# Multi-Branch Network for Imagery Emotion Prediction

Quoc-Bao Ninh  0009-0002-0341-3540 1 Faculty of Information Technology,
University of Science, Ho Chi Minh City, Vietnam 2 Vietnam National
University, Ho Chi Minh City, Vietnam<EMAIL_ADDRESS>,  Hai-
Chan Nguyen  0009-0007-5732-8671 1 Faculty of Information Technology,
University of Science, Ho Chi Minh City, Vietnam 2 Vietnam National
University, Ho Chi Minh City, Vietnam<EMAIL_ADDRESS>,  Triet
Huynh  0009-0007-4277-1881 1 Faculty of Information Technology, University of
Science, Ho Chi Minh City, Vietnam 2 Vietnam National University, Ho Chi Minh
City, Vietnam<EMAIL_ADDRESS>and  Trung-Nghia Le
0000-0002-7363-2610 1 Faculty of Information Technology, University of
Science, Ho Chi Minh City, Vietnam 2 Vietnam National University, Ho Chi Minh
City, Vietnam<EMAIL_ADDRESS>
(2023)

###### Abstract.

For a long time, images have proved perfect at both storing and conveying rich
semantics, especially human emotions. A lot of research has been conducted to
provide machines with the ability to recognize emotions in photos of people.
Previous methods mostly focus on facial expressions but fail to consider the
scene context, meanwhile scene context plays an important role in predicting
emotions, leading to more accurate results. In addition, Valence-Arousal-
Dominance (VAD) values offer a more precise quantitative understanding of
continuous emotions, yet there has been less emphasis on predicting them
compared to discrete emotional categories. In this paper, we present a novel
Multi-Branch Network (MBN), which utilizes various source information,
including faces, bodies, and scene contexts to predict both discrete and
continuous emotions in an image. Experimental results on EMOTIC dataset, which
contains large-scale images of people in unconstrained situations labeled with
26 discrete categories of emotions and VAD values, show that our proposed
method significantly outperforms state-of-the-art methods with 28.4% in mAP
and 0.93 in MAE. The results highlight the importance of utilizing multiple
contextual information in emotion prediction and illustrate the potential of
our proposed method in a wide range of applications, such as effective
computing, human-computer interaction, and social robotics. Source code:
https://github.com/BaoNinh2808/Multi-Branch-Network-for-Imagery-Emotion-
Prediction

Imagery Emotion Prediction, Multi-Branch Network.

*Equal contributors

**Corresponding author. Email address<EMAIL_ADDRESS>
††journalyear: 2023††copyright: acmlicensed††conference: The 12th
International Symposium on Information and Communication Technology; December
7–8, 2023; Ho Chi Minh, Vietnam††booktitle: The 12th International Symposium
on Information and Communication Technology (SOICT 2023), December 7–8, 2023,
Ho Chi Minh, Vietnam††price: 15.00††doi: 10.1145/3628797.3628954††isbn:
979-8-4007-0891-6/23/12††ccs: Computing methodologies Supervised learning by
regression††ccs: Computing methodologies Supervised learning by
classification††ccs: Computing methodologies Neural networks

##  1\. Introduction

An estimated 4.59 billion people in 2022 are using social media worldwide
(Dixon, 2023), with approximately 3.2 billion photos being shared across all
social media platforms every day (Tsvetkova, 2023). Algorithms analyze and
process large amounts of information from images, enabling social media
platforms to develop strategies that effectively attract users for increasing
profits (NARAYANAN, 2023). Among them, analyzing the emotions and moods of
humans in images plays an extremely important role (Haleem et al., 2022). It
helps improve the user experience, analyze user behavior and trends, and
develop appropriate marketing strategies and advertising campaigns. Therefore,
it poses both motivation and challenges for the field of emotion analysis from
images, particularly human images.

(a) Crying due to joy on her wedding day.

(b) Crying due to the death of a COVID-19 patient.

Figure 1. Crying in different context.

Deep learning is a transformative field of machine learning that uses neural
networks to solve complex problems. It has revolutionized computer vision,
natural language processing, and speech recognition(Dong et al., 2021). Deep
learning models have the ability to efficiently learn features from images,
enabling them to achieve high effectiveness in object recognition and
classification tasks. Therefore, in the past decade, deep learning has been
widely applied in research and development of applications in the field of
emotion recognition and analysis from images. Recently, Transformer models
have expanded their applications from natural language processing to computer
vision, demonstrating outstanding performance in emotion prediction of image
and opening up a new direction for deep learning models in this field.

This paper focuses on the issue of predicting emotions conveyed in images
based on contextual information. Concretely, we aim to identify the apparent
emotional state of a person depicted in a given image. This problem has been
extensively studied in the field of computer vision, with a particular
emphasis on two approaches: analyzing facial expressions and analyzing body
posture and gestures. These approaches have yielded notable results and
practical applications in real-life scenarios (Noroozi et al., 2018). However,
they only concentrate on analyzing the information expressed by the human in
the images, neglecting other equally important information, such as the
contextual background of the image. The importance of the contextual
background of the image is demonstrated in (Barrett et al., 2011), (Barrett,
2017). For example, in Fig. 1, the person may be crying tears of joy at a
wedding day. However, on the contrary, the person may also be crying out of
frustration or anger due to the devastating impact of the Covid-19 pandemic.
Therefore, considering the images shared on social media platforms today,
which often encompass both human and their surrounding contexts, solely
relying on these traditional approaches to predict emotions may not fulfill
the need for high-quality information on social media platforms.

Emotions can be identified by discrete categories or represented as
coordinates on a multi-dimensional continuous scale. Classifying emotions into
different categories is convenient for ease of expression and visualization,
but it only represents emotions in a qualitative manner and lacks information
(Young et al., 1997). Even for the same type of emotion, its intensity can
vary depending on different situations. Furthermore, an emotion does not
solely belong to a single category and can belong to multiple categories,
making it difficult and ambiguous to determine. Using a multi-dimensional
continuous scale, such as Valence-Arousal-Dominance (VAD) (see Fig. 2),
provides more information about a specific emotion, such as its pleasantness,
arousal level, and level of control (Harmon-Jones et al., 2017). For example,
an emotion such as anger may have a low value for valence (unpleasant), a high
level of arousal (extremely stimulating), and a high level of dominance (a
feeling of being hard to control). This approach quantifies emotions and helps
to determine them more accurately, making it convenient for computational or
further processing tasks. However, it is less commonly used due to its
complexity and lack of popularity. In addition, evaluating emotions on a
continuous scale is often overlooked due to the lack of labeled data on the
continuous scale, whereas there is a wealth of labeled data for discrete
emotions in image datasets.

To overcome the limitations of traditional methods in predicting human
emotions, we propose an effective yet straightforward solution. Our proposed
Multi-Branch Network (MBN) consists of three branches that analyze various
contextual information, such as person’s face, and body, and scene context.
These extracted features significantly contribute to the prediction of both
discrete and continuous emotions.

Experiments conducted on the EMOTIC dataset (Kosti et al., 2017) show that our
MBN significantly outperforms the state-of-the-art method (Kosti et al.,
2020). Particularly, the proposed method achieved 0.2837 and 0.9256 in terms
of mAP and MAE on the EMOTIC dataset, respectively. We also analyzed the
performance of MBN on different backbones.

Our contributions are as follows:

  * •

We present an effective method for human emotion prediction from images. Our
method integrates information extracted from various sources, including
person’s face and body, and scene context. The combination of facial features
and attention to scene contexts helps improve the performance of both discrete
and continuous emotion prediction.

  * •

We provide extensive experiments and analysis on emotion prediction using the
continuous VAD scale, which can serve as a baseline for future studies.
Experimental results contribute to the limited existing research on emotion
recognition models using the continuous VAD scale.

Figure 2. Distribution of the six emotions in valance-arousal-dominance
(V.A.D) space (Mitruț et al., 2019).

##  2\. Related work

###  2.1. Imagery Emotion Prediction

Emotional prediction has been a long-standing area of research. In 1967,
Mehrabian showed that 7% of knowledge moves between people verbally, 38%
through voice, and 55% through facial expression (Marechal et al., 2019).
Facial expressions have long been considered a key component in understanding
and predicting human emotions. The most successful CNN models in this field
commonly employ popular backbones such as ResNet (Wen et al., 2023) and
EfficientNet (Savchenko, 2021). In recent years, with the emergence of
Transformer models, a new approach has been created with great potential for
development in the field of facial expression recognition. Jia Li, et al.
achieved high results on the AffectNet dataset by using Vision Transformer (Ma
et al., 2021).

Another approach to emotion prediction is body gesture and posture;
researchers have also turned to emotional body language, i.e., the expression
of emotions through human body pose and body motion (Schindler et al., 2008).
An implicit assumption common to the work on emotional body language is that
body language is only a different means of expressing the same set of basic
emotions as facial expressions. Other studies have combined the recognition of
body gestures, posture, and facial expressions to achieve more accurate
prediction results (Behera et al., 2020).

However, we cannot predict exactly the emotions of an image solely based on
the facial expressions or body gestures of people, but rather have to take
into account the context of the image. A study conducted by Google AI and UC
Berkeley analyzed the contexts in which people use a variety of facial
expressions across 144 countries (Blog, 2021). They found that each kind of
facial expression had distinct associations with a set of contexts that were
70% preserved across 12 world regions . This suggests that context plays a
significant role in how facial expressions are interpreted and understood.
Several studies have shown that incorporating contextual features into models
can improve their accuracy in predicting emotions. For example, a study on
sentiment and emotion classification using bidirectional LSTM found that by
incorporating contextual information among utterances in the same video, their
proposed attention-based multimodal contextual fusion model outperformed
existing methods by over 3% in emotion classification (Huddar et al., 2021).
Another study found that perceives make use of a wide variety of perceptible
cues including social context, facial expression, and tone of voice to infer
what emotions others are feeling (Barrett et al., 2016).

Figure 3. Architecture of our proposed Multi-Branch Network (MBN) for image
emotion prediction.

In recent methods for predicting the emotion of people in images, one of the
most common approaches is combining facial expression and scene context
information. The two most effective approaches are the EmotiCon algorithm
(Mittal et al., 2020) and the fusion model (Kosti et al., 2020). The fusion
model consists of two feature extraction modules and a fusion network for
jointly estimating the discrete categories and continuous dimensions.
EmotiCon, which is based on Frege’s Principle, uses three interpretations of
context to perform perceived emotion recognition. They use multiple modalities
of faces and gaits, background visual information, and sociodynamic inter-
agent interactions to infer the perceived emotion (Mittal et al., 2020).
Taking inspiration from the EmotiCon algorithm (Mittal et al., 2020) and the
fusion model (Kosti et al., 2020), we propose an multi-branch deep network for
image emotion prediction.

