# GCF: Graph Convolutional Networks for Facial Expression Recognition

Hozaifa Kassab, Mohamed Bahaa, Ali Hamdi  MSA University
Giza, Egypt
{hozaifa.fadl, mohamed.bahaa4<EMAIL_ADDRESS>
###### Abstract

Facial Expression Recognition (FER) is vital for understanding interpersonal
communication. However, existing classification methods often face challenges
such as vulnerability to noise, imbalanced datasets, overfitting, and
generalization issues. In this paper, we propose GCF, a novel approach that
utilizes Graph Convolutional Networks for FER. GCF integrates Convolutional
Neural Networks (CNNs) for feature extraction, using either custom
architectures or pretrained models. The extracted visual features are then
represented on a graph, enhancing local CNN features with global features via
a Graph Convolutional Neural Network layer. We evaluate GCF on benchmark
datasets including CK+, JAFFE, and FERG. The results show that GCF
significantly improves performance over state-of-the-art methods. For example,
GCF enhances the accuracy of ResNet18 from 92% to 98% on CK+, from 66% to 89%
on JAFFE, and from 94% to 100% on FERG. Similarly, GCF improves the accuracy
of VGG16 from 89% to 97% on CK+, from 72% to 92% on JAFFE, and from 96% to
99.49% on FERG. We provide a comprehensive analysis of our approach,
demonstrating its effectiveness in capturing nuanced facial expressions. By
integrating graph convolutions with CNNs, GCF significantly advances FER,
offering improved accuracy and robustness in real-world applications. The code
is available at: https://github.com/4qlaa7/GCF

###### Index Terms:

Graph Convolutional Networks, FER, CNN

##  I Introduction

Facial Expression Recognition (FER) is a key area in computational psychology
and human-computer interaction, focusing on interpreting the emotional
language conveyed through facial cues. FER involves the automated detection
and classification of human emotions, encompassing a range of feelings such as
anger, disgust, fear, happiness, sadness, and surprise. This field has
significant applications in areas such as social robotics, healthcare[4], and
security[13]. However, existing FER classification methods face challenges
such as noise from environmental factors, imbalanced datasets, and issues with
overfitting and generalization[20]. Despite these obstacles, efforts to
improve FER continue, motivated by the potential to enhance human-machine
interaction, support assistive technologies, and deepen our understanding of
human behavior and cognition.

Deep learning has significantly advanced various domains of computer vision
[8, 7, 10, 2], including FER [21, 17]. CNNs have become the standard for image
processing tasks due to their proficiency in capturing spatial hierarchies of
features in images. By automating the feature extraction process, CNNs
eliminate the need for manual feature engineering and allow models to learn
directly from vast amounts of data, improving scalability and performance.
However, CNN-based FER requires extensive labeled datasets and substantial
computational power, which can be limiting when resources are constrained or
when data is scarce. Hybrid models, integrating CNNs with other machine
learning or neural network frameworks, have enhanced the robustness and
efficiency of FER. For example, combining CNNs with Recurrent Neural Networks
(RNNs) has proven effective for video-based FER by capturing dynamic changes
in expressions over time. Additionally, using Support Vector Machines (SVMs)
as classifiers on features extracted by CNNs has improved decision boundaries
for better classification accuracy. We present GCF, a pioneering methodology
that integrates CNNs with Graph Convolutional Networks (GCNs) to enhance FER.
Our approach leverages the robust feature extraction capabilities of CNNs and
the relational modeling strengths of GCNs to analyze distinct facial regions.
This synergy provides a comprehensive understanding of both local nuances and
global context in facial expressions, enabling nuanced analysis of emotional
states. By exploiting the structural information in GCNs, our model navigates
the complexities of facial expressions, surpassing the limitations of
traditional FER methodologies [15]. We evaluate our CNN-GCN model using
benchmark FER datasets, including CK+, FERG, and RAF, which cover diverse
expression scenarios. Our results show that the proposed model exceeds state-
of-the-art performance, outperforming CNN-only and other hybrid systems. The
paper is structured as follows: We first review related works in FER, then
describe our framework, followed by dataset descriptions, experiments and
results analysis, discussion, and conclusions with implications for future
research.

Figure 1: GCF, the proposed model design for EFR using Graph Convolutional
Neural Networks.

##  II Related Work

This section reviews the progression of methodologies employed in FER,
highlighting the evolution from CNNs to hybrid models integrating CNNs with
other computational techniques, and highlighting the need graph-based methods.

CNNs have been widely employed across various computer vision tasks, including
Facial Expression Recognition (FER). Multiple studies demonstrated the
robustness of CNNs to changes in face location and scale variations,
outperforming Multilayer Perceptrons (MLPs) particularly in scenarios
involving unseen face poses [21]. Another study [22] utilized CNNs to tackle
challenges such as subject independence, as well as translation, rotation, and
scale invariance in facial expression recognition. Early applications of CNNs
in FER often involved training models from scratch. For instance, a custom CNN
architecture significantly outperformed traditional machine learning
approaches by automatically learning features directly from facial images
[17]. The advent of pre-trained models brought about a paradigm shift, with
researchers leveraging models trained on large, diverse datasets to improve
FER accuracy. The work in [6] demonstrated that features extracted from deep
layers of pre-trained CNNs could be repurposed for various vision tasks,
including emotion recognition. This approach not only improved the accuracy
but also reduced the training time and resource consumption. VGG model pre-
trained on ImageNet for FER, achieving robust performance across multiple
datasets.

Incorporating Graph Convolutional Neural Networks (GCNs) alongside
Convolutional Neural Networks (CNNs) offers a promising avenue for advancing
Facial Expression Recognition (FER). By fusing CNNs’ robust feature extraction
capabilities with GCNs’ capacity to model relational data among facial
regions, this approach enables a more comprehensive understanding of both
local nuances and global context within facial expressions. The synergy
between CNNs and GCNs addresses the subtleties and complexities inherent in
facial expressions, leading to improved FER performance compared to
traditional methods[14],[5].

Recent advancements in Facial Expression Recognition (FER) have significantly
benefited from hybrid models that integrate Convolutional Neural Networks
(CNNs) with Graph Convolutional Networks (GCNs), effectively leveraging the
strengths of both methodologies. The MER-GCN framework enhances micro-
expression recognition by modeling relational dependencies among facial action
units (AUs) using GCNs, with spatial-temporal features extracted by a 3D
ConvNet and processed through a stacked GCN, resulting in notable improvements
in recognizing subtle and transient expressions[16]. However, the reliance on
co-occurrence probabilities to form the adjacency matrix can be limited by the
quality and representativeness of the training data, potentially leading to
biased or inaccurate relationships being modeled. Another approach constructs
an undirected graph from facial images using fixed and random points, where
GCNs handle non-Euclidean data structures, capturing complex relationships
between facial features to enhance recognition accuracy[25]. Despite its
innovative approach, the method’s effectiveness heavily depends on the initial
graph construction, which may not always capture the most relevant features
for FER. Additionally, a novel method for video-based FER employs a GCN to
represent each video frame as a node, using a dynamic adjacency matrix to
model dependencies and refine features through a Bidirectional Long Short-Term
Memory (BiLSTM) network, thus capturing both spatial and temporal variations
effectively[14]. While this approach addresses dynamic expression variations
well, it can be computationally intensive, requiring substantial resources for
real-time applications. These models demonstrate the powerful synergy of CNNs
and GCNs in advancing FER, but they also highlight the challenges in balancing
complexity, data dependency, and computational efficiency.

