# Synthetic Dataset Generation for Adversarial Machine Learning Research

Xiruo Liu  Shibani Singh  Cory Cornelius  Colin Busho  Mike Tan  Anindya Paul
Jason Martin

###### Abstract

Existing adversarial example research focuses on digitally inserted
perturbations on top of existing natural image datasets. This construction of
adversarial examples is not realistic because it may be difficult, or even
impossible, for an attacker to deploy such an attack in the real-world due to
sensing and environmental effects. To better understand adversarial examples
against cyber-physical systems, we propose approximating the real-world
through simulation. In this paper we describe our synthetic dataset generation
tool that enables scalable collection of such a synthetic dataset with
realistic adversarial examples. We use the CARLA simulator to collect such a
dataset and demonstrate simulated attacks that undergo the same environmental
transforms and processing as real-world images. Our tools have been used to
collect datasets to help evaluate the efficacy of adversarial examples, and
can be found at https://github.com/carla-simulator/carla/pull/4992.

Machine Learning, ICML

##  1 Introduction

Deep Neural Networks (DNN) have revolutionized the field of artificial
intelligence with significant success in many emerging fields like computer
vision and natural language processing. As the production and deployment of
DNN models are on the rise in security and safety critical applications,
numerous studies have shown that DNN models are susceptible to adversarial
examples (Szegedy et al., 2018; Carlini & Wagner, 2017; Goodfellow et al.,
2015; Papernot et al., 2016).

(a) Green Patch in CARLA

(b) Digital Patch

(c) DAPRICOT Patch

(d) Rendered Adversarial Patch

Figure 1: A comparison of patch insertion methods. 1(b) shows a digital
adversarial patch that is composed directly onto the image and obviously does
not blend into the surrounding environment. In 1(c), a DAPRICOT (Threet et
al., 2021) patch is crafted to mimic environmental effects, but artifacts are
still noticeable. In 1(a), a green patch is first inserted into CARLA, then an
adversarial texture is streamed onto it as shown in 1(d). Hence, this patch is
rendered by CARLA in the same way as other objects in the scene.

In the computer vision domain, although the robustness of defense schemes are
often analyzed through digital perturbation attacks, these attacks are very
difficult to deploy in practice as they manipulate images directly after the
inputs have been captured and digitized inside the system. Digital insertion
attacks are not realistic as they do not account for real-world environmental
transforms, such as shading, lighting, occlusions and sensor noise. Figure 1
shows an exemplar comparison between the realism of three types of patch
attacks. Other work has shown that physical adversarial examples need to be
robust against multi-distance and multi-perspective view of the camera in
order to have an impact in a real-world scenario (Lu et al., 2017). Hence, it
is not a fair evaluation for many defense schemes, if any at all, with these
types of digital attacks.

We propose a simulation tool that is able to generate large scale synthetic
datasets with realistic adversarial examples. By leveraging the capabilities
of CARLA (Dosovitskiy et al., 2017), our simulated datasets have the following
distinct attributes: multi-modal sensory measurements, instance segmentation
labels, diverse backgrounds with varying weather conditions, and multi-
perspective views of the scenes.

##  2 Related Work

There has been a significant amount of literature on generating synthetic
datasets for various applications, including autonomous driving through
diverse scenes with synthetic pedestrians and objects (Geiger et al., 2012;
Cordts et al., 2016; Richter et al., 2016). For example, Sim 200k (Johnson-
Roberson et al., 2017) dataset consists of $200$k synthetic images of vehicles
driving in different times of the day under diverse weather and lighting
conditions. A lot of recent work relies on synthetic data to create
benchmarks, or to compensate for the lack of training data with data
augmentation. SynDataGeneration (Dwibedi et al., 2017) cut and paste object
patches in scenes to create images thus focusing on patch-level realism. This
technique requires preexisting object masks which are generated using pixel-
objectness models with bi-linear pooling.

Prior work (Brown et al., 2017) have shown that it is possible to generate
scene-independent adversarial patches which can be placed anywhere and is able
to alter the behaviour of the targeted model. Generating patches, which work
under various lighting conditions and transfer to different target models, is
a computationally intensive optimization process. More recent work (Pintor et
al., 2022) released a small adversarial patch dataset for fast robustness
evaluation, where patches are applied with translation and rotation and
composed onto images from ImageNet. Imagenet-C (Hendrycks & Dietterich, 2019)
and WILDS (Koh et al., 2021) datasets address distribution shifts in the
dataset. Imagenet-C has image corruptions (e.g., blurring, jitter) and WILDS
addresses domain generalization and sub-population shifts.

APRICOT open source dataset (Braunegg et al., 2020) was the first step to
provide a benchmark to evaluate the robustness of object detection models
against realistic attacks. APRICOT photographed printed adversarial patches on
real objects and scenes in real-world environments. However, there are several
drawbacks of this dataset: not every object in the image is labelled; there is
no complexity to the attacks since the attacks are static and not adaptive in
nature; not all patches work against all models. Moreover, this manual data
collection is time-consuming, not configurable to new environmental factors
and does not capture multi-modal inputs beside RGB images. Dynamic APRICOT
(DAPRICOT) (Threet et al., 2021) is one step further towards a realistic
adversarial dataset using CARLA with customized assets. Customized green
screen patches, which serve as the place holders for adversarial patches, are
inserted into the scene. Then, color transforms these patches undergo are
taken and used for adjusting adversarial patches such that the adversarial
patches blend into the simulated environment more realistically. While a step
closer to realistic attacks leveraging simulation, this attack cannot map
textures to 3D objects. Prior work (Cornelius et al., 2019) showed how to use
CARLA to demonstrate the efficacy of ShapeShifter (Chen et al., 2018) attacks
against driving scenarios. That work also highlighted the importance of
validating realistic attacks by reproducing scenarios across varying
environmental conditions, which CARLA already supports.

There are other advancements of placing adversarial attacks physically and
capturing images in different weather conditions. For example, with the sign
embedding attack (Sitawarin et al., 2018), traffic signs are modified to
improve robustness to noisy transformations happened during the image capture
stage. This attack also exemplifies the difficulty in creating datasets with
realistic attacks.

Privacy concerns also raise for real-world datasets collected from public
places, especially for tasks like pedestrian detection and tracking. To
address privacy concerns, generating a large diverse synthetic dataset using a
rendering game engine is explored in (Fabbri et al., 2021). The dataset has
temporally consistent bounding boxes, instance segmentation, depth maps and
pose occlusion information along with varied environment conditions, camera
viewpoints, object textures, lighting conditions, weather, seasonal changes,
and object identities.

##  3 Motivation

While aforementioned datasets maybe useful for the purposes of augmenting
benign datasets, there is an imminent need for frameworks that enable the
exploration of realistic adversarial attacks with varying threat models.

