzeev, and I. Zahed, Entanglement entropy in a time-dependent holographic Schwinger pair creation, Phys. Rev. D 108, 126014 (2023b), arXiv:2310.12042 [hep-th] .
  * Kou and Chen [2024] W. Kou and X. Chen, Exploring quantum entanglement in chiral symmetry partial restoration with 1+1 string model, Phys. Lett. B 853, 138675 (2024), arXiv:2401.16673 [hep-ph] .
  * Lowenstein and Swieca [1971] J. H. Lowenstein and J. A. Swieca, Quantum electrodynamics in two-dimensions, Annals Phys. 68, 172 (1971).
  * Coleman  _et al._ [1975] S. R. Coleman, R. Jackiw, and L. Susskind, Charge Shielding and Quark Confinement in the Massive Schwinger Model, Annals Phys. 93, 267 (1975).
  * Hamer  _et al._ [1982] C. J. Hamer, J. B. Kogut, D. P. Crewther, and M. M. Mazzolini, The Massive Schwinger Model on a Lattice: Background Field, Chiral Symmetry and the String Tension, Nucl. Phys. B 208, 413 (1982).
  * Coleman [1975] S. R. Coleman, The Quantum Sine-Gordon Equation as the Massive Thirring Model, Phys. Rev. D 11, 2088 (1975).
  * Mandelstam [1975] S. Mandelstam, Soliton Operators for the Quantized Sine-Gordon Equation, Phys. Rev. D 11, 3026 (1975).
  * Singh  _et al._ [1993] V. Singh, D. A. Browne, and R. W. Haymaker, Structure of Abrikosov vortices in SU(2) lattice gauge theory, Phys. Lett. B 306, 115 (1993), arXiv:hep-lat/9301004 .
  * Schilling  _et al._ [1999] K. Schilling, G. S. Bali, and C. Schlichter, A Ginzburg-Landau analysis of the color electric flux tube, Nucl. Phys. B Proc. Suppl. 73, 638 (1999), arXiv:hep-lat/9809039 .
  * Chernodub  _et al._ [2001] M. N. Chernodub, F. V. Gubarev, M. I. Polikarpov, and V. I. Zakharov, Towards Abelian - like formulation of the dual gluodynamics, Nucl. Phys. B 600, 163 (2001), arXiv:hep-th/0010265 .
  * Chernodub  _et al._ [2005] M. N. Chernodub, K. Ishiguro, Y. Mori, Y. Nakamura, M. I. Polikarpov, T. Sekido, T. Suzuki, and V. I. Zakharov, Vacuum type of SU(2) gluodynamics in maximally Abelian and Landau gauges, Phys. Rev. D 72, 074505 (2005), arXiv:hep-lat/0508004 .
  * Suzuki  _et al._ [2009] T. Suzuki, M. Hasegawa, K. Ishiguro, Y. Koma, and T. Sekido, Gauge invariance of color confinement due to the dual Meissner effect caused by Abelian monopoles, Phys. Rev. D 80, 054504 (2009), arXiv:0907.0583 [hep-lat] .
  * Cea  _et al._ [2012] P. Cea, L. Cosmai, and A. Papa, Chromoelectric flux tubes and coherence length in QCD, Phys. Rev. D 86, 054501 (2012), arXiv:1208.1362 [hep-lat] .
  * Abrikosov [1957] A. A. Abrikosov, On the Magnetic properties of superconductors of the second group, Sov. Phys. JETP 5, 1174 (1957).
  * Nielsen and Olesen [1973] H. B. Nielsen and P. Olesen, Vortex Line Models for Dual Strings, Nucl. Phys. B 61, 45 (1973).
  * Witten [1985] E. Witten, Superconducting Strings, Nucl. Phys. B 249, 557 (1985).
  * Baker  _et al._ [2020] M. Baker, P. Cea, V. Chelnokov, L. Cosmai, F. Cuteri, and A. Papa, The confining color field in SU(3) gauge theory, Eur. Phys. J. C 80, 514 (2020), arXiv:1912.04739 [hep-lat] .
  * Clem [1975] J. R. Clem, Simple model for the vortex core in a type ii superconductor, Journal of Low Temperature Physics 18, 427 (1975).
  * Kaneko  _et al._ [2014] T. Kaneko, S. Aoki, G. Cossu, H. Fukaya, S. Hashimoto, and J. Noaki (JLQCD), Large-scale simulations with chiral symmetry, PoS LATTICE2013, 125 (2014), arXiv:1311.6941 [hep-lat] .
  * Iritani  _et al._ [2013] T. Iritani, G. Cossu, and S. Hashimoto, Lattice QCD study of partial restoration of chiral symmetry in the flux-tube, PoS Hadron2013, 159 (2013), arXiv:1401.4293 [hep-lat] .