##  III The Proposed Framework

In this section, we provide a comprehensive overview of our proposed
methodology for facial expression recognition using the GCF model, inspired by
[9] [11] [12].

###  III-A CNN for Feature Extraction

Initially, a CNN is employed to extract relevant features from facial images,
utilizing either pre-trained models like VGG or ResNet, or a custom-designed
architecture. The CNN processes input images $X$ to produce a feature
representation $F_{\text{CNN}}(X)$ that captures various facial
characteristics, such as edges, textures, and patterns associated with
different expressions.The output of the CNN, $F_{\text{CNN}}(X)$, is then
sliced into nine vectors:

$F_{\text{CNN}}(X)=[f_{1},f_{2},\ldots,f_{9}]$

where $f_{i}$ represents the feature vector corresponding to the $i$-th facial
region. Pre-trained CNNs or other models are favoured for feature extraction
in machine learning and computer vision due to their ability to learn rich
hierarchical representations. These models, trained on large datasets like
ImageNet, allow practitioners to benefit from learned features without needing
to train entire models from scratch, significantly reducing computational
costs and training time. Fine-tuning these models on task-specific datasets
enhances their effectiveness, tailoring representations to specific tasks.
This strategy boosts performance, improves generalization, and accelerates the
development of state-of-the-art methodologies.

###  III-B Graph Convolutional Neural Network (GCN)

Each of the nine vectors $f_{i}$ is fed into a node within a Graph
Convolutional Neural Network (GCN). GCN is a type of neural network tailored
for analyzing and processing graph-structured data. Its mechanism revolves
around adapting convolutional operations from traditional image-based
Convolutional Neural Networks (CNNs) to operate directly on graph structures.

Let $A$ be the adjacency matrix representing the graph structure, and
$H^{(0)}=[f_{1},f_{2},\ldots,f_{9}]$ be the initial node features. A single
GCN layer can be represented as:

$H^{(l+1)}=\sigma\left(D^{-\frac{1}{2}}AD^{-\frac{1}{2}}H^{(l)}W^{(l)}\right)$

where:

  * •

$H^{(l)}$ is the feature matrix at layer $l$,

  * •

$W^{(l)}$ is the trainable weight matrix at layer $l$,

  * •

$D$ is the degree matrix of $A$,

  * •

$\sigma$ is an activation function, typically ReLU.

This aggregation process enables the network to extract hierarchical
representations of the graph, where deeper layers capture increasingly
abstract features.

###  III-C Feature Concatenation and Classification

After processing through the GCN, the output features from the nine nodes are
aggregated into a single feature vector $H_{\text{GCN}}$. This vector is then
concatenated with the initial output feature vector from the CNN before
slicing. Let $F_{\text{CNN-initial}}$ be the initial output feature vector
from the CNN before slicing. The concatenated feature vector is:

$F_{\text{concat}}=[F_{\text{CNN-initial}},H_{\text{GCN}}]$

This concatenation creates a comprehensive feature representation that
includes both detailed local features and broader contextual information. The
final feature vector $F_{\text{concat}}$ is then fed into a classifier (e.g.,
a fully connected layer followed by a softmax function) to predict the facial
expression:

$y=\text{softmax}(W_{\text{fc}}F_{\text{concat}}+b_{\text{fc}})$

where $W_{\text{fc}}$ and $b_{\text{fc}}$ are the weights and biases of the
fully connected layer. By leveraging the strengths of both CNN and GCN, our
proposed framework efficiently captures intricate facial features and their
relationships, enhancing the accuracy and robustness of facial expression
recognition systems across various datasets. The effectiveness of our proposed
methodology, which combines feature extraction via a CNN with subsequent
processing by a Graph Convolutional Network (GCN), represents a significant
advancement in facial expression recognition. By first utilizing a CNN, either
custom-designed or pre-trained, our approach efficiently captures intricate
facial features at multiple scales. These features are then fed into the GCN,
which excels in handling structured data, allowing for the modeling of
relationships between various facial regions through its graph-based nature.
This dual-stage processing not only enhances the representational power of the
system but also significantly improves the accuracy and robustness of
expression recognition across diverse datasets. Through rigorous
experimentation on benchmark datasets such as CK+, JAFFE, and FERG, our
proposed model demonstrates superior performance, achieving higher accuracy
rates compared to state-of-the-art methods. The results validate the efficacy
of integrating CNNs with GCNs, offering a promising direction for future
research in the domain of facial expression recognition.

Figure 2: Basic Emotions From CK+, JAFEE, and, FERG dataset

##  IV Datasets

In our study, we conduct a comprehensive experimental analysis of our proposed
model using several well-known facial expression recognition datasets: the
Extended Cohn-Kanade (CK+), the Facial Expression Research Group Database
(FERG), and the Real-world Affective Faces Database (RAF). Below is a detailed
overview of these databases, which are crucial for evaluating the
effectiveness of our approach.
Cohn-Kanade The Extended Cohn-Kanade dataset(CK+), commonly referred to as
CK+, is a widely used public dataset in the realm of action unit and emotion-
specified facial expression recognition. This dataset encompasses $593$
sequences from $123$ subjects, featuring both posed and spontaneous
expressions. In most research, including ours, the last frame of these
sequences is extracted and utilized for static facial expression recognition,
capturing the peak of the emotional expression. Facial Expression Research
Group Database (FERG) The FERG database consists of $55,767$ annotated images
representing six stylized characters designed using MAYA, a 3D animation
software. These characters are depicted in a range of seven different
expressions, providing a unique challenge in recognizing facial expressions
from cartoon-style visuals. This dataset allows us to test the adaptability
and performance of our model on non-realistic, stylized facial expressions,
which is essential for applications in animated films and video games.Japanese
Female Facial Expression(JAFFE) The JAFFE dataset consist of 213 facial images
of different facial expressions, sourced from 10 distinct Japanese female
subjects. Each participant was instructed to convey seven facial expressions,
encompassing six fundamental emotional states along with a neutral expression.
These images were meticulously annotated with mean semantic ratings for each
emotional expression, as assessed by a panel of 60 annotators[18],[19].

##  V Experimental Design and Results

In this section, we present and discuss the experimental results obtained from
the evaluation of the proposed Graph Convolutional Network-based Facial
Expression Recognition (GCF) model. The results are compared against several
state-of-the-art approaches using the aforementioned benchmark datasets. The
methods included in the comparison represent a range of traditional and deep
learning-based approaches for facial expression recognition. We split our
dataset into 80% for training and validation and 20% for testing.

We designed our extensive experimental work as follows:

  * •

Enhancing the state-of-the-art CNN pre-trained models using the proposed GCF.

  * •

Benchmarking the proposed model against the state-of-the-art FER models.

  * •

Varying the GCF design in an ablation study.

###  V-A Enhancing the State-of-the-art Models using GCF

Table I compares the performance of the GCF method against baseline and state-
of-the-art convolutional neural network (CNN) models on the JAFFE dataset. The
results highlight the effectiveness of our approach in achieving higher
accuracy rates across various architectures. In each comparison, the GCF
method outperformed the baseline CNN models. For instance, the GCF method
achieved an accuracy rate of 95% with VGG16, improving upon the CNN baseline
accuracy of 92%, while for VGG19, the GCF method achieved 92% accuracy,
significantly outperforming the CNN baseline of 72%. Similarly, with ResNet18,
the GCF method reached an accuracy of 89%, a notable improvement from the
baseline CNN accuracy of 66%. In more complex models like EfficientNetB0, the
GCF method achieved 90% accuracy, up from the CNN baseline of 87%, while with
InceptionV3, the GCF method achieved 99.2%, surpassing the CNN baseline of
97%. The results highlight the effectiveness of utilizing graph-based
representations for facial expression recognition, as the GCF approach
consistently achieves higher accuracy rates, even in highly optimized models
like EfficientNetB0 and complex architectures like InceptionV3 and DenseNet.
This consistent enhancement, with improvements ranging from a few percentage
points to over 20% in certain cases, underscores the potential of graph
convolutional networks in advancing the field of facial expression
recognition.