We define realistic attacks as those attacks that use the same end-to-end
processing pipeline as the defense models. In Figure 2, the complete pipeline
for image classification tasks typically includes four stages. Attacks may
come in at different stages. For example, digital attacks such as Projected
Gradient Descent (PGD) (Madry et al., 2017) are injected at the final
recognition stage, while DAPRICOT attacks are performed at preprocessing
stage. Physical attacks and static physical attacks are injected at the very
front and applied to real-world objects, which makes them realistic. However,
these physical attacks are very expensive and hence are costly to scale.
Therefore, we adopt an alternative approach by replacing the real-world with a
simulated world created by CARLA. As a result, we can apply adversarial
textures onto 3D objects, which then undergo all environmental transforms in
the simulation as shown in Figure 2.

Figure 2: Comparison of performing attacks in the real world versus in the
simulation. The end-to-end image classification pipeline includes four stages.
As shown at the top, in the real world, a realistic attack needs to be applied
to a real object to undergo the complete pipeline, which makes it very
challenging. However, with simulation, it is convenient to apply a adversarial
texture to the target stop sign and force the attack to undergo all the same
environmental transformations and processing as the defense model.

##  4 Data Collection Tool

Motivated by the lack of large scale dataset with realistic adversarial
examples, we developed a scalable data collection tool that is built on top of
CARLA. We open sourced this tool at https://github.com/carla-
simulator/carla/pull/4992.

This tool enables the collection of multimodal data (as shown in Figure 3) and
different scenes from sensors placed at different locations. A wide range of
scenarios can be configured to collect data with varying weather patterns,
times of day, traffic, and crowd conditions. The tool provides accurate labels
and minimizes the costs involved in generating annotations for the dataset.
The annotations include bounding boxes, classification labels, instance
segmentation masks and frame sequence order for temporal tasks.

Figure 3: Multimodality is well supported in CARLA, while it may be very
difficult to obtain synced data from different sensors in a real-world
dataset. From left to right, the depth image, the RGB image and the ground
truth (i.e., instance segmentation masks) provide well aligned data for
research that explores the relations among different modalities.

CARLA supports various atmospheric conditions and illumination settings with
tunable configurations for position and color of the sun, the intensity and
color of diffuse sky radiation, ambient occlusion, atmospheric fog,
cloudiness, precipitation, as well as lighting conditions by time of day.
CARLA allows configurable placements of sensors from different modalities,
such as RGB, depth, LIDAR, RADAR, optical flow, event based dynamic vision
sensor and ground-truth sensors (instance segmentation and semantic
segmentation). Taking advantage of these features from CARLA, we developed a
data collection tool, which supports various configuration of synchronized
sensor suites, weather and lighting conditions, traffic and crowd settings.

The tool consists of two major components: a data saver and an annotator. The
data saver takes a configuration file (details in LABEL:lst:config_yaml) as an
input and then instantiates a CARLA simulation accordingly. We support three
kinds of sensor placements: statically placed sensors, sensors attached to
vehicles or pedestrians, and sensors that move.

The three supported sensor movement patterns are:  (1) linear motion, where
the sensor moves straight from the source position to the destination
position;  (2) rotation, where the sensor rotates within a predefined angle
range;  (3) jitter, where the sensor jitters randomly within a configured
range.  All the three types of movement can be applied together to a moving
sensor. Also, jitter and rotation can be added to a “static” sensor too.

The annotator supports two types of annotation formats: kwcoco (Crall, 2020)
and Multi Object Tracking and Segmentation (MOTS) (Voigtlaender et al., 2019).
The kwcoco annotation format is an extension of the COCO format with COCOAPI
support. We choose this format to support temporal annotations that are not
supported in COCO. The MOTS format has two forms, text and PNG, to support
object tracking and segmentation scenarios. Figure 4 shows a RGB image and the
visualization of its annotations generated by our tool.

(a) Scene

(b) Annotated scene

Figure 4: An image captured by a RGB camera from CARLA using the data saver
and the visualization of the generated annotations using the annotator tool:
4(a) scene and 4(b) annotated scene.

We focus on five specific aspects that we find lacking in the existing
datasets and synthetic data generation frameworks: configurability with
different settings via a YAML configuration file, reproducibility of data with
a seed and configuration, the enablement of realistic attacks that go through
the same environmental transforms as the defenses, extensibility to augment a
existing dataset to accommodate new requirements, ease of use with docker to
bypass the complexity from CARLA and Unreal Engine installations and simplify
running simulations, and scalability to be able to run multiple simulations in
containers simultaneously on different GPUs.

##  5 Capabilities to Enable Benign and Adversarial Machine Learning Research

Configurability and extensibility have been of huge importance to the
development of datasets that advance machine learning research. Quite often,
when attempting to deep dive into complex problems and decouple a set of
mingled factors, researchers find out that it is very challenging to obtain
new customized data from the real world that can help them conduct experiments
and pinpoint the root cause. They may have to use the initial real world
dataset due to the heavy cost in collecting and labelling that specific data.

However, with our tool, extending an existing dataset with customized
configurations to meet new research requirements become feasible. Because the
simulation can isolate one environmental factor at a time, researchers can
determine specific effects that lighting, weather, object orientation, or time
of day have on a model’s ability to correctly perform its task by easily
creating new data that only changes a chosen environmental factor. As a
result, researchers can have better understanding of the model and hence
improve the model correspondingly. Moreover, as the data collection is defined
by configurations in our tool, it is easy to extract the statistics of the
dataset so that we can better understand the characteristics and limitations
of the models trained on the dataset. And if more data is needed at a later
time, it is also convenient to generate more data with new customized
configurations.

Figure 5: An example of how the simulation tool helps a patch attack
generalizes better with more object tracking data. The left figure shows that
the attack overfits to a single pedestrian tracking video. In the middle
figure, the attack improves as it is computed over five tracking videos. The
attack shown in the right figure is optimized with ten diverse tracking videos
and hence performs the best among the three.

From an adversarial perspective, an attacker can also enjoy all the benefits.
Just as model training may overfit to a given training dataset, an attack may
overfit to a set of data as well. By being able to easily create additional
data and change environmental factors, an attacker can generalize the attack
better to a given situation. Figure 5 shows an example of how a patch attack
may overfit to an object tracking model in a scenario, where a child walks by
a wall with an adversarial patch on it. As more diverse data is created from
simulations, however, the attack improves and generalizes better as it is able
to attack the tracking model in case of any person walking in front of the
adversarial patch on the wall from any directions.