###  2.2. Discrete and Continuous Emotions

Emotions are an important part of human life. They can help us better
understand ourselves and those around us, as well as help us make informed
decisions. Based on the existing literature in the fields of psychology and
neuroscience, one’s emotional state can be described following two distinct
representational approaches, namely the categorical and the dimensional
approaches.

The categorical approach is based on research on basic emotions, pioneered by
Darwin, interpreted by Tomkins, and supported by the findings of Ekman (Ekman
et al., 1987). There are a small number of basic emotions, include happiness,
sadness, fear, anger, surprise, and disgust. Within basic emotions, the term
discrete emotion means that this emotion represents its own category. For
example, basic theorists view fear, anger, and disgust as three separate
discrete emotions. Rage, annoyance, and anger would be feelings that could be
categorized within the discrete emotion of anger.

But the 6 basic emotions cannot fully describe the full range of human
emotions, so some new theories have divide into more types of emotions. A new
study identifies 27 categories of emotion and shows how they blend together in
our everyday experience (Cowen and Keltner, 2017). This findings will help
other scientists and engineers more precisely capture the emotional states
that underlie moods, brain activity, and expressive signals, leading to
improved psychiatric treatments, an understanding of the brain basis of
emotion, and technology responsive to our emotional needs (Horikawa et al.,
2020).

According to the dimensional approach, affective states are not independent of
one another; rather, they are related to one another systematically. In this
approach, the majority of affect variability is covered by three dimensions:
valence, arousal, and potency (dominance) (Davitz, 1964) (see Fig. 2). The
valence dimension refers to how positive or negative the emotion is, and
ranges from unpleasant feelings to pleasant feelings of happiness. The arousal
dimension refers to how excited or apathetic the emotion is and it ranges from
sleepiness or boredom to frantic excitement. The power dimension refers to the
degree of power or sense of control over the emotion. The field concerned with
the psychological study of emotion has recently witnessed much debate
regarding the structure of emotion.

In particular, scientists have argued over whether emotions are best described
along dimensions of valence and arousal or as discrete entities (Harmon-Jones
et al., 2017). However, we note that each approach has its own advantages and
disadvantages. Therefore, we predict the emotion of an image based on both
discrete emotion and VAD values.

##  3\. Propose Method

###  3.1. Overview

Figure 4. Face extraction and enhancement from body image.

Figure 3 shows the architecture of our proposed Multi-Branch Network (MBN).
The network is divided into two main parts. The first part extracts features
from the body, the face, and context of the image, referred to as body image,
face image, and context image. It consists of three branches to exploit
emotions from subjects and background. We remark that the face region is
extracted from the body image. The second part is a fusion network combining
the features extracted from the three branches to predict the discrete
emotions and VAD values of each person in the image.

###  3.2. Feature Extraction Branches

Our proposed network consists of three feature extraction branches, utilizing
different deep learning models trained on suitable datasets to efficiently
make predictions on human and scene images.

Body Feature Extraction Branch.  This branch is designed to extract features
from body image. It takes an image that contains only the person’s body,
without the surrounding scene, and its objective is to extract critical
information from the input. To achieve this goal, the backbone of the Body
Feature Extraction Branch must work efficiently with images that contain only
the person’s body. Therefore, we use some deep learning backbones such as
ResNet-18, ResNet-50 (He et al., 2016), and SwinT(Liu et al., 2021), which
come with ImageNet(Deng et al., 2009) pre-trained weights since ImageNet is a
large-scale dataset of human images. Additionally, we also use some self-
retrained models. We retrained the ResNet-50 and SwinT models using the EMOTIC
dataset. We utilize these backbones to build a distinct model from scratch for
predicting the gender and age range of individuals in the image. These models
were trained for 15 epochs, employing the body image of the EMOTIC dataset as
both the training and testing datasets. The results showed that both models
achieved an mAP score above 0.33, indicating satisfactory performance. We
believe that retraining these models on the EMOTIC dataset can improve their
effectiveness.

Context Feature Extraction Branch.  This branch is utilized to extract
features from the scene context of an image. Similarly, we use backbones that
perform well in scene images for this branch. To achieve the goal, we use the
ResNet-18 and ResNet-50 models with weights pre-trained on the Places365
dataset (Zhou et al., 2017), which consists of more than 1.8 million images
from 365 scene classes.

Face Feature Extraction Branch.  The face branch is one of the most essential
parts of our proposed network and it represents our additional improvement
over the previous approach (Kosti et al., 2020). As we mentioned, the key to
predicting emotion in a human image is facial expression. Therefore, we
utilize different deep learning models pre-trained on FER-2013 dataset
(Radford et al., 2013), a facial expression recognition dataset consisting of
30,000 facial images with 7 different expressions: angry, disgust, fear,
happy, sad, surprise, and neutral. The FER-2013 dataset is also a popular
benchmark for facial expression recognition algorithms. Particularly, we
employ two pre-trained models on the FER-2013 dataset by Shangeth (Shangeth,
2022) and Balmukund (balmukund, 2022), denoted by S-FER and B-FER,
respectively.

Face Extraction and Enhancement.  We employ a face detection (Google, 2021)
method to locate the bounding box of the face within the body image and then
crop the image accordingly. Subsequently, we resize the cropped images to
48x48.

As body image was not originally of high quality, resizing face image to
different height and width ratios after cropping may significantly reduce the
quality of the facial image. To ensure optimal results from the Face Feature
Extraction Branch, we first extend the bounding box to ensure equal width and
height before cropping. In cases where the extension is not possible (e.g.,
faces near the image edges), we add padding after cropping to achieve equal
width and height. These small preprocessing steps ensure that the face image
are preserved with improved quality.

Nevertheless, since the extracted facial images from the body image still have
suboptimal quality, it necessary to enhance their quality before inputting
them into the Face Feature Extraction Branch. So, after cropping and resizing
the face images to 48x48, we further improve their quality by performing
denoising and sharpening(see Fig. 4). We use the filter2D function of OpenCV
to make the face image appear sharper and maintain its quality.

###  3.3. Fusion Network

Architecture of the fusion network is adapted from the work of Ronak et al.
(Kosti et al., 2020), which is combined of two fully-connected layers. The
first fully-connected layer is to reduce the dimension of the input, which is
taken from the last layer of the predecessor models to 256, and then the
second fully-connected layer is used to predict the discrete emotions and VAD
value of the emotions.

To connect the fusion network with the three predecessors, we cut off the last
layer of all the element models, then rebuild the first three by creating
three sequential networks of all the layers of each original model, except for
the last being cut off. Then we pass the number of input features from the
last layer of three models into a linear layer to construct the first fully-
connected layer of the third part. The output of this layer is a vector with
size of 256. The second layer is a linear layer, which takes in a vector with
size of 256 and outputs two vectors, first has 26 dimensions denoting 26 kinds
of discrete emotions, and the second has 3 dimensions denoting VAD value.

###  3.4. Loss function

####  3.4.1. Criterion for Discrete Categories

The problem of predicting discrete emotions for people in images is a
multilabel prediction task, because each person is labeled with multiple
different emotions from among the N kinds of discrete emotions. Additionally,
the issue of data imbalance (see Fig. 5) poses difficulties in training a
model to produce the most accurate predictions.

Following instructions of the EMOTIC dataset (Kosti et al., 2020), we employ a
weighted Euclidean loss during the training process. This loss function is
deemed more appropriate for the task than Kullback-Leibler divergence or a
multi-class multi-classification hinge loss (Kosti et al., 2020). Concretely,
for each prediction $\hat{y}=[\hat{y}_{1},\hat{y}_{2},...,\hat{y}_{N}]$, the
weighted Euclidean loss is defined as follows:

(1) $L^{disc}(\hat{y})=\sum_{i=1}^{N}w_{i}(\hat{y}_{i}-y_{i})^{2},$

where N is the number of categories, $\hat{y}_{i}$ is the prediction for the
$i^{th}$ class and $y_{i}$ is the true label of this class. The parameter
$w_{i}$ is the weight assigned to each class. $w_{i}=\frac{1}{ln(c+p_{i})}$,
where $p_{i}$ is the propability of the class $i^{th}$ and c is the parameter
to control the range of valid values for $w_{i}$.

####  3.4.2. Criterion for Continuous Dimensions

The task of predicting values on a continuous scale is formally treated as a
regression problem. Consequently, the adoption of L2 Loss as the preferred
loss function is deemed appropriate for conducting this particular task. L2
Loss (Euclidean Loss or Mean Squared Error) is a common loss function in
machine learning used for regression tasks. It calculates the squared
difference between predicted and actual values, aiming to minimize overall
error and create a smoother model output. By penalizing large errors more, it
helps achieve greater accuracy and precision in regression models. The
optimization process aims to find the model’s parameters that result in the
minimum L2 Loss, which leads to a more accurate and precise model for
regression tasks. Concretely, for each prediction
$\hat{y}=[\hat{y}_{1},\hat{y}_{2},...,\hat{y}_{N}]$, the L2 loss is defined as
follows:

(2) $L^{cont}(\hat{y})=\sum_{i=1}^{N}v_{i}(\hat{y}_{i}-y_{i})^{2},$

where N is the number of dimensions, $\hat{y}_{i}$ is the prediction for the
$i^{th}$ dimension value and $y_{i}$ is the true label of this dimension. The
parameter $v_{i}\in\\{0,1\\}$ is a binary weight to represent the error
margin. $v_{i}=0$ if $(\hat{y}_{i}-y_{i})<\theta$ else $v_{i}=1$. If the
prediction of each dimension is too close to the ground truth, i.e., the error
is less than $\theta$, then the loss for this dimension is effectively crossed
out or set to zero.

##  4\. EXPERIMENTS

###  4.1. Implementation Details

We used Google Colab as the platform to conduct experiments, taking advantage
of the free resources it provided. Specifically, we had access to 78GB of disk
space, 12GB of RAM, and 12GB of T4 GPU RAM. Our deep learning model was
implemented with the PyTorch library. We conducted the experiments using a
batch size of 52 for the discrete emotion prediction tasks and a batch size of
26 for the continuous emotion prediction tasks. Our model trained for 15
epochs with an initial learning rate of 0.001. We conducted tests with various
initial learning rates and determined that 0.001 is the most suitable one. The
learning rate was decayed by 10% every 7 epochs to prevent overfitting. The
optimization algorithm used was stochastic gradient descent (SGD) with
momentum of 0.9.

###  4.2. Dataset

We evaluate methods on the EMOTIC (i.e., EMOTions In Context) dataset (Kosti
et al., 2020), a database of images of people in real environments, annotated
with their apparent emotions. The images are annotated with an extended list
of 26 emotion categories combined with the three common continuous dimensions
Valence, Arousal and Dominance. It consists of approximately 23,500 images
collected from websites, social media, and other public datasets. Each image
contains one or many people, and each person is labeled with Gender (Male or
Female), Age (adult, kid, or teenager), a bounding box (top-left, bottom-
right) of full body and VAD (valence-arousal-dominance) values ranging from 0
to 10 and labeled to some of 26 discrete emotion categories. The 26 discrete
emotion categories include: Peace, Affection, Esteem, Anticipation,
Engagement, Confidence, Happiness, Pleasure, Excitement, Surprise, Sympathy,
Doubt/Confusion, Disconnection, Fatigue, Embarrassment, Yearning, Disapproval,
Aversion, Annoyance, Anger, Sensitivity, Sadness, Disquietment, Fear, Pain,
Suffering.