Models | CNN | GCF (Ours)
---|---|---
VGG16 | 92% | 95%
VGG19 | 83% | 92%
RESNET18 | 70% | 89%
RESNET34 | 82% | 87%
RESNET50 | 77.3% | 83%
EfficientNetB0 | 87% | 90%
InceptionV3 | 97% | 99.2%
densenet121 | 91.2% | 98%
densenet169 | 83% | 89.2%
mobilenetv2 | 97% | 100%
TABLE I: FER benchmarck on the JAFFE Dataset.

The performance evaluation on the CK+ dataset, as presented in Table II,
highlights the superior accuracy achieved by the GCF method compared to
baseline convolutional neural network (CNN) models. The GCF method
consistently outperformed the CNN counterparts across various architectures.
For the VGG models, GCF achieved 97% and 98% accuracy for VGG16 and VGG19,
respectively, marking significant improvements of 8% and 5% over the baseline.
Similarly, with the ResNet models, the GCF method improved accuracy rates to
98%, 96%, and 94% for ResNet18, ResNet34, and ResNet50, respectively,
outperforming the baselines by 6%, 3%, and 7%. The GCF method also achieved a
perfect accuracy of 100% with EfficientNetB0, surpassing the CNN baseline of
98%. In the case of InceptionV3 and DenseNet121, the GCF method achieved 99.2%
and 100%, respectively, slightly improving upon the CNN baselines.
Furthermore, with MobileNetV2, the GCF method attained 98%, surpassing the CNN
baseline by 3%. These consistent enhancements across diverse architectures
emphasize the potential of our graph-based approach for FER, solidifying its
effectiveness on the CK+ dataset and showcasing the advantages of graph
convolutional networks in this domain.

Models | CNN | GCF (Ours)
---|---|---
VGG16 | 89% | 97%
VGG19 | 93% | 98%
RESNET18 | 92% | 98%
RESNET34 | 93% | 96%
RESNET50 | 87% | 94%
EfficientNetB0 | 98% | 100%
InceptionV3 | 97% | 99.2%
densenet121 | 97% | 100%
densenet169 | 97% | 98%
mobilenetv2 | 95% | 98%
TABLE II: Performance comparison on CK+ dataset

The performance evaluation on the FERG dataset, as presented in Table III,
underscores the robust accuracy achieved by GCF method compared to baseline
convolutional neural network (CNN) models. Across different architectures, the
GCF method consistently outperformed the CNN counterparts. For the VGG models,
GCF achieved 99.6% and 99.4% accuracy for VGG16 and VGG19, respectively,
representing improvements of 1.8% and 3.3%. The ResNet models also benefited
from the GCF approach, with ResNet18 achieving a perfect 100% accuracy, an
improvement of 5.6%, and ResNet50 achieving 99.7%, a slight improvement over
the baseline. Even with more efficient architectures, such as EfficientNetB0,
the GCF method achieved 99.97%, surpassing the baseline accuracy of 98%. The
InceptionV3 and DenseNet architectures also saw improvements with GCF,
achieving up to 96% and 98.7% accuracy, respectively, outperforming their CNN
baselines. Finally, with MobileNetV2, the GCF method achieved 99%, surpassing
the CNN baseline by 3%. These results not only show the impact of the GCF upon
diverse architectures but also highlight the adaptability and superior
performance of graph convolutional networks for facial expression recognition
on the FERG dataset.

Models | CNN | GCF (Ours)
---|---|---
VGG16 | 97.8% | 99.6%
VGG19 | 96.1% | 99.4%
RESNET18 | 94.4% | 100%
RESNET34 | 93% | 96%
RESNET50 | 98% | 99.7%
EfficientNetB0 | 98% | 99.97%
InceptionV3 | 93% | 96%
densenet121 | 97% | 98.7%
densenet169 | 96% | 98%
mobilenetv2 | 96% | 99%
TABLE III: Performance comparison on FERG dataset.

###  V-B Benchmark with the State-of-the-art Models

The comprehensive performance comparison across the CK+, JAFFE, and FERG
datasets, as presented in Table IV, highlights the robust accuracy achieved by
the GCF method compared to several state-of-the-art models. In this benchmark
with the state-of-the-art models, our GCF method consistently outperformed
prominent approaches. On the CK+ dataset, our GCF method achieved a perfect
accuracy rate of 100%, outperforming notable models such as DeepEmotion (98%),
Nonlinear Evaluation on SL + SSL Puzzling (98.23%), FAN (99.7%), and ViT + SE
(99.8%). Similarly, on the JAFFE dataset, the GCF method also achieved a
perfect accuracy rate of 100%, surpassing strong baselines like DeepEmotion
(92.8%), ViT (94.83%), ARBEx (96.67%), and TLE (99.52%). The performance trend
continued on the FERG dataset, where the GCF method achieved an impressive
accuracy rate of 99.98%, outperforming DeepEmotion (99.3%) and ARBEx (98.18%).
These results underscore the efficacy and robustness of our GCF method across
diverse facial expression recognition datasets, highlighting its superior
performance over both traditional and cutting-edge deep learning approaches.
The consistently high accuracy rates achieved across different datasets
demonstrate the adaptability and effectiveness of the GCF method, making it a
promising approach for facial expression recognition tasks.

CK+ | JAFFE | FERG
---|---|---
Method | Accuracy |  | Method | Accuracy |  | Method | Accuracy |
DeepEmotion[23] | 98% |  | DeepEmotion[23] | 92.8% |  | DeepEmotion[23] | 99.3% |
Nonlinear eval on SL + SSL puzzling | 98.23% |  | ViT[3] | 94.83% |  | ARBEx[24] | 98.18% |
FAN [22] | 99.7% |  | ARBEx[24] | 96.67% |  |  |  |
ViT + SE [3] | 99.8% |  | TLE[1] | 99.52% |  |  |  |
GCF (Ours) | 100% |  | GCF (Ours) | 100% |  | GCF (Ours) | 99.98% |
TABLE IV: Comparison of accuracy across CK+, JAFFE, and FERG datasets. Figure
3: GCF versions.

###  V-C Ablation Study

The focus of this research is on the movements of the graph approach and its
significance when the direction of edges in the graph is taken into account.
Three models are identified: GCF-V1 with bidirectional edges, GCF-V2 with
cross-linked edges from left to right, and GCF-V3 with cross-linked edges from
right to left. The results summarize the accuracy rates of facial expression
recognition models across three key datasets: JAFFE, CK+, and FERG. The study
considers pretrained models including VGG16, ResNet18, and InceptionV3 and
utilizes them for the three versions of the Graph Convolutional Facial
Expression (GCF) distorted dataset. This work is proof of the power of certain
neural network architectures for recognizing diverse facial patterns indicated
in several datasets and versions of algorithms. GCF variations between
directional arrangements can be used to acquire the quantitative effect of
different organizational schemes on model performance.