  * Iritani  _et al._ [2014a] T. Iritani, G. Cossu, and S. Hashimoto, Analysis of topological structure of the QCD vacuum with overlap-Dirac operator eigenmode, PoS LATTICE2013, 376 (2014a), arXiv:1311.0218 [hep-lat] .
  * Iritani  _et al._ [2014b] T. Iritani, G. Cossu, and S. Hashimoto, Partial restoration of chiral symmetry inside hadrons, PoS LATTICE2014, 338 (2014b), arXiv:1412.2322 [hep-lat] .
  * Iritani  _et al._ [2015] T. Iritani, G. Cossu, and S. Hashimoto, Partial restoration of chiral symmetry in the color flux tube, Phys. Rev. D 91, 094501 (2015), arXiv:1502.04845 [hep-lat] .
  * Iritani  _et al._ [2016] T. Iritani, G. Cossu, and S. Hashimoto, Partial restoration of chiral symmetry inside hadrons, AIP Conf. Proc. 1701, 100009 (2016).
  * Cortés  _et al._ [2016] S. Cortés, A. Gómez Nicola, and J. Morales, Chiral Symmetry Restoration for the large-$N$ pion gas, Phys. Rev. D 94, 116008 (2016), arXiv:1609.07751 [hep-ph] .
  * Dobado [2017] A. Dobado, Spontaneous symmetry breaking and the Unruh effect, in  _4th International Conference on the Nature and Ontology of Spacetime_ (2017) pp. 161–173, arXiv:1703.05675 [gr-qc] .
  * Casado-Turrión and Dobado [2019] A. Casado-Turrión and A. Dobado, Triggering the QCD phase transition through the Unruh effect: chiral symmetry restoration for uniformly accelerated observers, Phys. Rev. D 99, 125018 (2019), arXiv:1905.11179 [hep-ph] .
  * Gerber and Leutwyler [1989] P. Gerber and H. Leutwyler, Hadrons Below the Chiral Phase Transition, Nucl. Phys. B 321, 387 (1989).
  * Luscher  _et al._ [1981] M. Luscher, G. Munster, and P. Weisz, How Thick Are Chromoelectric Flux Tubes?, Nucl. Phys. B 180, 1 (1981).
  * Gliozzi  _et al._ [2010] F. Gliozzi, M. Pepe, and U. J. Wiese, The Width of the Color Flux Tube at 2-Loop Order, JHEP 11, 053, arXiv:1006.2252 [hep-lat] .
  * Feilmair and Markum [1992] W. Feilmair and H. Markum, Flux tube profiles with dynamical quarks and finite temperatures, Nucl. Phys. B 370, 299 (1992).
  * Castorina  _et al._ [2007] P. Castorina, D. Kharzeev, and H. Satz, Thermal Hadronization and Hawking-Unruh Radiation in QCD, Eur. Phys. J. C 52, 187 (2007), arXiv:0704.1426 [hep-ph] .
  * Bicudo  _et al._ [2019] P. Bicudo, N. Cardoso, and M. Cardoso, Pure gauge QCD flux tubes and their widths at finite temperature, Nucl. Phys. B 940, 88 (2019), arXiv:1702.03454 [hep-lat] .
  * Chagdaa  _et al._ [2021] S. Chagdaa, B. Purev, and E. Galsandorj, Flux tubes in full QCD at high temperature, J. Phys. G 48, 125001 (2021).
  * Galsandorj  _et al._ [2023] E. Galsandorj, S. Chagdaa, and B. Purev, Color Screening from Flux Tube in (2 + 1)-Flavour QCD, Phys. Part. Nucl. Lett. 20, 10 (2023).
  * Antonov  _et al._ [2003] D. Antonov, L. Del Debbio, and A. Di Giacomo, A Model for string breaking in QCD, JHEP 08, 011, arXiv:hep-lat/0302015 .
  * Bonati and Morlacchi [2020] C. Bonati and S. Morlacchi, Flux tubes and string breaking in three dimensional SU(2) Yang-Mills theory, Phys. Rev. D 101, 094506 (2020), arXiv:2003.07244 [hep-lat] .
  * Maezawa  _et al._ [2009] Y. Maezawa, S. Aoki, S. Ejiri, T. Hatsuda, N. Ishii, K. Kanaya, H. Ohno, and T. Umeda (WHOT-QCD), Free energies of heavy quarks in full-QCD lattice simulations with Wilson-type quark action, Nucl. Phys. A 830, 247C (2009), arXiv:0907.4203 [hep-lat] .