Researchers can test and challenge assumptions for machine learning defenses
and attacks, when given the ability to create synthetic data in this manner.
As an example, Figure 6 shows a machine learning defense that assumes a static
background. The figures in the first row show the scene where an adversarial
patch is placed on the background wall. The figures in the second row show the
defense mechanism of ablating the background in order to remove the patch. In
the static setting, it is clear that this defense scheme is quite effective by
masking out the background. However, a more advanced attack may challenge this
static assumption and test how well the defense holds up under a different
assumption. The third row of Figure 6 shows a set of images, which is a moving
scene as the camera changes its transform slightly. The figures in the fourth
row show how the defense performs in this dynamic setting. The adversarial
patch is not removed and the defense model fails to ablate the background as
expected. This experiment shows that even a slight shift in the background of
the scene causes the background ablation defense to be largely ineffective.
This type of study would be time consuming and expensive, if at all possible,
in a real world setting. A researcher would need to recollect a scene, ideally
keeping most of the environmental factors as similar as possible. Using the
data collection tool we introduced allows the recollection of a scene with a
simple change (e.g. adding sensor jitter movement) to a configuration file.
This allows researchers to better understand what makes their machine learning
defenses generalize better.

Figure 6: Generalization of the adversarial patch defense using background
ablation. Figures at top two rows show that an adversarial attack on a video
with a static background, and the defense successful ablates the adversarial
patch. However, as shown by the figures in the bottom two rows, an attack on a
tracking video with dynamic background causes the defense fail to ablate the
adversarial patch. Figure 7: Generalization of adversarial patch defense using
physical constraints (Feng et al., 2022). The left column includes a low
resolution RGB image and the visualization after applying the defense, while
the right column is a high resolution RGB image with the visualized defenses.
The first row shows two adversarial images with an adversarial patch in the
top left quadrant. The second row shows masks of pixels with anomalous colors.
The third row shows masks in regions with high frequency contents. And the
fourth row shows masks in regions with high hue and saturation pixel values.
This defense, which was trained on low resolution images, performs well on the
left column images as it is able to identify the patch location. However, it
does not generalize well on high resolution images as shown in the right
column where the patch is located at wrong position or false positives occur.

As another example, Figure 7 demonstrates how an attack’s efficacy may vary
when placed in different scenes by taking into consideration of physical
constraints in the scene. An adversarial patch defense may attempt to locate a
patch by detecting anomalous natural scene statistics. The success of the
defense largely depends on where other objects are located within the scene as
well as the natural environmental statistics of the scene. Figure 7 provides a
comparison on how a patch defense performs on low resolution images and high
resolution images. As it was trained on low resolution images, the defense
performs well on the RGB image in the left column and can locate the patch as
shown in the other three images in the left column. However, given a high
resolution RGB image in the right column, the defense performance degrades as
the patch is located at a wrong place in the third image in the right column.
While the defense still identifies the patch in the second image, it also
generates many false positives as well. These failures indicate potential
generalizability issues of this defense that uses physical constraints.

Our tool can help better generalize defenses by enabling researchers to easily
generate additional data with variability for different scenarios. On the
other hand, attackers could also receive benefits as they can take this
information and use it to their advantage. For example, an adversarial patch
can be placed in a location that better “blends in” to the scene. Or the patch
may be optimized to more closely match the current environment where it is
placed.

Multimodal machine learning is one of the most vibrant research areas that
aims to process and learn related information from different modalities. As it
is similar to how humans sense and learn the world, multimodal learning shows
increasing importance and extraordinary potential. However, a big challenge in
this research area is the lack of high quality, well aligned data from
different modalities. It is very difficult to obtain real-world datasets with
data captured from different types of sensors at the exact same location and
perspective. On the other side, it is easy to generate a multimodal synthetic
dataset using CARLA, which currently supports RGB camera, depth camera,
optical flow camera, LIDAR and RADAR sensors. Figure 3 shows an example where
a depth camera and a RGB camera are perfectly aligned (i.e., placed at the
exact same location with the same rotation) and share the same ground truth.

In the adversarial machine learning domain, digital perturbation attacks, such
as PGD, are widely used for evaluating the robustness of models. However,
these attacks are often too powerful in the sense that they directly attack
the image pixels after the image has been captured and hence do not undergo
the environmental transforms and the processing pipeline that a defense scheme
needs to consider for accuracy and robustness.

We show a comparison between varying methods of inserting an adversarial patch
into a scene captured by CARLA in Figure 1. The digital patch in Figure 1(b)
is composed directly onto the image, which bypasses all the environmental
transforms. Note that there are no shadows on the patch even though it is
located in the shadowy area of the sidewalk. Figure 1(c) shows a DAPRICOT
patch that estimates the environmental transforms and calculates a color
correction to be applied to the patch before digitally inserting into the
scene. Finally, using CARLA, a green patch in Figure 1(a) is first inserted
into the scene as a place holder. Then the adversarial patch is applied as a
texture to the designated area in the simulation, and then rendered in CARLA
as shown in Figure 1(d). This approach makes the patch look very realistic as
it takes the environmental transforms (such as shadow, lighting, reflections)
and applies them to the patch in the same way as other objects in the scene.
To evaluate the robustness of machine learning models fairly and accurately,
it is critical to perform realistic attacks that undergo the complete
processing pipeline as normal models would take. Our synthetic dataset
generation has the capability to force the attacker to consider the effects of
environmental transformations. As a result, we are able to provide an even
playground for both the attacker and the defender.

Moreover, from the attacker’s perspective, in real life it is very expensive
and slow to compute and apply a physical adversarial perturbation. However,
with the simulation tool, it is much faster to iterate the perturbation
computations and evaluate the effectiveness of the realistic attack. The tool
also allows an attacker to choose any suitable technique to craft adversarial
examples for specific scenes. Figure 8 shows an example of validating a
ShapeShifter (Chen et al., 2018) attack, where an adversarial texture is
computed and then applied to a Tesla Model 3 car in CARLA. This attack is
successful as it causes the vehicle to be misclassified as an orange. Without
simulation, to validate this physically realizable attack, a car needs to be
painted with the computed adversarial texture to validate the effectiveness of
the attack in the real world. Even worse, the attack optimization may go
through this paint-validation-repaint process multiple times, which makes it
very expensive to conduct realistic attack research in real world. On
contrary, our simulation tool brings significant benefits for developing
realistic attack techniques, as it saves the heavy overhead of creating real
adversarial objects while at the same time ensures the attack undergoes the
same environmental transformations as the defense.

Figure 8: Validation of a ShapeShifter attack in CARLA. An adversarial texture
is computed and then applied to a Tesla Model 3 car in the simulation. This
attack is successful as it causes the vehicle to be misclassified as an
orange.

Hence, our tool provides a cheap and convenient alternative to the real-world
adversarial dataset generation, which has an prohibiting cost for realistic
adversarial examples and thus is very difficult to be of large scale, if even
possible. With the simulation tool, it becomes much easier to insert 2D or 3D
adversarial perturbations back into the simulation. And due to its low cost
and low time consumption, generating a large scale synthetic adversarial
dataset with realistic attack samples becomes feasible. Even further, as
discussed in Figure 1, streaming texture capability provided by CARLA, which
allows the texture of an object to be changed in real time during the
simulation, opens an door for the adversarial training with realistic
adversarial noises.