Dataset | Models | GCF-V1 | GCF-V2 | GCF-V3
---|---|---|---|---
JAFFE | VGG16 | 95% | 93% | 86%
VGG19 | 92% | 90% | 83%
RESNET18 | 89% | 63% | 83%
RESNET34 | 87% | 86% | 82%
RESNET50 | 83% | 77% | 80%
EfficientNetB0 | 90% | 78% | 86%
InceptionV3 | 99.2% | 91% | 88%
densenet121 | 98% | 90% | 88%
densenet169 | 89.2% | 83% | 91%
mobilenetv2 | 100% | 80% | 83%
CK+ | VGG16 | 97% | 98% | 99%
VGG19 | 98% | 99.4% | 96%
RESNET18 | 98% | 99% | 99.4%
RESNET34 | 96% | 98% | 98.6%
RESNET50 | 94% | 99.3% | 94%
EfficientNetB0 | 100% | 96% | 95.8%
InceptionV3 | 99.2% | 97% | 99.1%
densenet121 | 100% | 97.7% | 98.4%
densenet169 | 98% | 95% | 99.6%
mobilenetv2 | 98% | 97% | 96.7%
FERG | VGG16 | 99.6% | 97% | 98%
VGG19 | 99.4% | 96% | 97%
RESNET18 | 100% | 95.6% | 96%
RESNET34 | 96% | 92% | 94%
RESNET50 | 99.7% | 98% | 96%
EfficientNetB0 | 99.97% | 98% | 98.8%
InceptionV3 | 96% | 94% | 94%
densenet121 | 98.7% | 97% | 97.5%
densenet169 | 98% | 96% | 93%
mobilenetv2 | 99% | 97% | 95%
TABLE V: Benchmark different GCF versions

##  VI Conclusion

In this paper, we introduced a novel approach to Facial Expression Recognition
(FER) by combining Convolutional Neural Networks (CNNs) and Graph
Convolutional Networks (GCNs). GCF model leverages the feature extraction
capabilities of CNNs and the relational modeling strengths of GCNs to capture
both local and global features of facial expressions. Our experiments on
benchmark datasets, including CK+, JAFFE, and FERG, demonstrated significant
improvements in accuracy, achieving up to 100% accuracy on the FERG dataset.
These results highlight that The GCF framework represents a promising
direction for future research in facial expression recognition, providing a
deeper understanding of human emotions.

##  VII Acknowledgment

We extend our heartfelt gratitude to AiTech for Artificial Intelligence &
Software Development (https://aitech.net.au) for providing the computational
resources essential for our experiments. Their support has been crucial to the
successful completion of this research.

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# Minkowski’s lost legacy and hadron electromagnetism

Yang Li  Department of Modern Physics, University of Science and Technology of
China, Hefei 230026, China  Wen-bo Dong  Department of Modern Physics,
University of Science and Technology of China, Hefei 230026, China  Yi-liang
Yin  Department of Modern Physics, University of Science and Technology of
China, Hefei 230026, China  Qun Wang  Department of Modern Physics, University
of Science and Technology of China, Hefei 230026, China  James P. Vary
Department of Physics and Astronomy, Iowa State University, Ames, IA 50010,
U.S.

###### Abstract

We revisit Minkowski’s lost legacy on relativistic electromagnetism in order
to resolve long-standing puzzles over the charge distribution of relativistic
systems like hadrons. Hadrons are unique relativistic electromagnetic systems
characterized by their comparable size and Compton wavelength
$r_{h}\sim\lambda_{C}$. As such, it was recently realized that the traditional
Sachs definition of the charge distribution based on a non-relativistic
formula is invalid. We explain that this is the same problem pursued by
Lorentz, Einstein and others, on the electromagnetism of a moving body. We
show how various charge distributions proposed in hadronic physics naturally
emerge as the multipole moment densities in the macroscopic theory of
relativistic electromagnetism.

##  I Introduction

At the turn of the 20th century, some of the greatest minds, Sir Thomson,
Lorentz and Einstein along with many others were pursuing – what now might be
regarded as a “wrong” – problem: the electromagnetic structure of the electron
[1, 2]. Later research revealed that the electron is a point-like particle
without internal structures [3]. Nevertheless, relativity, a pillar of modern
physics, was born from these endeavors. The relativistic theory of macroscopic
electromagnetism was fully established by Minkowski, Einstein and Laub in 1908
[4]. The final expressions Einstein, Minkowski and Laub derived for moving
media are identical to the macroscopic Maxwell equations applicable to media
at rest. Therefore, it is then not a surprise when the right kind of problem
finally arrived in the 1950s, the investigation of the proton electromagnetic
structure [5, 6], their work was largely forgotten. The proton charge
distribution was then defined based on the non-relativistic formula [7, 8],

$\rho_{\text{ch}}^{(\text{Sachs})}(\vec{x})\equiv\int\frac{{\mathrm{d}}^{3}q}{(2\pi)^{3}}\,e^{-i\vec{q}\cdot\vec{x}}G_{E}(-\vec{q}^{2}).$
(1)

In this formula, proposed by Sachs et al., $G_{E}(q^{2})$ is the charge form
factor obtained from the covariant decomposition of the hadron matrix element,
$\langle p^{\prime}|J^{\mu}(0)|p\rangle$, which can be measured in elastic
electron-proton scatterings. This expression is usually understood as defined
in the Breit frame $q^{0}=0$, $\vec{p}+\vec{p}^{\prime}=0$. It is known to be
ambiguous at large $Q^{2}=|q^{2}|$. For example, the Dirac form factor $F_{1}$
is an equally good choice as the charge form factor. It is probably an ad hoc
expression, since at that time, the accessible $Q^{2}$ was quite limited.
Nevertheless, this has become the textbook definition of the charge
distribution [10, 11, 13, 9, 12].

Only recently, the Sachs charge distribution was criticized as unphysical [14,
15, 16, 17, 18, 19, 20]. Miller et al. argued that a physically measurable
charge distribution should be obtained from the quantum expectation value
$\rho_{\Psi}(x)=\langle\Psi|J^{0}(x)|\Psi\rangle$, which depends on the hadron
wavepacket $\Psi(\vec{p})=\langle p|\Psi\rangle$. In relativistic quantum
mechanics (including quantum field theories), the Lorentz boosts are dynamical
[21]. As a result, the intrinsic charge distribution of the hadron cannot be
separated from the c.m. motion. For example, with a Gaussian wavepacket of a
width $R_{\text{w}}$, the r.m.s. charge radius $\langle
r^{2}_{\text{ch}}\rangle_{\Psi}=-6G^{\prime}_{E}(0)+3R^{2}_{\text{w}}$
diverges as the wavepacket approaches the plane wave $R_{\text{w}}\to\infty$,
which is the implicit wavepacket adopted in the Sachs definition [19].
Similarly, a localized wavepacket is not possible, either, since particles in
relativistic quantum mechanics cannot be localized [22, 23, 24, 25, 26, 27].
One could give up the 3D description in favor of a 2D light-front
distributions [28] with all its peculiarities [14]. Indeed, the majority of
the works in hadron structure are devoted to this formulation [29]. To resolve
these issues, Lorcé applied the Weyl-Wigner quasi-distributions defined in a
special reference frame, the elastic frame $q^{0}=0$. The Sachs distribution
and the light-front distribution appear as two slices of the quasi-
distributions [30]. Epelbaum et al. provided a definition based on sharply
localized wavepackets with a spherical symmetry [31].