In this paper, we conduct the experiment on both the 26 discrete emotion
categories and the VAD continuous emotion value. Figure 5 shows the
distribution of the number of people annotated across 26 emotion categories
and three continuous dimensions. The discrete distribution is uneven, with
Engagement being the most common (about 55%) and Embarrassment being the least
common (about 1%). While the VAD value is annotated quite evenly.

With a large number of people with various emotional states in the dataset,
deep learning models can learn to recognize more complex patterns and
relationships in the data, which can improve the accuracy and predictive
power. However, EMOTIC’s considerable imbalance (see Fig. 5) adversely impacts
the models’ learning performance for emotions with limited samples. As a
result, the models excel at recognizing emotions that have a higher number of
samples.

Figure 5.  Dataset Statistics. (a) Number of people annotated for each emotion
category; (b), (c) and (d) Number of people annotated for every value of the
three continuous dimensions : Valence, Arousal and Dominance (Kosti et al.,
2020).

###  4.3. Evaluate Metric

####  4.3.1. Discrete Prediction Evaluation Metric

The metric used to evaluate the prediction of the discrete results is mAP
(mean Average Precision). mAP is used to evaluate the model’s performance in
multi-class classification tasks. The higher mAP value, the better the model
achieves in object recognition.

The AP for an object class is calculated as follows:

(3) $AP=\sum_{n}(R_{n}-R_{n-1})\cdot P_{n},$

where where $P_{n}$ and $R_{n}$ are the precision and recall at the $n^{th}$
threshold. For inference, we chose the threshold for each class at which the
precision and recall are equal.

To calculate mAP, we calculate AP for each emotion category and take the
average of those AP values. The mAP is calculated as follows:

(4) $mAP=\frac{1}{N}\sum_{i=1}^{N}AP_{i},$

where $N$ is the number of categories. $AP_{i}$ is the AP value of this
discrete emotion prediction.

####  4.3.2. Continuous Prediction Evaluation Metric

For the continuous dimension emotion VAD, we evaluate the result by the mean
absolute error (MAE), which is the mean absolute error regression loss, which
one lower is the better. The MAE value for each dimension is calculated as
below:

(5) $MAE=\frac{1}{N}\sum_{i=1}^{N}(|\hat{y_{i}}-y_{i}|),$

where $\hat{y_{i}}$ denotes the prediction value for a dimension of the
$i^{th}$ image and $y_{i}$ is the true continuous value for that dimension of
the image number $i$. The MAE of N dimensions is then calculated as the mean
of N values of MEA of each dimension.

###  4.4. Experiment Results

Table 1. Discrete and continuous results using only body or context or facial
feature.

BACKBONE | mAP | MAE
---|---|---
Body Branch | Context Branch | Face Branch
| Resnet-18
---
(ImageNet weight)
- | - | 0.2376 | 0.9757
| Resnet-50
---
(ImageNet weight)
- | - | 0.2465 | 0.9606
| Resnet-50
---
(Fine-tunning on EMOTIC)
- | - | 0.2445 | 0.9590
| SwinT
---
(ImageNet weight)
- | - | 0.2671 | 0.9403
| SwinT
---
(Fine-tunning on EMOTIC)
- | - | 0.2666 | 0.9356
- |  | Resnet-18
---
(Places365 weight)
- | 0.2538 | 0.9673
- |  | Resnet-50
---
(Places365 weight)
- | 0.2598 | 0.9616
- | - | S-FER | 0.1840 | 1.0385
- | - | B-FER | 0.2061 | 1.0259

Table 2. Discrete and continuous results using a combination of body, context,
and facial features.

BACKBONE | mAP | MAE
---|---|---
Body Branch | Context Branch | Face Branch
| Resnet-18
---
(ImageNet weight)
| Resnet-18
---
(Places365 weight)
- | 0.2585 | 0.9615
| Resnet-50
---
(ImageNet weight)
Resnet-50 (Places365 weight) | - | 0.2645 | 0.9525
| Resnet-50
---
(Fine-tunning on EMOTIC)
- | 0.2640 | 0.9483
| SwinT
---
(ImageNet weight)
- | 0.2774 | 0.9411
| SwinT
---
(Fine-tunning on EMOTIC)
- | 0.2756 | 0.9422
- |  Resnet-18 (Places365 weight) | S-FER | 0.2590 | 0.9419
B-FER | 0.2615 | 0.9461
- |  Resnet-50 (Places365 weight) | S-FER | 0.2659 | 0.9435
B-FER | 0.2674 | 0.9471
| Resnet-18
---
(ImageNet weight)
- | S-FER | 0.2467 | 0.9545
| Resnet-50
---
(ImageNet weight)
- | 0.2554 | 0.9455

Table 3. Discrete and continuous results using body, context, and facial
features.

BACKBONE | mAP | MAE
---|---|---
Body Branch | Context Branch | Face Branch
Resnet-18 (ImageNet weight) |  Resnet-18 (Places365 weight) | S-FER | 0.2638 | 0.9491
B-FER | 0.2591 | 0.9479
Resnet50 (ImageNet weight) |  Resnet-50 (Places365 weight) | S-FER | 0.2689 | 0.9349
B-FER | 0.2726 | 0.9465
Resnet-50 (Fine-tuning on EMOTIC) |  Resnet-50 (Places365 weight) | S-FER | 0.2731 | 0.9337
B-FER | 0.2721 | 0.9478
SwinT (ImageNet weight) |  Resnet-50 (Places365 weight) | S-FER | 0.2821 | 0.9278
B-FER | 0.2828 | 0.9296
SwinT (Fine-tunning on EMOTIC) |  Resnet-50 (Places365 weight) | S-FER | 0.2837 | 0.9323
B-FER | 0.2829 | 0.9256

Figure 6. Visualization of our results. Images are chosen randomly with
corresponding ground-truth and predicted discrete and continuous emotion.

To demonstrate the effectiveness of our proposed method, we conducted
experiments using various backbones. Concretely, Body Feature Extraction
Branch used Resnet-18, Resnet-50, and SwinT; Context Feature Extraction Branch
used Resnet-18 and Resnet-50; and Face Feature Extraction Branch used S-FER
and B-FER. For each experiment, we performed predictions for both discrete and
continuous emotions. First, experiments were conducted using only one type of
image information to investigate the effectiveness of each backbone and the
amount of information contained in each type of image, as shown in Table 1.
Next, we used combined information from two out of three types of images to
demonstrate the effectiveness of combining more information, as well as to
provide a baseline for comparison with our proposed method, and verified the
effectiveness of the previously method (Kosti et al., 2020), as shown in Table
2. Finally, we tested our proposed model on all three types of information, as
shown in Table 3.

####  4.4.1. Discretes Emotion Prediction

Table 1 shows that using information from body image or context image results
in superior performance compared to using information from face image, due to
the lower quality of facial images used in our experiments. The Body Feature
Extraction Branch with the SwinT backbone achieved the highest mAP score of
0.2671. Despite this, the Context Feature Extraction Branch still performs
well with an mAP score above 0.25, offering more consistent results without
requiring a large model such as SwinT.

The results presented in Table 2 provide evidence that combining multiple
types of information yields better performance than using individual
information alone, resulting in an average increase of 11.2% in mAP score.
Additionally, stronger backbones were found to produce better results for each
experiment. These experiments also demonstrate that all three types of images
contain significant information for predicting emotions, as combining any two
types of images consistently improved performance. When using two types of
information, the method presented in (Kosti et al., 2020), which combines
information extracted from body image and context image, yields the best
results.

Combining information from all three types of images significantly improved
prediction results, with an mAP score above 0.28, using powerful models
including SwinT, Resnet-50, and FER models. Beside, combining different
backbones also proves effective, with an mAP score above 0.26. Compared to
using only two types of information, the results are improved by 4.1%.

####  4.4.2. Continous Emotion Prediction

The results for continuous emotion prediction followed the same patterns as
discrete emotion prediction. Specifically, MAE decreased by 3.5% when using
two types of information compared to using only one type, and decreased by
1.1% when using three types of information compared to using two types.
Moreover, our proposed model achieving the remarkable result with the lowest
MAE of 0.9256 by combining SwinT, Resnet-50, and B-FER. These findings
demonstrate the importance of combining multiple types of contextual
information and powerful models for improving emotion prediction.

####  4.4.3. Overall

Our proposed method outperformed the approach presented in (Kosti et al.,
2020), which utilizes a combination of information from body image and context
image, in both discrete and continuous emotion prediction. Specifically, the
mAP score increased by 2.3% and the MAE score decreased by 1.2%. Beside, our
study demonstrated that all three types of images (e.g., body image, context
image and face image) contain important information for emotion prediction.
Additionally, utilizing appropriate and stronger backbones leads to more
accurate results.

##  5\. Conclusion

In summary, this paper provides a Multiple -Branch Network that combines
information from the body, face, and context to enhance emotion prediction
from images. Our results on the EMOTIC dataset indicate that combining body,
face, and context images improves emotion prediction accuracy than just by
using one or two kinds of features. Our work also contributes by utilizing the
cutting-edge CNN models with pre-train weights on available datasets and
applying them to emotion prediction. We also provide further techniques to
optimize CNN’s accuracy for facial features extraction, such as denoising and
sharpening. Our proposed method achieved mAP scores above 0.26 and MAE scores
below 0.95. In additional, utilizing powerful deep learning models like SwinT,
ViT, etc., with our approach yields significant results. The Swin Transformer
backbone performed best with a mAP of 0.2837 and a MAE of 0.9256, but requires
more computational time due to its larger size.

Although we made efforts, this work has limitations. The inconsistent quality
of the dataset, which is the unequal in emotion distribution, could cause the
learning process into bias. Due to limited resources, we could only conduct
experiments on a limited number of models and datasets, which may lead to
incomplete or inaccurate evaluations. Moreover, the image processing and the
feature extraction techniques used were still in their early stages. Each
extraction model could be improved by training them to learn more in-depth
information relevant to their features, such as the body gestures for the body
branch or place attributes for the context branch, etc. Future research can
address these issues for better results.

###### Acknowledgements.

This research is supported by research funding from Faculty of Information
Technology, University of Science, Vietnam National University - Ho Chi Minh
City.
We would like to express our gratitude to Dr. Minh-Triet Tran for his valuable
suggestions and support that greatly contributed to the completion of this
research.