##  6 Conclusion

Adversarial machine learning research suffers from a lack of large scale, high
quality datasets with realistic, physically plausible adversarial examples. We
overcome these limitations by using simulation to insert adversarial examples
and generate synthetic data. Our tools enables a new way for distributing
datasets that are reproducible, configurable, extensible, and scalable. Rather
than downloading large, static archives, researchers can distribute a dataset
in the form of configuration files and locally re-create the dataset by
running our tools and CARLA. They can also easily extend the dataset with new
configurations based on their specific needs and dynamically insert
adversarial examples in a realistic manner. In future work, we would like to
conduct large scale studies on the efficacy of attacks from various sources
such as those shown in Figure 1. We believe this tool will enable more
realistic evaluation of adversarial examples.

## References

  * Braunegg et al. (2020) Braunegg, A., Chakraborty, A., Krumdick, M., Lape, N., Leary, S., Manville, K., Merkhofer, E., Strickhart, L., and Walmer, M.  APRICOT: A dataset of physical adversarial attacks on object detection.  In _European Conference on Computer Vision_ , pp. 35–50. Springer, 2020.
  * Brown et al. (2017) Brown, T., Mane, D., Roy, A., Abadi, M., and Gilmer, J.  Adversarial patch.  2017\.  URL https://arxiv.org/pdf/1712.09665.pdf.
  * Carlini & Wagner (2017) Carlini, N. and Wagner, D.  Towards evaluating the robustness of neural networks.  In _2017 IEEE Symposium on Security and Privacy (SP)_ , pp. 39–57, 2017.  doi: 10.1109/SP.2017.49.
  * Chen et al. (2018) Chen, S.-T., Cornelius, C., Martin, J., and Chau, D. H. P.  Shapeshifter: Robust physical adversarial attack on Faster R-CNN object detector.  In _Joint European Conference on Machine Learning and Knowledge Discovery in Databases_ , pp. 52–68. Springer, 2018.
  * Cordts et al. (2016) Cordts, M., Omran, M., Ramos, S., Rehfeld, T., Enzweiler, M., Benenson, R., Franke, U., Roth, S., and Schiele, B.  The cityscapes dataset for semantic urban scene understanding.  In _Proc. of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2016.
  * Cornelius et al. (2019) Cornelius, C., Chen, S.-T., Martin, J., and Chau, D. H.  Talk proposal: Towards the realistic evaluation of evasion attacks using CARLA.  _ArXiv_ , abs/1904.12622, 2019.
  * Crall (2020) Crall, J.  Kitware coco, 2020.  URL https://gitlab.kitware.com/computer-vision/kwcoco.
  * Dosovitskiy et al. (2017) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., and Koltun, V.  CARLA: An open urban driving simulator.  In _Proceedings of the 1st Annual Conference on Robot Learning_ , pp. 1–16, 2017.
  * Dwibedi et al. (2017) Dwibedi, D., Misra, I., and Hebert, M.  Cut, paste and learn: Surprisingly easy synthesis for instance detection.  In _Proceedings of the IEEE international conference on computer vision_ , pp. 1301–1310, 2017.
  * Fabbri et al. (2021) Fabbri, M., Brasó, G., Maugeri, G., Cetintas, O., Gasparini, R., Ošep, A., Calderara, S., Leal-Taixé, L., and Cucchiara, R.  MOTSynth: How can synthetic data help pedestrian detection and tracking?  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , pp. 10849–10859, 2021.
  * Feng et al. (2022) Feng, R., Jha, S., and Prakash, A.  Constraining the attack space of machine learning models with distribution clamping preprocessing.  _arXiv preprint arXiv:2205.08989_ , 2022.
  * Geiger et al. (2012) Geiger, A., Lenz, P., and Urtasun, R.  Are we ready for autonomous driving? the kitti vision benchmark suite.  In _Computer Vision and Pattern Recognition (CVPR), 2012 IEEE Conference on_ , pp. 3354–3361. IEEE, 2012.  URL https://ieeexplore.ieee.org/abstract/document/6248074.
  * Goodfellow et al. (2015) Goodfellow, I., Shlens, J., and Szegedy, C.  Explaining and harnessing adversarial examples.  In _International Conference on Learning Representations_ , 2015.  URL http://arxiv.org/abs/1412.6572.
  * Hendrycks & Dietterich (2019) Hendrycks, D. and Dietterich, T.  Benchmarking neural network robustness to common corruptions and perturbations.  _arXiv preprint arXiv:1903.12261_ , 2019.
  * Johnson-Roberson et al. (2017) Johnson-Roberson, M., Barto, C., Mehta, R., Sridhar, S. N., Rosaen, K., and Vasudevan, R.  Driving in the matrix: Can virtual worlds replace human-generated annotations for real world tasks?  In _2017 IEEE International Conference on Robotics and Automation (ICRA)_ , pp. 746–753, 2017.  doi: 10.1109/ICRA.2017.7989092.
  * Koh et al. (2021) Koh, P. W., Sagawa, S., Marklund, H., Xie, S. M., Zhang, M., Balsubramani, A., Hu, W., Yasunaga, M., Phillips, R. L., Gao, I., et al.  Wilds: A benchmark of in-the-wild distribution shifts.  In _International Conference on Machine Learning_ , pp. 5637–5664. PMLR, 2021.
  * Lu et al. (2017) Lu, J., Sibai, H., Fabry, E., and Forsyth, D. A.  No need to worry about adversarial examples in object detection in autonomous vehicles.  _ArXiv_ , abs/1707.03501, 2017.
  * Madry et al. (2017) Madry, A., Makelov, A., Schmidt, L., Tsipras, D., and Vladu, A.  Towards deep learning models resistant to adversarial attacks.  _arXiv preprint arXiv:1706.06083_ , 2017.
  * Papernot et al. (2016) Papernot, N., McDaniel, P., Jha, S., Fredrikson, M., Celik, Z. B., and Swami, A.  The limitations of deep learning in adversarial settings.  In _2016 IEEE European Symposium on Security and Privacy (EuroS P)_ , pp. 372–387, 2016.  doi: 10.1109/EuroSP.2016.36.
  * Pintor et al. (2022) Pintor, M., Angioni, D., Sotgiu, A., Demetrio, L., Demontis, A., Biggio, B., and Roli, F.  Imagenet-Patch: A dataset for benchmarking machine learning robustness against adversarial patches.  _ArXiv_ , abs/2203.04412, 2022.
  * Richter et al. (2016) Richter, S. R., Vineet, V., Roth, S., and Koltun, V.  Playing for data: Ground truth from computer games.  _ArXiv_ , abs/1608.02192, 2016.
  * Sitawarin et al. (2018) Sitawarin, C., Bhagoji, A. N., Mosenia, A., Mittal, P., and Chiang, M.  Rogue signs: Deceiving traffic sign recognition with malicious ads and logos.  _arXiv preprint arXiv:1801.02780_ , 2018.
  * Szegedy et al. (2018) Szegedy, C., Zaremba, W., Sutskever, I., Bruna, J., Erhan, D., Goodfellow, I., and Fergus, R.  Intriguing properties of neural networks.  In _Joint European Conference on Machine Learning and Knowledge Discovery in Databases_ , pp. 52–68. Springer, 2018.
  * Threet et al. (2021) Threet, M., Busho, C., Harguess, J., Jutras, M., Lape, N., Leary, S., Manville, K., Tan, M., and Ward, C.  Physical adversarial attacks in simulated environments.  In _2021 IEEE Applied Imagery Pattern Recognition Workshop (AIPR)_ , pp. 1–5, 2021.  doi: 10.1109/AIPR52630.2021.9762099.
  * Voigtlaender et al. (2019) Voigtlaender, P., Krause, M., Osep, A., Luiten, J., Sekar, B. B. G., Geiger, A., and Leibe, B.  Mots: Multi-object tracking and segmentation.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp. 7942–7951, 2019.