Jaffe further clarified that, in order for the Sachs definition to be valid, a
chain of inequalities have to be satisfied [18] (cf. Refs. [33, 34, 32])

$r_{h}\gg\lambda_{\gamma}\gg\lambda_{h}\geq\lambda_{\text{C}},$ (2)

where, $r_{h}$ is the hadron radius, $\lambda_{\gamma}\sim Q^{-1}$ is the
wavelength of the probing photon; $\lambda_{h}$ is the de Broglie wavelength
of the hadron, which is bounded by its Compton wavelength
$\lambda_{\text{C}}=M^{-1}_{h}$. Here, $M_{h}$ is the hadron mass and we have
adopted the natural units $\hbar=c=1$. Unfortunately, the above inequalities
do not hold, since for relativistic systems like hadrons, their size and their
Compton wavelength are comparable $r_{h}\sim\lambda_{\text{C}}$. For example,
the proton charge radius is $r_{p}\simeq 0.84\,\text{fm}$, while its Compton
wavelength is $M^{-1}_{p}\simeq 0.2\,\text{fm}$ [35]. The mass radius of the
proton is even smaller $r_{p}^{\text{mass}}\simeq 0.55\,\text{fm}$ [57, 37].
Similarly, the pion charge radius is $r_{\pi}\simeq 0.67\,\text{fm}$ whereas
its Compton wavelength is $M^{-1}_{\pi}\simeq 1.4\,\text{fm}$ [35].

An immediate implication of Jaffe’s argument is that a proton is a
relativistic matter wave, as far as the electromagnetic probe
($\lambda_{\gamma}\lesssim\lambda_{h}$) is concerned. This view takes us back
to Einstein and Minkowski’s problem, now with the right kind of system to
investigate. Minkowski, Einstein and Laub’s theory of relativistic
electromagnetism tells us what kind of quantities can be extracted from the
measurements. For example, the matter wave as a medium is characterized by an
antisymmetric medium polarization tensor $M^{\alpha\beta}$ whose components
give the polarization vector $\vec{{P}}^{i}=M^{0i}$ and the magnetization
vector $\vec{{M}}^{i}=-\frac{1}{2}\epsilon^{ijk}M^{jk}$. The full current is
the sum of the free current – the current of a poin-like source – and the
polarized current,
$j^{\beta}=j^{\beta}_{\text{f}}+\partial_{\alpha}M^{\alpha\beta}$. The
coupling to the classical electromagnetic field is determined by the
Minkowski-Maxwell equations
$\partial_{\alpha}H^{\alpha\beta}=j^{\beta}_{\text{f}}$, where the field
induction tensor $H^{\alpha\beta}=F^{\alpha\beta}-M^{\alpha\beta}$.

The macroscopic charge and current densities are associated with these
quantities, e.g. the polarization charge density
$\rho_{\text{pol}}=-\nabla\cdot\vec{{P}}$, the effective magnetic charge
density $\rho_{\text{mag}}=-\nabla\cdot\vec{{M}}$, and the magnetization
current $\vec{j}_{\text{mag}}=\nabla\times\vec{{M}}$. One interesting
consequence of this formalism is that a magnetized but unpolarized hadron,
e.g. the neutron, will acquire an electric dipole moment under relativistic
motion [4, 14, 30]. For media in motion, it is useful to introduce the co-
moving polarization 4-vector and the co-moving magnetization 4-vector:
$M^{\alpha\beta}=u^{\alpha}\mathcal{P}^{\beta}-u^{\beta}\mathcal{P}^{\alpha}+\varepsilon^{\alpha\beta\rho\sigma}u_{\rho}\mathcal{M}_{\sigma}$,
and the associated macroscopic densities,
$\varrho_{\text{pol}}(x)=-\partial_{\alpha}{\mathcal{P}^{\alpha}}$,
$\varrho_{\text{mag}}(x)=-\partial_{\alpha}{\mathcal{M}^{\alpha}}$, where
$u^{\alpha}$ is a timelike unit frame vector, whose definition, however, is
not unique [50, 55, 54].

For electromagnetic media comprised of composite particles, the multipole
moment densities describe the internal electromagnetic structures of these
composite particles [38, 39]. We will see that these multipole moment
densities bridge the macroscopic observables and the conventional quantities
like the Sachs distribution. On the other hand, they also lead to new
characterizations of the system. It is worth mentioning that what we discuss
in this work has nothing to do with the proton radius puzzle [40, 41, 42].

While we focus on the electromagnetic structures here, the formulation can be
readily generalized to hadron structures under the weak, gravitational and
Higgs probes, e.g. [56, 57]. In particular, the gravitational form factors
have been analyzed using similar frameworks as we proposed here [58, 59, 60].

##  II Densities in quantum field theory

Consider a relativistic quantum system, whose internal dynamics is described
by an underlying relativistic quantum field theory. For simplicity, we start
with a spin-0 hadron. The full current is given by,

$j^{\mu}(x)=\langle\Phi|J^{\mu}(x)|\Phi\rangle=\int\frac{{\mathrm{d}}^{3}p}{(2\pi)^{3}2p^{0}}\int\frac{{\mathrm{d}}^{3}p^{\prime}}{(2\pi)^{3}2p^{\prime
0}}\\\
\times\widetilde{\Phi}^{*}(\vec{p}^{\prime})\widetilde{\Phi}(\vec{p})2P^{\mu}F_{\text{ch}}(q^{2})e^{iq\cdot
x},$ (3)

where $p$ and $p^{\prime}$ are on-shell momenta with
$p^{0}=\sqrt{\vec{p}^{2}+M^{2}}$, $q=p^{\prime}-p$ and
$P=\frac{1}{2}(p^{\prime}+p)$. The charge form factor $F_{\text{ch}}(q^{2})$
is defined from the hadron matrix element of the current operator $J^{\mu}$,
viz. $\langle
p^{\prime}|J^{\mu}(0)|p\rangle=(p+p^{\prime})^{\mu}F_{\text{ch}}(q^{2})$. The
hadron state vector is normalized as $\langle\Phi|\Phi\rangle=1$ and $\langle
p^{\prime}|p\rangle=2p^{0}(2\pi)^{3}\delta^{3}(p-p^{\prime})$. As such, the
wavepacket $\widetilde{\Phi}(\vec{p})=\langle p|\Phi\rangle$ is also
normalized as,

$\int\frac{{\mathrm{d}}^{3}p}{(2\pi)^{3}2p^{0}}\widetilde{\Phi}^{*}(\vec{p})\widetilde{\Phi}(\vec{p})=1.$
(4)

We can introduce a Lorentz covariant “coordinate space wave function” as the
Fourier transform of the momentum space wave function,

$\Phi(x)=\int\frac{{\mathrm{d}}^{3}p}{(2\pi)^{3}2p^{0}}\widetilde{\Phi}(\vec{p})e^{-ip\cdot
x},$ (5)

which satisfies the Klein-Gordon (KG) equation. In contrast to their momentum
space counterpart, the KG wave function cannot be normalized, since it is not
a true wave function. Instead, it is normalized as a current,

$\int{\mathrm{d}}^{3}x\,\Phi^{*}(x)i\tensor{\partial}_{t}\Phi(x)=1,$ (6)

where, $f\tensor{\partial}g\equiv f\partial g-(\partial f)g$. This fact is
closely related to the non-existence of the coordinate operator in
relativistic quantum theory. Using the KG wave function, we define the free
current as the convective current of a point-like particle [43],
$j^{\mu}_{\text{f}}(x)=\Phi^{*}(x)i\tensor{\partial}^{\mu}\Phi(x)$. The medium
polarization tensor can be constructed as,