## References

  * (1)
  * balmukund (2022) balmukund. 2022\.  FER-2013 pytorch implementation.  https://www.kaggle.com/code/balmukund/fer-2013-pytorch-implementation?fbclid=IwAR3xaZrtY7-RDZiXHGcjf6ytJ5Nk4wMDxGhsQs0pg2R0ul7GNv7lgS3ePI8
  * Barrett (2017) Lisa Feldman Barrett. 2017\.  _How emotions are made: The secret life of the brain_.  Pan Macmillan.
  * Barrett et al. (2016) Lisa Feldman Barrett, Ralph Adolphs, Stacy Marsella, Aleix M Martinez, and Seth D Pollak. 2016\.  Emotional expressions reconsidered: Challenges to inferring emotion from human facial movements.  _Psychological Science in the Public Interest_ 17, 1 (2016), 1–68.
  * Barrett et al. (2011) Lisa Feldman Barrett, Batja Mesquita, and Maria Gendron. 2011\.  Context in emotion perception.  _Current directions in psychological science_ 20, 5 (2011), 286–290.
  * Behera et al. (2020) Ardhendu Behera, Peter Matthew, Alexander Keidel, Peter Vangorp, Hui Fang, and Susan Canning. 2020\.  Associating facial expressions and upper-body gestures with learning tasks for enhancing intelligent tutoring systems.  _International Journal of Artificial Intelligence in Education_ 30 (2020), 236–270.
  * Blog (2021) Google AI Blog. 2021\.  Understanding Contextual Facial Expressions Across the Globe.  https://ai.googleblog.com/2021/05/understanding-contextual-facial.html
  * Cowen and Keltner (2017) Alan S Cowen and Dacher Keltner. 2017.  Self-report captures 27 distinct categories of emotion bridged by continuous gradients.  _Proceedings of the national academy of sciences_ 114, 38 (2017), E7900–E7909.
  * Davitz (1964) Joel R Davitz. 1964\.  Expression of Emotion in Man and Animals.  _New York: McGraw-Hill_ (1964).
  * Deng et al. (2009) Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. 2009\.  Imagenet: A large-scale hierarchical image database. In _2009 IEEE conference on computer vision and pattern recognition_. Ieee, 248–255.
  * Dixon (2023) S. Dixon. 2023.  _Number of global social network users 2017-2027_.  https://www.statista.com/statistics/278414/number-of-worldwide-social-network-users/
  * Dong et al. (2021) Shi Dong, Ping Wang, and Khushnood Abbas. 2021.  A survey on deep learning and its applications.  _Computer Science Review_ 40 (2021), 100379.
  * Ekman et al. (1987) Paul Ekman, Wallace V Friesen, Maureen O’Sullivan, Arthur Chan, Irini Diacoyanni-Tarlatzis, Klaus Heider, Richard Krause, William A LeCompte, Tracy Pitcairn, and Paolo E Ricci-Bitti. 1987.  Universals and cultural differences in the judgments of facial expressions of emotion.  _Journal of Personality and Social Psychology_ 53, 4 (1987), 712–723.  https://doi.org/10.1037/0022-3514.53.4.712
  * Google (2021) Google. 2021.  Face Detection.  https://colab.research.google.com/github/dortmans/ml_notebooks/blob/master/face_detection.ipynb.  Accessed: 2023-12-09.
  * Haleem et al. (2022) Abid Haleem, Mohd Javaid, Mohd Asim Qadri, Ravi Pratap Singh, and Rajiv Suman. 2022.  Artificial intelligence (AI) applications for marketing: A literature-based study.  _International Journal of Intelligent Networks_ (2022).
  * Harmon-Jones et al. (2017) Eddie Harmon-Jones, Cindy Harmon-Jones, and Elizabeth Summerell. 2017\.  On the importance of both dimensional and discrete models of emotion.  _Behavioral Sciences_ 7, 4 (2017), 66.  https://doi.org/10.3390/bs7040066
  * He et al. (2016) Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. 2016\.  Deep residual learning for image recognition. In _Proceedings of the IEEE conference on computer vision and pattern recognition_. 770–778.
  * Horikawa et al. (2020) Tomoyasu Horikawa, Alan S Cowen, Dacher Keltner, and Yukiyasu Kamitani. 2020.  The neural representation of visually evoked emotion is high-dimensional, categorical, and distributed across transmodal brain regions.  _Iscience_ 23, 5 (2020), 101060.
  * Huddar et al. (2021) Mahesh G Huddar, Sanjeev S Sannakki, and Vijay S Rajpurohit. 2021\.  Attention-based multimodal contextual fusion for sentiment and emotion classification using bidirectional LSTM.  _Multimedia Tools and Applications_ 80, 9 (2021), 13059–13076.
  * Kosti et al. (2017) Ronak Kosti, Jose M Alvarez, Adria Recasens, and Agata Lapedriza. 2017.  EMOTIC: Emotions in Context dataset. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops_. 61–69.
  * Kosti et al. (2020) Ronak Kosti, Jose M. Alvarez, Adria Recasens, and Agata Lapedriza. 2020.  Context Based Emotion Recognition Using EMOTIC Dataset.  _IEEE Transactions on Pattern Analysis and Machine Intelligence_ 42, 11 (2020), 2755–2766.  https://doi.org/10.1109/TPAMI.2019.2916866
  * Liu et al. (2021) Ze Liu, Yutong Lin, Yue Cao, Han Hu, Yixuan Wei, Zheng Zhang, Stephen Lin, and Baining Guo. 2021.  Swin transformer: Hierarchical vision transformer using shifted windows. In _Proceedings of the IEEE/CVF international conference on computer vision_. 10012–10022.
  * Ma et al. (2021) Fuyan Ma, Bin Sun, and Shutao Li. 2021.  Facial expression recognition with visual transformers and attentional selective fusion.  _IEEE Transactions on Affective Computing_ (2021).
  * Marechal et al. (2019) Catherine Marechal, Dariusz Mikolajewski, Krzysztof Tyburek, Piotr Prokopowicz, Lamine Bougueroua, Corinne Ancourt, and Katarzyna Wegrzyn-Wolska. 2019.  Survey on AI-Based Multimodal Methods for Emotion Detection.  _High-performance modelling and simulation for big data applications_ 11400 (2019), 307–324.
  * Mitruț et al. (2019) Oana Mitruț, Gabriela Moise, Livia Petrescu, Alin Moldoveanu, Marius Leordeanu, and Florica Moldoveanu. 2019.  Emotion Classification Based on Biophysical Signals and Machine Learning Techniques.  _Symmetry_ 12 (12 2019), 21.  https://doi.org/10.3390/sym12010021
  * Mittal et al. (2020) Trisha Mittal, Pooja Guhan, Uttaran Bhattacharya, Rohan Chandra, Aniket Bera, and Dinesh Manocha. 2020.  EmotiCon: Context-Aware Multimodal Emotion Recognition using Frege’s Principle.  arXiv:2003.06692 [cs.AI]
  * NARAYANAN (2023) ARVIND NARAYANAN. 2023\.  _Understanding Social Media Recommendation Algorithms_.  https://knightcolumbia.org/content/understanding-social-media-recommendation-algorithms
  * Noroozi et al. (2018) Fatemeh Noroozi, Ciprian Adrian Corneanu, Dorota Kamińska, Tomasz Sapiński, Sergio Escalera, and Gholamreza Anbarjafari. 2018\.  Survey on emotional body gesture recognition.  _IEEE transactions on affective computing_ 12, 2 (2018), 505–523.
  * Radford et al. (2013) Alec Radford, Luke Metz, and Ilya Sutskever. 2013.  Facial Expression Recognition 2013 Dataset.  https://www.kaggle.com/datasets/msambare/fer2013
  * Savchenko (2021) Andrey V Savchenko. 2021\.  Facial expression and attributes recognition based on multi-task learning of lightweight neural networks. In _2021 IEEE 19th International Symposium on Intelligent Systems and Informatics (SISY)_. IEEE, 119–124.
  * Schindler et al. (2008) Konrad Schindler, Luc Van Gool, and Beatrice De Gelder. 2008\.  Recognizing emotions expressed by body pose: A biologically inspired neural model.  _Neural networks_ 21, 9 (2008), 1238–1246.
  * Shangeth (2022) Akash Shangeth. 2022\.  Facial Emotion Recognition PyTorch ONNX.  https://github.com/shangeth/Facial-Emotion-Recognition-PyTorch-ONNX
  * Tsvetkova (2023) Roza Tsvetkova. 2023\.  _99 Amazing Social Media Statistics and Facts_.  https://www.brandwatch.com/blog/amazing-social-media-statistics-and-facts/
  * Wen et al. (2023) Zhengyao Wen, Wenzhong Lin, Tao Wang, and Ge Xu. 2023\.  Distract your attention: multi-head cross attention network for facial expression recognition.  _Biomimetics_ 8, 2 (2023), 199.
  * Young et al. (1997) Andrew W Young, Duncan Rowland, Andrew J Calder, Nancy L Etcoff, Anil Seth, and David I Perrett. 1997.  Facial expression megamix: Tests of dimensional and category accounts of emotion recognition.  _Cognition_ 63, 3 (1997), 271–313.
  * Zhou et al. (2017) Bolei Zhou, Agata Lapedriza, Aditya Khosla, Aude Oliva, and Antonio Torralba. 2017.  Places: A 10 million Image Database for Scene Recognition.  _IEEE Transactions on Pattern Analysis and Machine Intelligence_ (2017).


# Chiral sum rule on the light front and the 3D image of the pion

Yang Li<EMAIL_ADDRESS>Department of Modern Physics, University of
Science and Technology of China, Hefei 230026, China  Pieter Maris  Department
of Physics and Astronomy, Iowa State University, Ames, IA 50011  James P. Vary
Department of Physics and Astronomy, Iowa State University, Ames, IA 50011

###### Abstract

The lightest meson, the pion, features two faces – one is the elementary
Goldstone boson of QCD and the other is the structured bound state of quarks
and gluons. To accommodate both in a single light-front wave function in the
valence space, we obtain a sum rule by analyzing the conserved axial-vector
current and the general structures of the wave functions. Using an analytic
model motivated by holography, we show this sum rule is consistent with
requirements of chiral symmetry breaking in AdS/QCD. Within this model, we
find a remarkable feature of the pion, namely that the density is mostly
uniform inside its radius; furthermore, we obtain good agreement with the
experimental pion form factor at spacelike momenta.

##  I Introduction

It is well known that the pion, the lightest meson, serves a dual role in
Nature. On the one hand, it is the Goldstone boson of the spontaneously broken
chiral symmetry [1, 2], as described in detail by the chiral effective field
theory [3]. On the other hand, it is a composite particle – like all other
hadrons – quarks and gluons bound together by the strong force. Perturbative
QCD analysis of hard exclusive processes firmly establishes dominant
contributions in the pion to the valence (quark and antiquark) Fock sector
[4].

How does the structure of the pion accommodate these seemingly opposing
perspectives? Since confinement and chiral symmetry breaking are the two most
fundamental features of QCD in the non-perturbative regime, the understanding
of the strong interaction entails an answer to this question.