##  Appendix A Appendix

The synthetic data collection tool described in Section 4 consists of two
major components, a data saver and an annotator. The data saver takes a
configuration file as input to configure the CARLA simulation scenario. This
configuration includes CARLA server configurations, weather settings, and
configurations for spawning actors (i.e., vehicles and pedestrians) and
sensors for data capture. LABEL:lst:config_yaml shows an example of such a
configuration that captures $300$ frames from three sensors (RGB camera, depth
camera, and instance segmentation sensor) with two pedestrians. These sensors
are statically placed to capture the two pedetrians as they walk past these
sensors. The two pedestrians that walk past each other are specified by their
the start positions, destination positions, and walking speeds. Finally, the
configuration sets a seed to enable repeated generations of a particular
scenario. Figure 9 shows example frames captured from three different captures
under different weather conditions using configurations shown in
LABEL:lst:config_yaml and LABEL:lst:weather_config. Our tool enables
reproducible capture under a variety of conditions like weather and lighting.

(a) Sunny

(b) Rainy

(c) Foggy

Figure 9: Our tool can repeatably collect data over a weather distribution
shift. We collected frames from the same scenario for different weather
conditions: 9(a) sunny, 9(b) rainy, and 9(c) foggy. Listing 1: An example
config.yaml file that generates 2 pedestrians with 3 static sensors (RGB,
depth and instance segmentation).

⬇

1carla

2 host "127.0.0.1"

3 port 2000

4 timeout 5.0

5 sync

6 fps 30

7 timeout 2.0

8 seed 30

9 townmap "Town10HD"

10 traffic_manager_port 8000

11 retry 10

12

13output_dir "_out"

14max_frames 300

15

16# Sunny weather

17weather

18 cloudiness 0.0

19 precipitation 0.0

20 precipitation_deposits 0.0

21 wind_intensity 0.0

22 sun_azimuth_angle 0.0

23 sun_altitude_angle 10.0

24 fog_density 0.0

25 fog_distance 0.0

26 wetness 0.0

27

28spawn_actors

29 # Spawn Pedestrian 1

30 - blueprint

31 name "walker.pedestrian.*"

32 attr {role_name "hero1", is_invincible "false"}

33 speed 1.4 # Between 1 and 2 m/s (default is 1.4 m/s).

34 transform

35 location {x -91, y 170, z 0.6}

36 rotation {yaw -90.0}

37 destination_transform

38 location {x -91, y 150, z 0.6}

39 # Spawn statically placed sensors - RGB, Depth and Instance Segmentation

40 - blueprint

41 name "sensor.camera.rgb"

42 attr {"image_size_x" "800", "image_size_y" "600"}

43 transform

44 location {x -95, y 160, z 1.6}

45 rotation {yaw 0.0}

46 - blueprint

47 name "sensor.camera.depth"

48 attr {"image_size_x" "800", "image_size_y" "600"}

49 transform

50 location {x -95, y 160, z 1.6}

51 rotation {yaw 0.0}

52 - blueprint

53 name "sensor.camera.instance_segmentation"

54 attr {"image_size_x" "800", "image_size_y" "600"}

55 transform

56 location {x -95, y 160, z 1.6}

57 rotation {yaw 0.0}

58 # Spawn Pedestrian 2

59 - blueprint

60 name "walker.pedestrian.*"

61 attr {role_name "hero2", is_invincible "false"}

62 speed 2.0

63 transform

64 location {x -91, y 150, z 0.6}

65 rotation {yaw 90.0}

66 destination_transform

67 location {x -91, y 170, z 0.6}

Listing 2: An example to show 2 different weather settings used to create the
examples shown in Figure 9

⬇

1# Rainy weather

2weather

3 cloudiness 60.0

4 precipitation 60.0

5 precipitation_deposits 60.0

6 wind_intensity 60.0

7 sun_azimuth_angle -1.0

8 sun_altitude_angle 15.0

9 fog_density 3.0

10 fog_distance 0.75

11 wetness 0.0

12

13# Foggy weather

14weather

15 cloudiness 0.0

16 precipitation 0.0

17 precipitation_deposits 0.0

18 wind_intensity 0.0

19 sun_azimuth_angle 0.0

20 sun_altitude_angle 10.0

21 fog_density 100.0

22 fog_distance 1.0

23 wetness 0.0


# Experimental verification of the temperature coefficient of resistivity

Robert D. Polak Department of Physics, Loyola University Chicago, Chicago, IL
60660, USA Michael R. Harris Department of Physics, Loyola University Chicago,
Chicago, IL 60660, USA Kiet A. Nguyen Department of Physics, Loyola University
Chicago, Chicago, IL 60660, USA Anthony Kearns Department of Physics, Loyola
University Chicago, Chicago, IL 60660, USA

##

We have created an experimental procedure for determining the temperature
coefficient of resistivity, $\alpha_{R}$, for introductory physics
laboratories. As in the procedure from Henry [1], this method examines the
relationship between temperature and resistivity to establish $\alpha_{R}$
within 10% of the accepted value.

Electrical resistivity, $\rho$, varies with temperature according to:

$\rho=\rho_{o}(1+\alpha_{R}(T-T_{o}))$ (1)

where $\rho_{o}$ is the resistivity for a given temperature $T_{o}$, $T$ is
the temperature of the material, and $\alpha_{R}$ is the temperature
coefficient of resistivity. For a wire of length, $L$, and cross-sectional
area, $A$, the resistance of a wire, $R$, is defined accordingly as

$R=\rho\frac{L}{A}\ .$ (2)

While resistance will increase as a product of both increased length and
resistivity, the increase in length provides a negligible increase in
resistance. This is evident from observing that the thermal coefficient of
resistivity is approximately two orders of magnitude larger than the
coefficient of thermal expansion. As such, the change in resistivity is
primarily responsible for the increase in resistance. Hence, $R$ will vary
similarly with

$R=R_{o}(1+\alpha_{R}(T-T_{o}))$ (3)

where $R_{o}$ is the resistance of the wire at a temperature $T_{o}$.