$M^{\mu\nu}(x)=\int\frac{{\mathrm{d}}^{3}P}{(2\pi)^{3}}\int\frac{{\mathrm{d}}^{3}q}{(2\pi)^{3}}\widetilde{\Phi}^{*}({\vec{P}}+{\frac{1}{2}}\vec{q})\widetilde{\Phi}({\vec{P}}-{\frac{1}{2}}\vec{q})\\\
\times\frac{q^{\mu}P^{\nu}-q^{\nu}P^{\mu}}{2p^{0}p^{\prime
0}}\frac{F_{\text{ch}}(q^{2})-1}{iq^{2}}e^{iq\cdot x}.$ (7)

From this expression, we obtain the macroscopic charge and current densities
$\rho_{\text{pol}}$, $\rho_{\text{mag}}$ and $\vec{j}_{\text{mag}}$. For
example,

$\rho_{\text{pol}}(x)=\int\frac{{\mathrm{d}}^{3}P}{(2\pi)^{3}}\int\frac{{\mathrm{d}}^{3}q}{(2\pi)^{3}}\widetilde{\Phi}^{*}({\vec{P}}+{\frac{1}{2}}\vec{q})\widetilde{\Phi}({\vec{P}}-{\frac{1}{2}}\vec{q})\\\
\times\frac{P^{0}}{2p^{0}p^{\prime
0}}\big{(}F_{\text{ch}}(q^{2})-1\big{)}e^{iq\cdot x}.$ (8)

For the co-moving densities, we need to define the frame vector $u^{\alpha}$
first. Different schemes exist. A very simple choice is
$u^{\alpha}=i\tensor{\partial}^{\alpha}/\sqrt{-\tensor{\partial}\cdot\tensor{\partial}}=2P^{\alpha}/\sqrt{4M^{2}-{q^{2}}}$.
The corresponding polarization charge density is,

$\varrho_{\text{pol}}(x)=\int\frac{{\mathrm{d}}^{3}p}{(2\pi)^{3}2p^{0}}\widetilde{\Phi}(\vec{p})\int\frac{{\mathrm{d}}^{3}p^{\prime}}{(2\pi)^{3}2p^{\prime
0}}\widetilde{\Phi}^{*}(\vec{p}^{\prime})\\\
\times\sqrt{4M^{2}-q^{2}}\big{(}F_{\text{ch}}(q^{2})-1\big{)}e^{iq\cdot x}.$
(9)

Similarly, for a spin-1/2 particle, the full current is,

$j^{\mu}(x)=\int\frac{{\mathrm{d}}^{3}p}{(2\pi)^{3}2p^{0}}\int\frac{{\mathrm{d}}^{3}p^{\prime}}{(2\pi)^{3}2p^{\prime
0}}\widetilde{\Psi}_{s^{\prime}}^{*}(\vec{p}^{\prime})\widetilde{\Psi}_{s}(\vec{p})\\\
\times\bar{u}_{s^{\prime}}(p^{\prime})\Big{[}\gamma^{\mu}F_{1}(q^{2})+\frac{i\sigma^{\mu\nu}q_{\nu}}{2M}F_{2}(q^{2})\Big{]}u_{s}(p)e^{iq\cdot
x},$ (10)

where, $F_{1},F_{2}$ are the Dirac and Pauli form factors, respectively. The
free current is obtained from taking the point particle limit,

$j_{\text{f}}^{\mu}(x)=\frac{q_{N}}{2M}\overline{\Psi}(x)i\tensor{\partial}^{\mu}\Psi(x)+\mu_{N}\partial_{\nu}\big{[}\overline{\Psi}(x)\sigma^{\mu\nu}\Psi(x)\big{]},$
(11)

where, $M$ is the spinor mass, $q_{N}=F_{1}(0)$ is its charge number and
$\mu_{N}=[F_{1}(0)+F_{2}(0)]/2M$ is its magnetic moment. The Lorentz covariant
“wave function”

$\Psi(x)=\sum_{s}\int\frac{{\mathrm{d}}^{3}p}{(2\pi)^{3}2p^{0}}\widetilde{\Psi}_{s}(\vec{p})u_{s}(p)e^{-ip\cdot
x}$ (12)

satisfies the Dirac equation. The medium polarization tensor can be
constructed as,

$M^{\mu\nu}=\int{\mathrm{d}}^{3}r_{1}\int{\mathrm{d}}^{3}r_{2}\,\overline{\Psi}(\vec{r}_{1},t)\int\frac{{\mathrm{d}}^{3}P}{(2\pi)^{3}}e^{i{\vec{P}}\cdot(\vec{r}_{1}-\vec{r}_{2})}\\\
\times\int\frac{{\mathrm{d}}^{3}q}{(2\pi)^{3}}\Big{\\{}(q^{\mu}\gamma^{\nu}-q^{\nu}\gamma^{\mu})\frac{F_{1}(q^{2})-F_{1}(0)}{iq^{2}}\\\
-\frac{\sigma^{\mu\nu}}{2M}\big{[}F_{2}(q^{2})-F_{2}(0)\big{]}\Big{\\}}e^{-i\vec{q}\cdot(\vec{x}-\frac{\vec{r}_{1}+\vec{r}_{2}}{2})}\Psi(\vec{r}_{2},t).$
(13)

Figure 1 shows the charge density of the pion with selected Gaussian
wavepackets as compared with the Sachs distribution (1). Neither the
localization limit ($R_{w}\to 0$) or the plane wave limit ($R_{w}\to\infty$)
reproduces the Sachs distribution. This comparison may reinforce the
traditional impression that the Sachs distribution is only an approximation
valid in the nonrelativistic limit. However, as we will see later, the Sachs
distribution is in fact an exact quantity, the electric monopole density.

Figure 1: Charge distribution $\rho(r)=j^{0}(r)$ with Gaussian wavepackets
$\widetilde{\Phi}(\vec{p})=N\exp(-\frac{1}{2}R_{w}^{2}\vec{p}^{2})$, as
compared with the Sachs distribution. We adopt a dipole ansatz for the pion
form factor. The dependence on the wavepacket is shown. For the pion, the
charge distribution in neither the localization limit ($R_{w}\to 0$) or the
plane wave limit ($R_{w}\to\infty$) reproduces the Sachs distribution which,
as we explain later in the paper, is in fact the electric monopole density.

##  III Hadronic multipole moment densities

Figure 2: The full current can be viewed as the convolution of the hadronic
multipole density and the wavepacket. $\vec{r}$ and $\vec{P}$ are the
coordinate and the momentum of the hadron (shown as a Lorentz contracted
ellipsoid), respectively. $\vec{b}$ is the coordinate of the internal charge,
conjugate to $\vec{q}$.