Quark confinement from lattice gauge theory (LGT) is based on simulation of
pure gauge theory with infinitely heavy quarks [5]. For chiral fermions,
lattice simulations are computationally formidable, and no simple picture has
emerged for the inner structure of the pion as a Goldstone boson [6]. Careful
analysis of the axial-vector Ward-Takahashi identity (AVWTI) in Dyson-
Schwinger equations (DSEs) shows that the dominant (pseudoscalar) component of
the pion Bethe-Salpeter amplitude is equal to the dynamically generated scalar
quark self-energy [7]; in addition, the AVWTI provides rigorous constraints on
the other covariant structures of the pion as well. Indeed, models based on
this analysis are very successful in describing light meson properties [8].
However, the access to the partonic structure of the pion is still indirect,
since the DSEs (as well as LGT) are defined in the Euclidean spacetime. An
interesting perspective is provided by AdS/QCD [9]. In this model, the two
faces of the pion, confinement and chiral symmetry breaking, are encoded in
the large-$z$ and small-$z$ behaviors in terms of the fifth dimension $z$,
respectively, which is thought of as the (inverse) renormalization scale [10].

Our modern understanding of the hadron structure is based on the parton
picture, which corresponds to how the pion is “seen” in high-energy scattering
experiments [12, 11, 13]. In particular, the partonic structure of the pion is
unambiguously encoded in its wave functions defined at fixed light-front time
$x^{+}=t+z/c$. These light-front wave functions (LFWFs) are amplitude-
probabilistic, Minkowskian and frame independent. There have been major
efforts to extract these quantities from Euclidean quantum field theories,
e.g. LGT and DSEs [14, 15, 16, 18, 17]. On the other hand, in principle, the
LFWFs can be directly obtained from diagonalizing the QCD Hamiltonian on a
null plane $\omega\cdot x=0$ with $\omega_{\mu}\omega^{\mu}=0$ [12]. This
approach has recently been revived to solve quantum field theories in lower
dimensions [19].

Quark confinement entails confining interactions between partons at large
separations in the pion. Then the question becomes: where is the chiral
symmetry breaking facet in the pion wave functions? Inspired by the exact
relations from the covariant analysis of the AVWTI in DSEs, we show there is
also a relation, a chiral sum rule, on the pion LFWFs. The sum rule developed
below reads,

$\int\frac{{\mathrm{d}}x}{\sqrt{x^{3}(1-x)^{3}}}\int\frac{{\mathrm{d}}^{2}k_{\perp}}{(2\pi)^{3}}k_{\perp}^{2}\psi_{\pi}(x,\vec{k}_{\perp})=0,$
(1)

in the chiral limit ($m_{q}\to 0$), where $\psi_{\pi}(x,\vec{k}_{\perp})$ is
the pion spin-singlet valence sector LFWF which depends on the relative
momenta $(x,\vec{k}_{\perp})$ between the quark and the antiquark (see
definition below). We further show, in the context of light-front holography
[20], this sum rule is equivalent to the chiral condensate in AdS/QCD.

Note that here we neglected the difference in the bare (current) $u,d$ quark
masses and consider the unflavored case with $m_{q}=\frac{1}{2}(m_{u}+m_{d})$.
We defer the fully consistent treatment of the UV including the
renormalization to future work and focus on the non-perturbative part of the
physics, confinement and chiral symmetry breaking. Therefore, we have
suppressed the explicit dependence on the renormalization scheme and scale.

##  II Formalism

The key point for obtaining this sum rule (1) is the covariant decomposition
of the pion LFWFs. In Fock space, the state vector of a pseudoscalar $P$ can
be decomposed as,

$|P(p)\rangle=\sum_{s,\bar{s}}\int\frac{{\mathrm{d}}x}{2x(1-x)}\int\frac{{\mathrm{d}}^{2}k_{\perp}}{(2\pi)^{3}}\psi_{s\bar{s}/P}(x,\vec{k}_{\perp})\\\
\times\frac{1}{\sqrt{N_{C}}}\sum_{i}b^{\dagger}_{si}(p_{1})d^{\dagger}_{\bar{s}i}(p_{2})|0\rangle+\cdots$
(2)

where the ellipsis represents the higher Fock sector wave functions. Here
$N_{C}=3$ is the number of colors. $p_{1}$ and $p_{2}$ are the quark and
antiquark momenta, respectively. We adopt the standard light-front
coordinates, where the components of a 4-vector $v$ are written as
$v^{\pm}=v^{0}\pm v^{3}$ and $\vec{v}_{\perp}=(v^{1},v^{2})$, corresponding to
$\omega^{-}=2$, $\omega^{+}=\omega_{\perp}=0$ [12]. The longitudinal momentum
fraction of the quark (antiquark) is $x=p_{1}^{+}/p^{+}$
($1-x=p_{2}^{+}/p^{+}$). The relative transverse momentum
$\vec{k}_{\perp}=\vec{p}_{1\perp}-x\vec{p}_{\perp}$. Momentum conservation
implies, $p^{+}=p^{+}_{1}+p_{2}^{+}$ and
$\vec{p}_{\perp}=\vec{p}_{1\perp}+\vec{p}_{2\perp}$. The most general form of
the pion valence ($q\bar{q}$) LFWF $\psi_{s\bar{s}/P}(x,\vec{k}_{\perp})$ that
satisfies all kinematical symmetries on the light front is given by the
covariant light-front dynamics (CLFD) [21],

$\psi_{s\bar{s}/P}(x,\vec{k}_{\perp})=\bar{u}_{s}(p_{1})\Big{[}\gamma_{5}\phi_{1}(x,k_{\perp})\\\
+\frac{\gamma_{5}\not{\omega}}{\omega\cdot
p}\hat{f}_{\chi}\phi_{2}(x,k_{\perp})\Big{]}v_{\bar{s}}(p_{2}).$ (3)

It depends on the orientation of the quantization surface (the light front),
as indicated by the null vector $\omega$. Note that the null vector $\omega$
is only defined up to a scaling factor. Therefore, the conformal invariant
structure $\not{\omega}/(\omega\cdot p)$ is mandatory. As a result, we need to
introduce a constant $\hat{f}_{\chi}$ with mass dimension to balance the
dimension. In CLFD, this constant is usually taken as the quark mass $m_{q}$
(or the pion mass $M_{P}$) which vanishes in the chiral limit [22, 23, 21,
24]. We instead take a constant non-vanishing in the chiral limit. Later, it
will be shown that a convenient choice is the pion decay constant. Eq. (3) can
be written explicitly as,

$\displaystyle\psi_{\uparrow\uparrow/P}(x,\vec{k}_{\perp})=\,$
$\displaystyle\psi^{*}_{\downarrow\downarrow/P}(x,\vec{k}_{\perp})$ (4)
$\displaystyle=-\frac{k_{\perp}e^{-i\arg\vec{k}_{\perp}}}{\sqrt{x(1-x)}}\phi_{1}(x,k_{\perp});$
$\displaystyle\psi_{\uparrow\downarrow-\downarrow\uparrow/P}(x,\vec{k}_{\perp})=\,$
$\displaystyle\frac{\sqrt{2}m_{q}}{\sqrt{x(1-x)}}\phi_{1}(x,k_{\perp})$ (5)
$\displaystyle-\hat{f}_{\chi}\sqrt{8x(1-x)}\phi_{2}(x,k_{\perp});$

where
$\psi_{\uparrow\downarrow-\downarrow\uparrow/P}=\big{[}\psi_{\uparrow\downarrow/P}-\psi_{\downarrow\uparrow/P}\big{]}/\sqrt{2}$.

##  III Chiral sum rule

The sum rule (1) follows directly from examining the partially conserved
axial-vector current (PCAC) [25],

$\partial_{\mu}J^{\mu}_{5}=2im_{q}\overline{q}\gamma_{5}q.$ (6)

Here, $J^{\mu}_{5}=\overline{q}\gamma^{\mu}\gamma_{5}q$ is the axial-vector
current, and $J_{5}=\bar{q}i\gamma_{5}q$ is known as the pseudoscalar current.
We have neglected the chiral anomaly term, which is irrelevant to our
discussion here. Consider the local vacuum-to-pseudoscalar matrix elements of
the operators,

$\displaystyle\langle 0|J^{\mu}_{5}(x)|P(p)\rangle=e^{-ip\cdot
x}ip^{\mu}f_{P},$ (7) $\displaystyle\langle 0|J_{5}(x)|P(p)\rangle=e^{-ip\cdot
x}g_{P}.$ (8)

The constant $f_{P}$ is known as the decay constant. Applying the PCAC (6), we
obtain the Gell-Mann-Oakes-Renner (GMOR) relation [26]:

$M_{P}^{2}f_{P}=2m_{q}g_{P},$ (9)

where $M_{P}$ is the mass of the pseudoscalar, $M_{P}^{2}=p^{2}$. In the LFWF
representation, these local matrix elements are,

$\displaystyle\langle 0|J^{\mu}_{5}(0)|P(p)\rangle=\,$
$\displaystyle\sqrt{N_{C}}\sum_{s,\bar{s}}\int\frac{{\mathrm{d}}x}{2x(1-x)}\int\frac{{\mathrm{d}}^{2}k_{\perp}}{(2\pi)^{3}}$
$\displaystyle\times\,$
$\displaystyle\psi_{s\bar{s}/P}(x,\vec{k}_{\perp})\bar{v}_{\bar{s}}(p_{2})\gamma^{\mu}\gamma_{5}u_{s}(p_{1}),$
(10) $\displaystyle\langle 0|J_{5}(0)|P(p)\rangle=\,$
$\displaystyle\sqrt{N_{C}}\sum_{s,\bar{s}}\int\frac{{\mathrm{d}}x}{2x(1-x)}\int\frac{{\mathrm{d}}^{2}k_{\perp}}{(2\pi)^{3}}$
$\displaystyle\times\,$
$\displaystyle\psi_{s\bar{s}/P}(x,\vec{k}_{\perp})\bar{v}_{\bar{s}}(p_{2})i\gamma_{5}u_{s}(p_{1}).$
(11)

Note that these expressions are exact, viz. the vacuum-to-pion matrix elements
only depend on the valence wave function. Applying the PCAC (6) and the
covariant decomposition of the LFWF (3), we obtain,

$\int\frac{{\mathrm{d}}x}{2x(1-x)}\int\frac{{\mathrm{d}}^{2}k_{\perp}}{(2\pi)^{3}}\Big{(}\frac{k_{\perp}^{2}+m_{q}^{2}}{x(1-x)}-M_{P}^{2}\Big{)}\\\
\times\Big{(}2m_{q}\phi_{1}(x,k_{\perp})-4x(1-x)\hat{f}_{\chi}\phi_{2}(x,k_{\perp})\Big{)}=0.$
(12)

In the chiral limit $m_{q}\to 0$, this expression reduces to,

$\int\frac{{\mathrm{d}}x}{2x(1-x)}\int\frac{{\mathrm{d}}^{2}k_{\perp}}{(2\pi)^{3}}k_{\perp}^{2}\phi_{2}^{(0)}(x,k_{\perp})=0,$
(13)

due to the GMOR relation (9), where we have used the superscript (0) to
indicate quantities in the chiral limit. From Eq. (5), the leading-twist
($L_{z}=0$) wave function
$\psi_{\pi}\equiv\psi^{(0)}_{\uparrow\downarrow-\downarrow\uparrow}\propto\sqrt{x(1-x)}\phi_{2}^{(0)}$
in the chiral limit. Hence, we obtain the chiral sum rule Eq. (1).