By applying a current through the wire, its temperature will also vary as a
result of Joule heating and the resistance of the wire can be measured based
on a given current, I, and difference in voltage, $\Delta V$, by

$R=\frac{\Delta V}{I}\ .$ (4)

Hence, by measuring the resistance as a function of temperature, we can
determine $\alpha_{R}$ by plotting $R$ vs. $T-T_{o}$ and performing a linear
fit using Eq. (3).

To perform the experiment, we created a closed circuit (see Fig. 1) in which a
carbon steel wire [2] is suspended under tension above a surface, as in most
stringed instruments. Two digital multimeters were used to record the voltage
across and current through a 0.016 inch (0.406 mm) diameter, 40-cm long wire.
Temperature measurements were taken using liquid crystal thermometers [3]
placed in thermal contact with the wire by fastening them to the wire with an
adhesive backing. Two thermometers were used, one ranging from $14-31^{o}C$
and the other from $32-49^{o}C$, to provide an overall temperature range of
$14-49^{o}C$. For the most accurate temperature readings, we found it is
essential to avoid all contact with the thermometers during the experiment. To
collect the data, we applied different currents to the wire, ranging from
$0.2A$ to $1.0A$. We used a BK Precision 1787B power supply that allows for
digital control of the current. We found that having initial steps of $0.2A$
and later reduced to $0.1A$ created consistent temperature changes in the wire
of $2-3^{o}C$. The experiment proved much more difficult to complete using an
analog power supply because of the difficulty in creating the precise changes
in current needed to have a well formed data set. After allowing around 30
seconds for the system to reach thermal equilibrium, the recorded temperature
of each trial was given as the uppermost visible temperature reading of the
liquid crystal thermometer, as seen in Fig. 2. We recorded the temperature of,
current through and voltage across the wire, and calculated its resistance
using Eq. (4).

By graphing the resistance of the wire as a function of $T-T_{o}$, where
$T_{o}$ is the temperature of the wire with the lowest current applied, we can
then apply a linear fit to the data with the y-intercept yielding $R_{o}$ and
slope giving $R_{o}\alpha_{R}$, according to Eq. (3). Figure 4 shows example
experimental results with the fit giving $R_{o}=0.744\Omega$ and
$\alpha_{R}=0.0039K^{-1}$. This is within 5% of the accepted value of
$\alpha_{R}=0.0041K^{-1}$ [4]. Repeated experiments found these results to be
reproducible with $\alpha_{R}$ consistently measured within 10% of the
accepted value.

To get the best results, we found that the resistance of the wire should be at
least $0.3\Omega$ to allow for accurate resistance measurements. Furthermore,
the wires need to be thick enough to support the thermometers. As such, we
achieved the best results when using steel as opposed to other materials with
lower resistivity and tensile strength, such as copper and aluminum.

This experiment can be completed in less than 2 hours and uses equipment that
is commonly present in a typical introductory physics lab, with the exception
of relatively low cost supplies such as music wire and liquid crystal
thermometers. It also reinforces key ideas from introductory physics such as
conservation of energy, where electrical energy becomes thermal energy, Ohm’s
Law, and the temperature dependence of resistance.

## References

1\. D. Henry, “Resistance of a wire as a function of temperature”, The Physics
Teacher 33, 96-97 (1995) https://doi.org/10.1119/1.2344149
2\. Precision Brand Music Wire (0.016 inch diameter); UPC. No. 21016
3\. TelaTemp reversible LCT strip model 416-2 ($14-31^{o}C$) and 416-3
($32-49^{o}C$)
4\. S. Yafei , N. Dongjie, and S. Jing, 4th IEEE Conference on Industrial
Electronics and Applications, 368-372 (2009).

Figure 1: The experimental setup (bottom) with the wire suspended above the
box, and a circuit diagram (top). The circuit consists of two multimeters set
up to measure current and voltage and a digital power supply. The liquid
crystal thermometers are affixed to the top of the wire. Figure 2: An example
of a temperature reading corresponding to $37^{\circ}C$ ($98^{\circ}F$),
according to the procedure of reading the uppermost visible indication of the
liquid crystal thermometer (the yellow bar spanning $37^{\circ}C$). Figure 3:
Plot of resistance against the associated change in temperature for currents
from 0.2 $A$ to 1.0 $A$. A linear fit of the data yields
$\alpha_{R}=0.0039K^{-1}$.


#
Towards a Model of Quarks and Leptons

Y. H. Ahn  Institute for Convergence Fundamental Study, Seoul National
University of Science and Technology,
Seoul 139-743, Korea  Sin Kyu Kang  School of Liberal Arts, Seoul National
University of Science and Technology, Seoul 139-743, Korea  Hyun Min Lee
Department of Physics, Chung-Ang University, Seoul 06974, Korea.

###### Abstract

We propose an extra-dimension framework on the orbifold $S^{1}/Z_{2}$ for
understanding the origin of the fermion mass and mixing hierarchies.
Introducing the flavor symmetry $G_{F}(=$non-Abelian$\times$Abelian) as well
as the extra gauged $U(1)$ symmetries through the bulk, we regard the $SU(2)$
singlet and doublet fermions in the Standard Model (SM) to be localized at the
separate 3-branes and let the extra $SU(2)$ singlet flavored fermions in the
bulk couple to the SM fermions at the 3-branes. The extra $U(1)$ symmetries
satisfy the $U(1)$ gravitational anomaly-free condition, playing a crucial
role in achieving the desirable fermion mass and mixing hierarchies and making
the flavored axion naturally light. The singlet scalar fields, the flavon
fields, are responsible for the spontaneous breaking of $G_{F}$ on the two
3-branes, while the $SU(2)$ singlet flavored fermions are integrated out to
give rise to the effective Yukawa couplings for the SM fermions, endowed with
the information of $G_{F}$ breaking in the two sectors. The flavored axion
from the PQ symmetry is also proposed for solving the strong CP problem and
being a dark matter candidate in our model.

##  I Introduction

The observed hierarchies in the masses and mixings of quarks and leptons are
one of the most puzzling problems in particle physics. A plausible explanation
is to introduce a new gauge symmetry which is spontaneously broken at some
ultraviolet (UV) scale and leaves behind its global subgroup. In this case, in
the low-energy effective theory, the symmetry structure is composed of the
Standard Model (SM) gauge symmetry $G_{\rm SM}=SU(3)_{C}\times SU(2)_{L}\times
U(1)_{Y}$ and a remnant global symmetry which plays an essential role in
making desirable flavor structure of quarks and leptons Ahn:2016typ . It is
known that flavor-dependent $U(1)$ gauge symmetries and/or non-Abelian
discrete symmetries can arise from the isometry of the compactified extra
dimensions in string theory. In addition, the compactification of the extra-
dimensions can be accompanied by certain 3-branes (4 dimensional (4D) surfaces
embedded in higher dimensional spaces)Horava:1995qa .