The macroscopic densities depend on the wavepacket of the hadron. A central
goal in hadronic physics is to extract the intrinsic information of the
subatomic particles in 3D. In Minkowski’s relativistic electromagnetism, the
internal distribution of the composite particles are characterized by the
multipole moment densities [39]. Schematically, the classical current may be
written as a convolution of the convective current and the multipole density
$\varTheta(\vec{r})$ (see Fig. 2),

$j^{\mu}(x)\overset{\cdot}{=}\int{\mathrm{d}}^{3}r\,\varTheta(\vec{x}-\vec{r})j^{\mu}_{\text{f}}(\vec{r},t).$
(14)

From Eqs. (3–5), we obtain,

$j^{\mu}(x)=\int{\mathrm{d}}^{3}r_{1}\Phi^{*}(\vec{r}_{1},t)\int{\mathrm{d}}^{3}r_{2}i\partial^{\mu}\Phi(\vec{r}_{2},t)\int\frac{{\mathrm{d}}^{3}P}{(2\pi)^{3}}\\\
\times
e^{i\vec{P}\cdot(\vec{r}_{1}-\vec{r}_{2})}\int\frac{{\mathrm{d}}^{3}q}{(2\pi)^{3}}F_{\text{ch}}(q^{2})e^{i\vec{q}\cdot(\frac{\vec{r}_{1}+\vec{r}_{2}}{2}-\vec{x})}+\text{H.c.}$
(15)

Recall, $q^{2}=(q^{0})^{2}-\vec{q}^{2}$, which depends on $\vec{P}$ as,

$q^{0}=\sqrt{(\vec{P}+{\frac{1}{2}}\vec{q})^{2}+M^{2}}-\sqrt{(\vec{P}-{\frac{1}{2}}\vec{q})^{2}+M^{2}}.$
(16)

Comparing to the classical case (14), we need to evaluate the
$\int{\mathrm{d}}^{3}P$ integral in Eq. (15) to obtain a Dirac-$\delta$,
provided $\vec{P}$ in the remainder of the expressions is replaced by
$(-i/2)\tensor{\nabla}_{\vec{r}}$. The resulting expression is,

$j^{\mu}(x)=\int{\mathrm{d}}^{3}r\,\Phi^{*}(\vec{r},t)\varTheta(\vec{x}-\vec{r})i\partial^{\mu}\Phi(\vec{r},t)+\text{H.c.}$
(17)

where,

$\varTheta(\vec{b})=\int\frac{{\mathrm{d}}^{3}q}{(2\pi)^{3}}F_{\text{ch}}(q^{2})e^{-i\vec{q}\cdot\vec{b}}.$
(18)

Again, it depends on $\vec{P}=(-{i}/{2})\tensor{\nabla}_{\vec{r}}$ via
$q^{0}$. This dependence has to be defined through a power series. For
example, the Taylor series of $F_{\text{ch}}(q^{2})$ around $|\vec{P}|=0$ is,

$\displaystyle q^{2}=\,$
$\displaystyle-\vec{q}^{2}+\frac{(\vec{q}\cdot\vec{P})^{2}}{M^{2}+\vec{q}^{2}/4}+\mathcal{O}(|\vec{P}|^{4}),$
(19) $\displaystyle F_{\text{ch}}(q^{2})=\,$ $\displaystyle
F_{\text{ch}}(-\vec{q}^{2})+\frac{F^{\prime}_{\text{ch}}(-\vec{q}^{2})}{M^{2}+\vec{q}^{2}/4}(\vec{q}\cdot\vec{P})^{2}+\mathcal{O}(|\vec{P}|^{4}).$

Then, the multipole densities $\mathfrak{m}^{i_{1}i_{2}\cdots i_{n}}_{n}$ are
defined as,

$\varTheta(\vec{b})=\sum_{n}\frac{(-i)^{n}}{2^{n}n!}\mathfrak{m}^{i_{1}i_{2}\cdots
i_{n}}_{n}(\vec{b})\tensor{\nabla}^{i_{1}}\tensor{\nabla}^{i_{2}}\tensor{\nabla}^{i_{3}}\cdots\tensor{\nabla}^{i_{n}}.$
(20)

With (19), the first two multipole densities are,

$\displaystyle\mathfrak{m}_{0}(\vec{r})=\,$
$\displaystyle\int\frac{{\mathrm{d}}^{3}q}{(2\pi)^{3}}F_{\text{ch}}(-\vec{q}^{2})e^{-i\vec{q}\cdot\vec{r}},$
(21) $\displaystyle\mathfrak{m}_{2}^{ij}(\vec{r})=\,$ $\displaystyle
2\int\frac{{\mathrm{d}}^{3}q}{(2\pi)^{3}}\frac{F^{\prime}_{\text{ch}}(-\vec{q}^{2})}{M^{2}+\vec{q}^{2}/4}q^{i}q^{j}e^{-i\vec{q}\cdot\vec{r}}.$
(22)

The monopole density (21) is just the Sachs charge distribution (1) – no
special frame is chosen here. Moreover, in addition to the monopole moment,
the scalar hadron also possesses higher moments – an effect of purely
relativistic origin: the hadron is Lorentz contracted in the direction of
motion within the wavepacket (see Fig. 2).

Note that the series expansion is not unique. A closely related issue is the
convergence of the series. Since both the wavepacket and the hadron density
span over the entire space, there will always be regions outside the radius of
absolute convergence. The hope is that the convoluted density is sufficiently
suppressed in those regions. Roughly speaking, $|\vec{P}|$ corresponds to the
inverse of the de Broglie wavelength, $|\vec{P}|\sim\lambda^{-1}_{h}$ and
similarly, $|\vec{q}|\sim\lambda^{-1}_{\gamma}$. Therefore, the series in (19)
converge if the width of the wavepacket is sufficiently large, approaching a
plane wave. Similarly, convergence of the series in terms of $1/|\vec{q}|$
requires a localized intrinsic charge density – not a viable choice for
hadrons since $r_{\text{ch}}\sim M^{-1}$.

An interesting alternative is the Taylor expansion in terms of $1/|\vec{P}|$,
which requires a wavepacket sufficiently localized in at least one spatial
direction. In this case,

$q^{2}=-\vec{q}_{\perp}^{2}-\frac{q_{\|}^{2}(M^{2}+\frac{1}{4}\vec{q}^{2}_{\perp})}{\vec{P}^{2}}+\mathcal{O}(|\vec{P}|^{-4}),$
(23)

where, $q_{\|}={\vec{q}\cdot\vec{P}}/{|\vec{P}|}$, and
$\vec{q}_{\perp}=\vec{q}-({\vec{q}\cdot\vec{P}}/{\vec{P}^{2}})\vec{P}$. The
resulting monopole density is the light-front distribution,

$\mathfrak{m}_{0}(\vec{r})=\delta(r_{\|})\int\frac{{\mathrm{d}}^{2}q_{\perp}}{(2\pi)^{2}}F_{\text{ch}}(-\vec{q}^{2}_{\perp})e^{-i\vec{q}_{\perp}\cdot\vec{r}_{\perp}}.$
(24)

We emphasize that the convergence of this multipole expansion does not require
a fully localized (i.e. localization in 3D) hadron, which is physically
problematic in relativistic quantum field theory (cf. [48, 31, 49]).

For spinors the multipole densities are not uniquely defined since they are
not Lorentz scalars and electric and magnetic quantities may be converted into
each other. This point is reflected in various choices of the electric and
magnetic form factors. One of the conventional choices is to associate the
electric part with the point-like free Dirac particle, i.e.
$\overline{\psi}\gamma^{\mu}\psi$ and to associate the magnetic part with the
spin $\partial_{\nu}(\overline{\psi}\sigma^{\mu\nu}\psi)$,

$j^{\mu}(x)=\int{\mathrm{d}}^{3}r\,\overline{\Psi}(\vec{r},t)\varTheta_{1}(\vec{x}-\vec{r})\gamma^{\mu}\Psi(\vec{r},t)\\\
+\int{\mathrm{d}}^{3}r\,\overline{\Psi}(\vec{r},t)\frac{\sigma^{\mu\nu}}{2M}\partial_{\nu}\varTheta_{2}(\vec{x}-\vec{r})\Psi(\vec{r},t).$
(25)

Here, $\varTheta_{1,2}$ are the multipole densities associated with the Dirac
and Pauli form factors $F_{1,2}$, respectively.