In the above derivation, there is an explicit assumption that $\hat{f}_{\chi}$
does not vanish in the chiral limit. If, instead, we take $\hat{f}_{\chi}\to
0$ as usually done in CLFD (cf. Ref. [27]), Eq. (12) will be automatically
satisfied in the chiral limit – for arbitrary pion LFWF. However, the pion
decay constant (16) also vanishes, and the pion mass is not required to be
zero (9). Indeed, this is the case when the chiral symmetry is not
spontaneously broken, and the pion is not a Goldstone boson regardless of its
mass.

In the vicinity of the chiral limit, we can expand the wave functions and
related quantities in terms of the quark mass $m_{q}$,

$\phi_{1,2}=\phi_{1,2}^{(0)}+m_{q}\phi_{1,2}^{(1)}+\cdots.$ (14)

Substituting this expansion into Eq. (12) and taking terms up to
$\mathrm{O}(m_{q})$, we obtain,

$-2m_{q}\int\frac{{\mathrm{d}}x}{2x(1-x)}\int\frac{{\mathrm{d}}^{2}k_{\perp}}{(2\pi)^{3}}\frac{2k_{\perp}^{2}}{x(1-x)}\phi_{1}^{(0)}(x,k_{\perp})\\\
=M_{P}^{2}4\hat{f}_{\chi}\int{\mathrm{d}}x\int\frac{{\mathrm{d}}^{2}k_{\perp}}{(2\pi)^{3}}\phi_{2}^{(0)}(x,k_{\perp}).$
(15)

The r.h.s. is just the wave function representation of the pseudoscalar decay
constant in the chiral limit [4]:

$f_{P}^{(0)}=4i\hat{f}_{\chi}\sqrt{N_{C}}\int{\mathrm{d}}x\int\frac{{\mathrm{d}}^{2}k_{\perp}}{(2\pi)^{3}}\phi_{2}^{(0)}(x,k_{\perp}),$
(16)

where the decay constant is extracted from the local matrix element $\langle
0|J^{+}_{5}|P(p)\rangle\equiv ip^{+}f_{P}$. Similarly, the l.h.s. is the wave
function representation of the pseudoscalar amplitude in the chiral limit [4]:

$g_{P}^{(0)}=-2i\sqrt{N_{C}}\int\frac{{\mathrm{d}}x}{2x(1-x)}\int\frac{{\mathrm{d}}^{2}k_{\perp}}{(2\pi)^{3}}\\\
\times\frac{k_{\perp}^{2}}{x(1-x)}\phi_{1}^{(0)}(x,k_{\perp}).$ (17)

Therefore, Eq. (15) is consistent with the GMOR relation (9),

$M_{P}^{2}f_{P}^{(0)}=2m_{q}g_{P}^{(0)}+\mathrm{O}(m_{q}^{2}).$ (18)

Note that the above relation also implies that the decay constants of the
excited pions vanish in the chiral limit [7].

##  IV Light-front holography

The effect of the chiral sum rule can be better pictured in a simple analytic
model. In the chiral limit, the light-front kinetic energy depends only on a
transverse vector $\vec{\kappa}_{\perp}=\vec{k}_{\perp}/\sqrt{x(1-x)}$. In
light-front holography (LFH), a semi-classical approach to QCD [20], it is
assumed that the dynamics in the chiral limit is solely dictated by the
transverse direction, and the wave function is separable [28, 29, 30, 31, 32,
33]. Taking this ansatz, we can rewrite $\phi_{2}$ for a pseudoscalar $P$ as,

$\phi_{2,P}(x,k_{\perp})=\frac{\pi}{2\sqrt{N_{C}}}\frac{\phi_{P}(x)}{x(1-x)f_{\pi}}\varphi_{P}(\kappa_{\perp}).$
(19)

Here, $\phi_{P}(x)\propto\sqrt{x(1-x)}$ is the light-cone distribution
amplitude with $\int{\mathrm{d}}x\,\phi_{P}(x)=f_{P}$.
$\varphi_{P}(\kappa_{\perp})$ is the transverse function, whose Fourier
transform,

$\widetilde{\varphi}_{P}(\zeta_{\perp})=\int\frac{{\mathrm{d}}^{2}\kappa_{\perp}}{(2\pi)^{2}}e^{i\vec{\kappa}_{\perp}\cdot\vec{\zeta}_{\perp}}\varphi_{P}(\kappa_{\perp}),$
(20)

where $\vec{\zeta}_{\perp}=\sqrt{x(1-x)}\vec{r}_{\perp}$. The chiral sum rule
(13) implies,

$f_{P}\nabla^{2}_{\perp}\widetilde{\varphi}_{P}(\zeta_{\perp}=0)=0.$ (21)

N.B. this relation is applicable to both the ground state, i.e. the pion, and
the excited states, i.e. excited pions. For the pion,
$\nabla^{2}_{\perp}\widetilde{\varphi}_{P}(\zeta_{\perp}=0)=0$ is required.
For the excited pions, the decay constants vanish in the chiral limit
$f_{P}=0$, as required by the GMOR relation (9) [7].

In LFH, the transverse wave function satisfies the semi-classical light-front
Schrödinger wave equation,

$\Big{[}-\nabla^{2}_{\perp}+U(\zeta_{\perp})\Big{]}\widetilde{\varphi}_{P}(\zeta_{\perp})=M^{2}_{P}\widetilde{\varphi}_{P}(\zeta_{\perp}).$
(22)

The sum rule (21) leads to $U(\zeta_{\perp}=0)=0$. It was shown that Eq. (22)
can be identified with the string equation of motion in the fifth dimension in
AdS/QCD if $\zeta_{\perp}$ is identified with the fifth coordinate $z$ in AdS5
[34]. The effective potential $U=(1/4)\Phi^{\prime
2}-(1/2)\Phi^{\prime\prime}-({3}/{2z})\Phi^{\prime}$ is related to the dilaton
profile $\Phi(z)$ introduced in AdS/QCD to generate confinement by breaking
the conformal invariance [34, 9]. Phenomenologically, the Regge scaling
requires $\Phi(z\to\infty)\to z^{2}$ [35]. The chiral condensate
$\Sigma=\langle\bar{q}q\rangle$ is encoded in the background scalar field $X$
at the UV boundary ($z=0$), viz. $\langle X(z\to 0)\rangle\to\frac{1}{2}\Sigma
z^{3}$ in the chiral limit [36]. The former produces a quadratic confining
potential, $U(\zeta_{\perp}\to\infty)\sim\zeta_{\perp}^{2}$ in the IR, while
the latter requires $U(\zeta_{\perp}\to 0)\sim-\Sigma^{2}\zeta^{4}_{\perp}$ in
the UV, consistent with the chiral sum rule (21).

The chiral sum rule does not uniquely determine the effective potential.
Nevertheless, it can be shown that such potentials have the shape of a
sombrero (see Fig. 1). From the AdS/QCD perspective, this shape is inherited
from the Higgs potential through the condensate [10]. The exact shape is
dictated by QCD. Instead of solving for such a potential, we can construct it
from some model pion wave function,
$U=\nabla_{\perp}^{2}\widetilde{\varphi}_{\pi}/\widetilde{\varphi}_{\pi}$. For
example, the following harmonic oscillator wave function satisfies the chiral
sum rule,

$\widetilde{\varphi}_{\pi}(\zeta_{\perp})=\Big{[}1+\frac{1}{2}(\kappa\zeta_{\perp})^{2}+\frac{c}{8}(\kappa\zeta_{\perp})^{4}\Big{]}e^{-\frac{1}{2}(\kappa\zeta_{\perp})^{2}},$
(23)

with a chiral condensate $\Sigma\sim(0.3\,\text{GeV})^{3}$. Here, $\kappa\sim
0.5\,\text{GeV}$ is the confining strength, related to the Regge slope, and
$0<c<2$ is a dimensionless free parameter unconstrained by the chiral sum
rule. Here we chose $c=0.3$ to obtain a good fit to the pion form factor. The
corresponding potential is in Padé form,

$U(\zeta_{\perp})=\kappa^{2}\frac{(\kappa\zeta_{\perp})^{2}\big{[}0.3(\kappa\zeta_{\perp})^{4}+(\kappa\zeta_{\perp})^{2}-11.2\big{]}}{0.3(\kappa\zeta_{\perp})^{4}+4(\kappa\zeta_{\perp})^{2}+8}.$
(24)

This is compared with the pion wave function and effective potential from
soft-wall AdS/QCD without implementing chiral symmetry breaking [34, 35],

$\displaystyle\widetilde{\varphi}_{\pi}^{\textsc{sw}}(\zeta_{\perp})=e^{-\frac{1}{2}(\kappa\zeta_{\perp})^{2}},$
(25) $\displaystyle
U_{\textsc{sw}}(\zeta_{\perp})=\kappa^{4}\zeta^{2}_{\perp}-2\kappa^{2}$ (26)

Figure 1 compares the wave function (23) with the Gaussian wave function (25)
obtained from the pure quadratic potential (26). The former has a distinctive
plateau as required by the chiral sum rule. Of course, our model can be
improved to accommodate better meson phenomenology [10, 11].

(a) wo. $\chi$SB: quadratic potential and LFWF Eq. (25)

(b) w. $\chi$SB: sombrero potential and LFWF Eq. (23)

Figure 1: (Colors online) Comparison of the pion wave functions Eq. (25)
obtained from a purely quadratic potential (26) without implementing chiral
symmetry breaking ($\chi$SB) (Top panel) and Eq. (23) obtained from a sombrero
potential (24) that implements $\chi$SB (Bottom panel). The plateau of the
bottom-panel wave function Eq. (23) near $\zeta_{\perp}=0$ is required by the
chiral sum rule (21). This feature resembles a structureless particle up to
the confinement scale.