Inspired by some string compactifications, in this paper, we propose a
mechanism for generating the fermion mass and mixing hierarchies in the extra-
dimension framework with the flavor symmetry. The extra gauged $U(1)$
symmetries are also introduced under the $U(1)$ gravitation anomaly-free
condition and they are crucial to achieve the desirable fermion mass and
mixing hierarchies in this scenario. These are the features in this work,
which are distinguishable from other approaches to tackle the flavor problem
in the extra-dimension framework Mirabelli:1999ks ; Gherghetta:2000qt ;
Kaplan:2000av ; Huber:2000ie ; Agashe:2004cp ; Casagrande:2008hr ;
Agashe:2008fe ; Archer:2012qa ; Frank:2015sua ; Ahmed:2019zxm ; Wojcik:2019kfq
; Kang:2020cxo . A simple toy model with vector-like leptons for seesaw
leptons has been recently proposed in light of the muon $g-2$ anomaly
Lee:2021gnw , but without involving a flavor symmetry.

For our purpose, we construct a higher dimensional theory compactified on the
orbifold $S^{1}/Z_{2}$ with a global symmetry group for flavors,
$G_{F}=U(1)\times$non-Abelian finite group, which might be originated from
certain string compactifications. A set of SM gauge singlet scalar fields
${\cal F}$, charged under $G_{F}$, the so-called flavon fields, are located at
two 3-branes in the extra dimension. In the 4D effective Lagrangian, the
flavor fields act on dimension-four(three) operators well-sewed by
$G_{F}\times G_{\rm SM}$ at different orders to generate the effective
interactions for the SM and the right-handed neutrinos, as follows;

$\displaystyle{\cal L}_{4D}=\tilde{c}_{1}{\cal O}_{3}\,{\cal F}\,\tilde{{\cal
X}}+{\cal O}_{4}\sum^{\rm finite}_{n=0}c_{n}\Big{(}\frac{\cal
F}{M_{5}}\Big{)}^{n}\,{\cal X}_{n}\,,$ (1)

where ${\cal O}_{3(4)}$ are dimension-3(4) operators, $\tilde{c}_{1}(c_{n}$)
are the complex coefficients of order unity, and $\tilde{{\cal X}}({\cal
X}_{n}$) are dimensionless parameters induced due to the non-local effects of
bulk messenger fields. ${\cal F}$ acquire vacuum expectation values (VEVs)
$\langle{\cal F}\rangle$ from some dynamics, thereby breaking $G_{F}$. Then,
the vacuum structure of the flavons plays a crucial role in achieving the SM
fermion mass and mixing hierarchies.

As a bonus in our scenario, the pseudo-Goldstone modes coming from the flavon
fields are localized on the 3-branes, becoming candidates for flavored-axions
$A_{i}$ (and a QCD axion) Ahn:2014gva  with decay constants determined by
$\langle{\cal F}\rangle$. However, it is well known that non-perturbative
quantum gravitational anomaly effects Kamionkowski:1992mf  could spoil the
axion solution to the strong CP problem. In order to keep the axion solution
in our scenario, we need to suppress the explicit breaking effects of the
axionic shift symmetry by gravity and consistently couple gravity to matter.
To this, we impose the $U(1)$-mixed gravitational anomaly-free conditions for
the extra gauged $U(1)$ symmetries, in turn, obtaining the constraints on the
$U(1)$ charges of quarks and leptons.

##  II Flavor physics embedded into 5D theory

We consider a 5D theory for flavors compactified on the orbifold $S^{1}/Z_{2}$
where the extra-dimension on the circle $S^{1}$ is identified $y$ with $-y$
Randall:1999ee . The orbifold fixed points at $y=0$ and $y=L$, the boundaries
of the 5D spacetime, are the locations of two 3-branes, We assume that all the
ordinary matter fields localized at either brane and they are charged under
the flavor symmetry $G_{F}$. We specify $G_{F}=SL_{2}(F_{3})\times U(1)_{X}$
where $SL_{2}(F_{3})$ is the symmetry group of the double
tetrahedronFeruglio:2007uu .

The metric solution to the 5D Einstein’s equations, respecting the 4D Poincare
invariance in the $x^{\mu}$ direction, is given by

$\displaystyle ds^{2}=e^{2\sigma(y)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}-dy^{2}$ (2)

where the extra dimension is compactified on an interval $y\in[0,L]$, the warp
factor is given by $\sigma(y)=ky$ with
$k=\sqrt{-\frac{\Lambda}{6M^{3}_{5}}}>0$ and $\Lambda$ being the bulk
cosmological constant, and the 4D Minkowski flat metric is $\eta_{\mu\nu}={\rm
diag}(+,-,-,-)$. Note that, in Eq.(2), we can always take $\sigma(0)=0$ by
rescaling the $x^{\mu}$ coordinates. Then, the 4D reduced Planck mass
$M_{P}\simeq 2.43\times 10^{18}$ GeV can be extracted in terms of the 5D
Planck mass $M_{5}$ as

$\displaystyle
M^{2}_{P}=M^{3}_{5}\int^{L}_{-L}dy\,e^{2\sigma(y)}=\frac{M^{3}_{5}}{k}\big{(}e^{2kL}-1\big{)}\,$
(3)

where $M_{5}$ is assumed to be higher than the electroweak scale, but $M_{5}$
and $k$ are lower than $M_{P}$. Then, the scale of flavor dynamics would be
given by the UV cutoff $M_{5}$.

For exchanging the information of the breaking of flavor symmetry $G_{F}$
between the two 3-branes, we introduce $SU(2)$ singlet flavored bulk fermions
$\Psi_{f_{i}}$ and their mirror partners $F_{f_{i}}$ charged under $G_{\rm
SM}\times G_{F}$, propagating in a 5D space with the metric Eq.(2). Here,
$f_{i}=u_{i}$ (up-type quark), $d_{i}$(down-type quark), $l_{i}$ (charged
lepton), and $F_{f_{i}}$ is introduced to keep the conservation of electric
charge. While $\Psi_{f_{i}}$ with hypercharges $Y_{f}$ and masses $M_{f_{i}}$
interact with the normal matter fields confined at $y=0$ or $L$ brane,
$F_{f_{i}}$ with hypercharges $-Y_{f}$ and masses $M_{f_{i}}$ does not do so
due to $U(1)_{Y}$ symmetry. Unlike Ref.Goldberger:1999uk , in this work, the
compactification length $L$ can be constrained thanks to the introduction of
flavored bulk fermions. In Table-1, for flavon fields ${\cal
F}=\Phi_{S},\Theta(\tilde{\Theta}),\Phi_{T},\rho,\eta,\chi,\tilde{\chi}$ and
flavored bulk fermions $\Psi_{f}$, we present the representations of
$SL_{2}(F_{3})$ and quantum charges under $U(1)_{X_{1}}\times
U(1)_{X_{2}}\times U(1)_{X_{T}}$.