Alternative partition of the electricity and magnetism is based on the
convection $\overline{\Psi}(x)i\tensor{\partial}^{\mu}\Psi(x)$ and non-
convection (i.e. spin) parts of the current [30], which leads to the following
multipole expansion,

$j^{\mu}(x)=\frac{1}{2M}\int{\mathrm{d}}^{3}r\,\overline{\Psi}(\vec{r},t)\varTheta_{E}(\vec{x}-\vec{r})i\partial^{\mu}\Psi(\vec{r},t)\\\
+\frac{i\varepsilon^{\mu\nu\alpha\beta}}{2M}\int{\mathrm{d}}^{3}r\,\overline{\Psi}(\vec{r},t)\gamma_{\beta}\gamma_{5}\Sigma_{\nu}(\vec{x}-\vec{r})i\partial_{\alpha}\Psi(\vec{r},t)\\\
+\text{H.c.}$ (26)

where, $\varTheta_{E}$ and $\Sigma^{\mu}=i\partial^{\mu}\varTheta_{M}$ are the
multipole (current) densities associated with the Sachs form factors $G_{E}$
and $G_{M}$, respectively. In this scheme, magnetic effects induced by charged
particle motion should vanish in its local rest frame. These different choices
are in parallel to the various choices of the fluid frames in spin
relativistic hydrodynamics [50, 51, 46, 52, 53], notably Landau-Lifshitz’s
energy frame [54] and Eckart’s particle frame [55].

##  IV Summary and Discussions

In this work, we revisited the macroscopic field theory description of the
electromagnetic structures of hadrons (e.g. proton, pion). This description
emerges because of the relativistic quantum nature of hadrons:
$r_{h}\sim\lambda_{C}$. The macroscopic fields are obtained from the quantum
average while the underlying dynamics is still dictated by quantum field
theory. Thus, they are physically measurable quantities.

The field description brings a salient yet fundamental viewpoint in the quest
of hadron structures in 3D. Minkowski, Einstein and Laub’s theory bridges the
macroscopic densities with the microscopic observables. The Sachs charge
distribution and the light-front charge distributions are clarified as two
types of the multipole moment expansion. These quantities can be measured from
any frame – no special frames are needed – provided the convergence conditions
of the corresponding multipole expansions are met. Of course, the inverse
problem of Eq. (14), known as deconvolution in signal processing, is highly
non-trivial even for Gaussian wavepackets [62]. In this regard, the light-
front formalism, exploiting (partially) localized wavepackets, has a clear
advantage in accessing the intrinsic structures of hadrons.

As we have mentioned, hadrons are distorted in phase space due to the Lorentz
contraction. This effect is dynamical. However, one may introduce a
rotationless Lorentz boost $\Omega$ with $\Omega\cdot P=(M,\vec{0})$ to define
“intrinsic” multipole densities
$\vartheta(\vec{r})=\varTheta({\vec{\Omega}}\cdot\vec{r})$ in a momentary rest
frame. As such, the kinematical part of the Lorentz contraction may be
removed. However, the choice of $\Omega$ is not unique [47], leading to
different intrinsic moments. One can show that with the canonical boost, only
the monopole density, i.e. the Sachs charge distribution, survives for scalar
particles in Eq. (18). N.B. both the spin and the current components will
change under the Lorentz boost. Indeed, Rinehimer and Miller showed that
boosting the current to the infinite momentum frame converts the Sachs form
factor $G_{E}$ to the Pauli form factor $F_{1}$ [61].

An immediate generalization of the present formalism is to the energy-momentum
tensor (EMT), which is the conserved Noether current of the diffeomorphism
invariance [63]. Using the same techniques, the full EMT current can be
written as two parts as well,
$t^{\alpha\beta}\equiv\langle\Phi|T^{\mu\nu}(x)|\Phi\rangle=t^{\alpha\beta}_{\text{f}}+\partial_{\sigma}\chi^{\mu\nu\sigma}$,
where the free EMT,

$t^{\alpha\beta}_{\text{f}}=\partial^{\alpha}\Phi^{*}\partial^{\beta}\Phi+\partial^{\beta}\Phi^{*}\partial^{\alpha}\Phi+Dg^{\alpha\beta}\big{(}\partial_{\sigma}\Phi^{*}\partial^{\sigma}\Phi\\\
-M^{2}\big{|}\Phi\big{|}^{2}\big{)}-\frac{1}{2}(D+1)\partial^{\alpha}\partial^{\beta}\big{|}\Phi\big{|}^{2},$
(27)

and the polarization tensor $\chi^{\alpha\beta\sigma}$,

$\chi^{\alpha\beta\sigma}(x)=\int\frac{{\mathrm{d}}^{3}p}{(2\pi)^{3}2p^{0}}\int\frac{{\mathrm{d}}^{3}p^{\prime}}{(2\pi)^{3}2p^{\prime
0}}\widetilde{\Phi}^{*}(\vec{p}^{\prime})\widetilde{\Phi}(\vec{p})\\\
\times\Big{\\{}2P^{\alpha}(P^{\beta}q^{\sigma}-P^{\sigma}q^{\beta})\frac{A(q^{2})-1}{iq^{2}}\\\
-2i\big{(}g^{\alpha\beta}q^{\sigma}-g^{\alpha\sigma}q^{\beta}\big{)}(D(q^{2})-D)\Big{\\}}.$
(28)

Here, $D=D(0)$ is a less known global charge, the druck term (or the
$D$-term), sometimes dubbed as the “cosmological constant” of the hadron.
Since the EMT is obtained from a gauge symmetry, it is defined up to a total
derivative term whose contributions vanishes upon coupling to the
gravitational field. We can choose the total derivative to be
$-\partial_{\sigma}\chi^{\alpha\beta\sigma}$, as such the EMT can always be
taken to be the free EMT current $t^{\alpha\beta}_{\text{f}}$. Similarly, for
spinors, the EMT can be written as the point-like particle EMT plus a total
derivative term irrelevant for gravitational coupling. Hence we obtain a
remarkable conclusion: the gravitational coupling does not distinguish between
elementary and composite particles. This is the equivalence principle (EP)
applied to relativistic quantum systems, and is closely tied to the low-energy
theorems of graviton [64, 65, 67, 66] which underlines Teryaev’s argument for
the vanishing of the anomalous gravitomagnetic moment [68].

Note that Eq. (27) differs from the free KG theory ($D=-1$) by some
$D$-dependent terms. Since the wavepacket is more or less arbitrary, we can
consider a static homogeneous wavepacket. In this setting,
$t^{00}_{\text{f}}=-DM^{2}|\Phi|^{2}$. The positivity of the energy
$t^{00}_{\text{f}}>0$ immediately leads to $D<0$, which has been speculated
for some time as the necessary condition for the stability of matter [57].
Unfortunately, the terms associated with $D$ are non-minimal coupling total
derivatives [69].

## Acknowledgements

The authors wish to thank Y. Chen, X.l. Sheng, S. Pu for valuable discussions.
This work was supported in part by the US Department of Energy (DOE) under
Grant Nos. DE-FG02-87ER40371 and DE-SC0018223 (SciDAC-4/NUCLEI). Q.W. is
supported in part by the National Natural Science Foundation of China (NSFC)
under Grants No. 12135011, 11890713, 12047502, and by the Strategic Priority
Research Program of the Chinese Academy of Sciences (CAS) under Grant No.
XDB34030102. Y.L. is supported by the new faculty startup fund of University
of Science and Technology of China.

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