In LFH, the pion valence LFWF [cf. (19)],

$\psi_{\pi}(x,\vec{k}_{\perp})=\chi(x)\varphi_{\pi}(\kappa_{\perp}).$ (27)

Chiral symmetry breaking with the ’t Hooft mechanism suggests a longitudinal
function $\chi(x)=\phi_{\pi}(x)/\sqrt{x(1-x)}\to\text{const.}$ [28], resulting
in an infinitely long pion [37]. The valence density,
$\widetilde{\rho}(x,\vec{\zeta}_{\perp})=N|\widetilde{\varphi}_{\pi}(\zeta_{\perp})|^{2}$,
then also acquires a plateau in the transverse direction
($\vec{\zeta}_{\perp}$) due to the chiral sum rule (21). Overall, the partons
appear uniformly distributed over the central region of the pion until
reaching the confining scale $\zeta_{\perp}\sim\kappa^{-1}$.

To explore the full 3D structure of the pion, we follow Miller and Brodsky to
introduce the coordinate-space pion LFWF as a 3D Fourier transform of the pion
LFWF [38],

$\widetilde{\psi}_{\pi}(\vec{r}_{\perp},\tilde{z})=\int\frac{{\mathrm{d}}x}{2\sqrt{x(1-x)}}\int\frac{{\mathrm{d}}^{2}k_{\perp}}{(2\pi)^{3}}\\\
\times
e^{i(x\tilde{z}-\vec{k}_{\perp}\cdot\vec{r}_{\perp})}\psi_{\pi}(x,\vec{k}_{\perp}).$
(28)

The third coordinate, $\tilde{z}=(1/2)p^{+}_{\pi}x^{-}$, is also known as the
Ioffe time. The 3-dimensional pion LFWFs in the coordinate space from these
two closely related AdS/QCD models Eqs. (23, 25) with and without the chiral
symmetry breaking, respectively, are compared in Fig. 2. When the chiral
symmetry breaking is taken into account in full 3 dimensions, the pion appears
as a uniform disk with long diffractive shadows, a unique impression for
subatomic particles yet common for laser beams.

Figure 2: (Colors online) Comparison of the 3D coordinate space pion valence
wave functions with (Left panel) and without (Right panel) incorporating the
chiral symmetry breaking. These wave functions are obtained from the 3D
Fourier transform (28) of the LFWFs Eqs. (23) and (25), respectively.
$\vec{r}_{\perp}=(x,y)$ are the relative transverse separation of the quark
and antiquark, $\tilde{z}$ is the Ioffe time of Miller and Brodsky. $\kappa$
is the strength of the confining potential. The pion appears infinitely long
in the longitudinal direction ($\tilde{z}$). The pion with the chiral symmetry
breaking (left) shows a signature plateau in the transverse direction, also as
depicted by the 3D cartoons at the bottom corners.

Figure 3 compares the pion electromagnetic form factors obtained using our
wave function (23) and the soft-wall AdS/QCD wave function (25). We adopt the
soft-wall holographic current following Ref. [39]. The results are compared
with the experimental measurements by CERN [40] and by the Jefferson Lab[41,
42], including a reanalysis of the DESY data [43]. Our results implementing
the $\chi$SB improve the pion form factor at large $Q^{2}$ as expected.

Figure 3: (Colors online) The pion electromagnetic form factor obtained using
wave function (23) with $\chi$SB and using the soft-wall AdS/QCD wave function
(25) without implementing $\chi$SB. For comparison we also show predictions
[44, 45] from the DSEs, which are frame-independent and consistent with
current conservation and $\chi$SB, as well as the experimental measurements
[40, 41, 42, 43] .

##  V Summary and discussions

Starting from the axial-vector current conservation and the general structures
of the LFWF, we derived a sum rule for the valence wave function of the pion.
This relation implies that the two faces of the pion, a quark-antiquark bound
state and a Goldstone boson, are encoded in its inner structure at the large
and small parton separation, respectively. This real-space image is consistent
with the picture from AdS/QCD in five dimensions. This picture is physically
reasonable, since the chiral breaking scale $\Lambda_{\chi}\sim 1\,\text{GeV}$
is larger than the confinement scale $\Lambda_{\textsc{qcd}}\sim
0.3\text{GeV}$.

Using a concrete analytic model, we showed that the valence partons are
uniformly distributed over the central region of the pion. In other words, the
pion is approximately a flat disk on the transverse plane with a Gaussian
edge. Therefore, in this analytic model, the pion is remarkably simple and
uniform while encoding both of its desired facets described at the outset.

The use of the general Lorentz structure of the LFWF is essential to obtain
this sum rule. In particular, the sum rule constrains the part of wave
function associated with the Lorentz structure $\gamma^{+}\gamma_{5}/p^{+}$.
Since this part does not appear to be covariant, it is often neglected in the
literature (see, e.g., Refs. [23, 27]). In fact, it can be written in a more
covariant form: $\not{p}\gamma_{5}$. The emergence of this term signifies the
spontaneous chiral symmetry breaking. It is also related to the leading-twist
pion wave function as well as the associated bi-local matrix element [46],

$\langle
0|\overline{q}(+{\frac{1}{2}}z)\gamma^{+}\gamma_{5}q(-{\frac{1}{2}}z)|P(p)\rangle_{z^{+}=0}\\\
=2\sqrt{2N_{C}}p^{+}\widetilde{\psi}_{\pi}(\vec{z}_{\perp},\tilde{z}).$ (29)

Here, $\tilde{z}=p\cdot z$ is a generalization of the Ioffe time of Miller and
Brodsky [38]. Only recently, it was realized that this part is important for
modeling the pion [24, 47, 48].

A proper renormalization is needed for the bare quark mass $m_{q}$ and the
local operators, e.g. $J^{\mu}_{5}$. The actual implementation of the
renormalization is beyond the scope of this work. However, it is important for
establishing a full picture of the pion, as well as for practical ab initio or
model calculations. Thanks to asymptotic freedom, the UV behavior of the pion
can be analyzed in perturbation theory, which very useful for the UV
renormalization. [49, 4, 8, 50, 51, 52].

The next important issue is the dynamical mass generation. In DSEs, the
dynamical quark mass (self-energy) is constrained by the AVWTI [7]. How the
AVWTI is realized on the light front is an interesting question. The problem
is that the light-front axial charge $Q_{5}$ does not create the pion pole
[53, 54]. It is likely that the proper relation that constrains the dynamical
mass generation on the light front comes from additional light-front chiral
current algebras [54].

There is a long-standing myth that the vacuum in light-front QCD is trivial
and hence the pion on the light front is not a Nambu-Goldstone boson. This
myth has been debunked many times (see, e.g. Ref. [54] for a recent review).
The key point is that the chiral condensate on the light front is non-local,
viz. $\langle
0|\bar{q}q|0\rangle=\frac{1}{2}m_{q}\big{\langle}0\big{|}\overline{q}{{\frac{\gamma^{+}}{\tensor{\partial}^{+}}}}q\big{|}0\big{\rangle}\neq
0$ in the chiral limit ($m_{q}\to 0$). However, as Refs. [55, 56] pointed out,
the relevant question is how the chiral condensate can be understood in terms
of the pion structure expressed in its light-front wave functions. We hope the
Janus-like character of the pion sketched here provides a pathway to an
answer.

## Acknowledgements

Y.L. thanks V.A. Karmanov for enlightening discussions on the covariant
structures of the light-front wave functions, and thanks L. Zhang for
conversations on the chiral symmetry breaking in AdS/QCD. This work was
supported in part by the US Department of Energy (DOE) under Grant Nos. DE-
FG02-87ER40371, DE-SC0018223 (SciDAC-4/NUCLEI), and DE-SC0023495
(SciDAC-5/NUCLEI). Y.L. is supported by the New faculty start-up fund of the
University of Science and Technology of China.

## References

  * [1] Y. Nambu and G. Jona-Lasinio, “Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. 1.,” Phys. Rev. 122, 345-358 (1961); 124, 246-254 (1961)
  * [2] J. Goldstone, A. Salam and S. Weinberg, “Broken Symmetries,” Phys. Rev. 127, 965-970 (1962) doi:10.1103/PhysRev.127.965
  * [3] S. Weinberg, “Phenomenological Lagrangians,” Physica A 96, no.1-2, 327-340 (1979) doi:10.1016/0378-4371(79)90223-1
  * [4] G. P. Lepage and S. J. Brodsky, “Exclusive Processes in Perturbative Quantum Chromodynamics,” Phys. Rev. D 22, 2157 (1980) doi:10.1103/PhysRevD.22.2157
  * [5] K. G. Wilson, “Confinement of Quarks,” Phys. Rev. D 10, 2445-2459 (1974) doi:10.1103/PhysRevD.10.2445
  * [6] Y. Aoki, T. Blum, G. Colangelo, S. Collins, M. Della Morte, P. Dimopoulos, S. Dürr, X. Feng, H. Fukaya and M. Golterman, et al. “FLAG Review 2021,” [arXiv:2111.09849 [hep-lat]].
  * [7] P. Maris, C. D. Roberts and P. C. Tandy, “Pion mass and decay constant,” Phys. Lett. B 420, 267-273 (1998) doi:10.1016/S0370-2693(97)01535-9 [arXiv:nucl-th/9707003 [nucl-th]].
  * [8] P. Maris and C. D. Roberts, “Dyson-Schwinger equations: A Tool for hadron physics,” Int. J. Mod. Phys. E 12, 297-365 (2003) doi:10.1142/S0218301303001326 [arXiv:nucl-th/0301049 [nucl-th]].
  * [9] J. Erlich, E. Katz, D. T. Son and M. A. Stephanov, “QCD and a holographic model of hadrons,” Phys. Rev. Lett. 95, 261602 (2005) doi:10.1103/PhysRevLett.95.261602 [arXiv:hep-ph/0501128 [hep-ph]].
  * [10] T. Gherghetta, J. I. Kapusta and T. M. Kelley, “Chiral symmetry breaking in the soft-wall AdS/QCD model,” Phys. Rev. D 79, 076003 (2009) doi:10.1103/PhysRevD.79.076003 [arXiv:0902.1998 [hep-ph]].
  * [11] E. M. Aitala et al. [E791], Phys. Rev. Lett. 86, 4768-4772 (2001) doi:10.1103/PhysRevLett.86.4768 [arXiv:hep-ex/0010043 [hep-ex]].
  * [12] S. J. Brodsky, H. C. Pauli and S. S. Pinsky, “Quantum chromodynamics and other field theories on the light cone,” Phys. Rept. 301, 299-486 (1998) doi:10.1016/S0370-1573(97)00089-6 [arXiv:hep-ph/9705477 [hep-ph]].
  * [13] D. Ashery and H. C. Pauli, Eur. Phys. J. C 28, 329-333 (2003) doi:10.1140/epjc/s2003-01187-3 [arXiv:hep-ph/0301113 [hep-ph]].
  * [14] L. Chang, I. C. Cloet, J. J. Cobos-Martinez, C. D. Roberts, S. M. Schmidt and P. C. Tandy, “Imaging dynamical chiral symmetry breaking: pion wave function on the light front,” Phys. Rev. Lett. 110, no.13, 132001 (2013) doi:10.1103/PhysRevLett.110.132001 [arXiv:1301.0324 [nucl-th]].
  * [15] C. Shi and I. C. Cloët, “Intrinsic Transverse Motion of the Pion’s Valence Quarks,” Phys. Rev. Lett. 122, no.8, 082301 (2019) doi:10.1103/PhysRevLett.122.082301 [arXiv:1806.04799 [nucl-th]].
  * [16] X. Ji, Y. S. Liu, Y. Liu, J. H. Zhang and Y. Zhao, “Large-momentum effective theory,” Rev. Mod. Phys. 93, no.3, 035005 (2021) doi:10.1103/RevModPhys.93.035005 [arXiv:2004.03543 [hep-ph]].
  * [17] W. de Paula, E. Ydrefors, J. H. A. Nogueira, T. Frederico and G. Salmè, “The parton distribution function in a pion with Minkowskian dynamics,” [arXiv:2203.07106 [hep-ph]].
  * [18] C. D. Roberts, D. G. Richards, T. Horn and L. Chang, “Insights into the emergence of mass from studies of pion and kaon structure,” Prog. Part. Nucl. Phys. 120, 103883 (2021) doi:10.1016/j.ppnp.2021.103883 [arXiv:2102.01765 [hep-ph]].
  * [19] N. Anand, A. L. Fitzpatrick, E. Katz, Z. U. Khandker, M. T. Walters and Y. Xin, “Introduction to Lightcone Conformal Truncation: QFT Dynamics from CFT Data,” [arXiv:2005.13544 [hep-th]].
  * [20] S. J. Brodsky, G. F. de Teramond, H. G. Dosch and J. Erlich, “Light-Front Holographic QCD and Emerging Confinement,” Phys. Rept. 584, 1-105 (2015) doi:10.1016/j.physrep.2015.05.001 [arXiv:1407.8131 [hep-ph]].
  * [21] J. Carbonell, B. Desplanques, V. A. Karmanov and J. F. Mathiot, “Explicitly covariant light front dynamics and relativistic few body systems,” Phys. Rept. 300, 215-347 (1998) doi:10.1016/S0370-1573(97)00090-2 [arXiv:nucl-th/9804029 [nucl-th]].
  * [22] Z. Dziembowski and L. Mankiewicz, “LIGHT MESON DISTRIBUTION AMPLITUDE: A SIMPLE RELATIVISTIC MODEL,” Phys. Rev. Lett. 58, 2175 (1987) doi:10.1103/PhysRevLett.58.2175
  * [23] H. M. Choi and C. R. Ji, “Mixing angles and electromagnetic properties of ground state pseudoscalar and vector meson nonets in the light cone quark model,” Phys. Rev. D 59, 074015 (1999) doi:10.1103/PhysRevD.59.074015 [arXiv:hep-ph/9711450 [hep-ph]].
  * [24] O. Leitner, J. F. Mathiot and N. A. Tsirova, “The Pion wave function in covariant light-front dynamics,” Eur. Phys. J. A 47, 17 (2011) doi:10.1140/epja/i2011-11017-4 [arXiv:1009.5484 [hep-ph]].
  * [25] M. Gell-Mann and M. Levy, “The axial vector current in beta decay,” Nuovo Cim. 16, 705 (1960) doi:10.1007/BF02859738
  * [26] M. Gell-Mann, R. J. Oakes and B. Renner, “Behavior of current divergences under SU(3) x SU(3),” Phys. Rev. 175, 2195-2199 (1968) doi:10.1103/PhysRev.175.2195
  * [27] W. Jaus, “Covariant analysis of the light front quark model,” Phys. Rev. D 60, 054026 (1999) doi:10.1103/PhysRevD.60.054026
  * [28] Y. Li and J. P. Vary, “Light-front holography with chiral symmetry breaking,” Phys. Lett. B 825, 136860 (2022) doi:10.1016/j.physletb.2021.136860 [arXiv:2103.09993 [hep-ph]].
  * [29] G. F. de Teramond and S. J. Brodsky, Phys. Rev. D 104, no.11, 116009 (2021) doi:10.1103/PhysRevD.104.116009 [arXiv:2103.10950 [hep-ph]].
  * [30] Y. Li and J. P. Vary, “Longitudinal dynamics for mesons on the light cone,” Phys. Rev. D 105, no.11, 114006 (2022) doi:10.1103/PhysRevD.105.114006 [arXiv:2202.05581 [hep-ph]].
  * [31] V. E. Lyubovitskij and I. Schmidt, Phys. Rev. D 105, no.7, 074009 (2022) doi:10.1103/PhysRevD.105.074009 [arXiv:2203.00604 [hep-ph]].
  * [32] M. Ahmady, S. Kaur, S. L. MacKay, C. Mondal and R. Sandapen, Phys. Rev. D 104, no.7, 074013 (2021) doi:10.1103/PhysRevD.104.074013 [arXiv:2108.03482 [hep-ph]].
  * [33] M. Ahmady, H. Dahiya, S. Kaur, C. Mondal, R. Sandapen and N. Sharma, Phys. Lett. B 823, 136754 (2021) doi:10.1016/j.physletb.2021.136754 [arXiv:2105.01018 [hep-ph]].
  * [34] S. J. Brodsky and G. F. de Teramond, “Hadronic spectra and light-front wavefunctions in holographic QCD,” Phys. Rev. Lett. 96, 201601 (2006) doi:10.1103/PhysRevLett.96.201601 [arXiv:hep-ph/0602252 [hep-ph]].
  * [35] A. Karch, E. Katz, D. T. Son and M. A. Stephanov, “Linear confinement and AdS/QCD,” Phys. Rev. D 74, 015005 (2006) doi:10.1103/PhysRevD.74.015005 [arXiv:hep-ph/0602229 [hep-ph]].
  * [36] L. Da Rold and A. Pomarol, “Chiral symmetry breaking from five dimensional spaces,” Nucl. Phys. B 721, 79-97 (2005) doi:10.1016/j.nuclphysb.2005.05.009 [arXiv:hep-ph/0501218 [hep-ph]].
  * [37] C. M. Weller and G. A. Miller, “Confinement in two-dimensional QCD and the infinitely long pion,” Phys. Rev. D 105, no.3, 036009 (2022) doi:10.1103/PhysRevD.105.036009 [arXiv:2111.03194 [hep-ph]].
  * [38] G. A. Miller and S. J. Brodsky, “Frame-independent spatial coordinate $\tilde{z}$: Implications for light-front wave functions, deep inelastic scattering, light-front holography, and lattice QCD calculations,” Phys. Rev. C 102, no.2, 022201 (2020) doi:10.1103/PhysRevC.102.022201 [arXiv:1912.08911 [hep-ph]].
  * [39] S. J. Brodsky and G. F. de Teramond, “Light-Front Dynamics and AdS/QCD Correspondence: The Pion Form Factor in the Space- and Time-Like Regions,” Phys. Rev. D 77, 056007 (2008) doi:10.1103/PhysRevD.77.056007 [arXiv:0707.3859 [hep-ph]].
  * [40] S. R. Amendolia et al. [NA7], Nucl. Phys. B 277, 168 (1986) doi:10.1016/0550-3213(86)90437-2
  * [41] T. Horn et al. [Jefferson Lab F(pi)-2], Phys. Rev. Lett. 97, 192001 (2006) doi:10.1103/PhysRevLett.97.192001 [arXiv:nucl-ex/0607005 [nucl-ex]].
  * [42] V. Tadevosyan et al. [Jefferson Lab F(pi)], Phys. Rev. C 75, 055205 (2007) doi:10.1103/PhysRevC.75.055205 [arXiv:nucl-ex/0607007 [nucl-ex]].
  * [43] G. M. Huber et al. [Jefferson Lab], Phys. Rev. C 78, 045203 (2008) doi:10.1103/PhysRevC.78.045203 [arXiv:0809.3052 [nucl-ex]].
  * [44] P. Maris and P. C. Tandy, Phys. Rev. C 62, 055204 (2000) doi:10.1103/PhysRevC.62.055204 [arXiv:nucl-th/0005015 [nucl-th]].
  * [45] P. Maris and P. C. Tandy, Nucl. Phys. B Proc. Suppl. 161, 136-152 (2006) doi:10.1016/j.nuclphysbps.2006.08.012 [arXiv:nucl-th/0511017 [nucl-th]].
  * [46] M. Burkardt, X. d. Ji and F. Yuan, “Scale dependence of hadronic wave functions and parton densities,” Phys. Lett. B 545, 345-351 (2002) doi:10.1016/S0370-2693(02)02596-0 [arXiv:hep-ph/0205272 [hep-ph]].
  * [47] M. Ahmady, F. Chishtie and R. Sandapen, “Spin effects in the pion holographic light-front wavefunction,” Phys. Rev. D 95, no.7, 074008 (2017) doi:10.1103/PhysRevD.95.074008 [arXiv:1609.07024 [hep-ph]].
  * [48] H. M. Choi and C. R. Ji, “Chiral anomaly and the pion properties in the light-front quark model,” Phys. Rev. D 102, no.3, 036005 (2020) doi:10.1103/PhysRevD.102.036005 [arXiv:2006.08034 [hep-ph]].
  * [49] S. J. Brodsky and G. R. Farrar, Phys. Rev. Lett. 31, 1153-1156 (1973) doi:10.1103/PhysRevLett.31.1153
  * [50] G. Beuf, Phys. Rev. D 94, no.5, 054016 (2016) doi:10.1103/PhysRevD.94.054016 [arXiv:1606.00777 [hep-ph]].
  * [51] T. Lappi and R. Paatelainen, Annals Phys. 379, 34-66 (2017) doi:10.1016/j.aop.2017.02.002 [arXiv:1611.00497 [hep-ph]].
  * [52] S. D. Glazek and K. G. Wilson, “Perturbative renormalization group for Hamiltonians,” Phys. Rev. D 49, 4214-4218 (1994) doi:10.1103/PhysRevD.49.4214
  * [53] M. H. Wu and W. M. Zhang, “Chiral symmetry in light front QCD,” JHEP 04, 045 (2004) doi:10.1088/1126-6708/2004/04/045 [arXiv:hep-ph/0310095 [hep-ph]].
  * [54] S. R. Beane, “Broken Chiral Symmetry on a Null Plane,” Annals Phys. 337, 111-142 (2013) doi:10.1016/j.aop.2013.06.012 [arXiv:1302.1600 [nucl-th]].
  * [55] S. J. Brodsky, C. D. Roberts, R. Shrock and P. C. Tandy, “Essence of the vacuum quark condensate,” Phys. Rev. C 82, 022201 (2010) doi:10.1103/PhysRevC.82.022201 [arXiv:1005.4610 [nucl-th]].
  * [56] S. J. Brodsky, A. Deur and C. D. Roberts, Nature Rev. Phys. 4, no.7, 489-495 (2022) doi:10.1038/s42254-022-00453-3 [arXiv:2202.06051 [hep-ph]].