Table 1:  Representations and quantum charges of SM singlet flavon fields and bulk fermions under $G_{F}$. Field | $SL_{2}(F_{3})$ | $U(1)_{X_{1}}$ | $U(1)_{X_{2}}$ | $U(1)_{X_{T}}$ | brane(y)
---|---|---|---|---|---
$\Phi_{S}$ : $\Theta(\tilde{\Theta})$ |  ${\bf 3}$: ${\bf 1}$ | $1$ | $0$ | $0$ | $0$
$\Phi_{T}$ : $\varrho$ |  ${\bf 3}$ : ${\bf 1}$ | $0$ | $0$ |  $1$, $-1$ | $L$
$\eta$ | ${\bf 2}^{\prime}$ | $0$ | $0$ | $0$ | $L$
$\chi$, $\tilde{\chi}$ | ${\bf 1}$ | $0$ | $1,-1$ | $0$ | $L$
$\Psi_{l}$ | ${\bf 3}$ | $-13/2$ | 0 | 0 |
$\Psi_{u_{1}}:\Psi_{d_{1}}$ | ${\bf 1}$ |  $-6$ : $-5$ | 0 | 0 |
$\Psi_{u_{2}}:\Psi_{d_{2}}$ | ${\bf 1^{\prime}}$ |  $-6$ : $-5$ | 0 | 0 |
$\Psi_{u_{3}}:\Psi_{d_{3}}$ | ${\bf 1^{\prime\prime}}$ | 0 : 3 | 0 | 0 |

All ordinary matter and Higgs fields charged under $G_{\rm SM}$ with +1 and 0
charges under $U(1)_{R}$, respectively, are localized at either brane. Then,
all the SM fermion mixings and masses can be generated by non-local effects
involving both branes and local breaking effects of $G_{F}$ due to flavon
fields. For the orbifold compactification, we set all elementary fermions form
a chiral set, and their group representations and quantum numbers are
summarized in Table-2. From the $U(1)_{X_{k}}\times[SU(3)_{C}]^{2}$ anomaly
coefficient defined by $\delta^{\rm
G}_{k}\delta^{ab}=2\sum_{\psi_{f}}X_{k\psi_{f}}{\rm Tr}(T^{a}T^{b})$ in the
QCD instanton backgrounds with $T^{a}$ and $X_{k}$ being $SU(3)_{C}$
generators and $U(1)_{X_{k}}$ charge, respectively, we get $\delta^{\rm
G}_{1}=-9$ and $\delta^{\rm G}_{2}=-11$ with the domain-wall number $N_{\rm
DW}=1$. In this model, $U(1)_{X_{1}}\times U(1)_{X_{2}}\equiv
U(1)_{\tilde{X}}$ is a pure axial symmetry $U(1)_{\rm PQ}$. A color anomaly
coefficient $N_{C}$ of $U(1)_{\tilde{X}}\times[SU(3)_{C}]^{2}$ and an
electromagnetic one $E_{A}$ of $U(1)_{\tilde{X}}\times[U(1)_{\rm EM}]^{2}$ are
defined by $N_{C}\equiv 2{\rm Tr}[\tilde{X}_{\psi_{f}}T^{a}T^{a}]=2\delta^{\rm
G}_{1}\delta^{\rm G}_{2}$ and
$E_{A}=2\sum_{\psi_{f}}\tilde{X}_{\psi_{f}}(Q^{\rm em}_{\psi_{f}})^{2}$ with
$Q^{\rm em}_{\psi_{f}}$ being the $U(1)_{\rm EM}$ charge of field $\psi_{f}$,
respectively. Then, their ratio becomes $E_{A}/N_{C}=-761/99$.

Table 2:  Representations of quark, lepton, and electroweak two Higgs $H_{u(d)}$ fields under $SL_{2}(F_{3})\times U(1)_{X_{i}}$ ($i=1,2,T$). All fields are left-handed particles/antiparticles. All of them have zero $U(1)_{R}$.  Field | $SL_{2}(F_{3})$ | $U(1)_{X_{1}}$ | $U(1)_{X_{2}}$ | $U(1)_{X_{T}}$ | brane
---|---|---|---|---|---
$Q_{1}$, $Q_{2}$, $Q_{3}$ |  ${\bf 1}$ ${\bf 1}^{\prime}$ ${\bf 1}^{\prime\prime}$ | $-8,-6,0$ | $0,0,0$ | $0,0,0$ | $y=0$
${\cal D}^{c}$, $b^{c}$ |  ${\bf 2}^{\prime}$ ${\bf 1}^{\prime}$ |  $5$, $-3$ |  $-14$, $18$ | $0,0$ | $y=L$
${\cal U}^{c}$, $t^{c}$ |  ${\bf 2}^{\prime}$ ${\bf 1}^{\prime}$ |  $6$, $0$ |  $-6$, $11$ | $0,0$ | $y=L$
$L$ | ${\bf 3}$ | $-\frac{15}{2}$ | $0$ | $0$ | $y=0$
$e^{c}$, $\mu^{c}$, $\tau^{c}$ |  ${\bf 1}$, ${\bf 1}^{\prime\prime}$, ${\bf 1}^{\prime}$ | $\frac{13}{2},\frac{13}{2},\frac{13}{2}$ |  $41$, $31$, $21$ |  $-2$, $1$, $1$ | $y=L$
$N^{c}$ | ${\bf 3}$ | $-\frac{1}{2}$ | $0$ | $0$ | $y=0$
$H_{u(d)}$ | ${\bf 1}$ | $0$ | $0$ | $0$ | $y=0$

The flavor symmetry $G_{F}$ is spontaneously broken by the nontrivial VEVs of
flavons. Note that the $U(1)$ charges of the fields are determined so as to
satisfy the $U(1)$ gravitation anomaly-free condition and the empirical
hierarchies of fermion masses and mixings. For instance, in a supersymmetric
model, the brane-localized superpotential for flavons with $G_{F}$ invariance
is given in Eq.(28). From the minimization conditions of the $F$-term scalar
potentials, the VEVs of $\Phi_{S}$ and $\tilde{\Theta}$ localized at $y=0$
brane are obtained

$\displaystyle\langle\Phi_{S}\rangle=\frac{v_{S}}{\sqrt{2}}(1,1,1)\,,\quad\langle\Theta\rangle=\frac{v_{\Theta}}{\sqrt{2}}\,,\quad\langle\tilde{\Theta}\rangle=0\,,$
(4)

with $\kappa=v_{S}/v_{\Theta}$ in supersymmetric limit. For $\Phi_{T}$,
$\varrho$, $\chi(\tilde{\chi})$, and $\eta$ localized at $y=L$ brane

$\displaystyle\langle\Phi_{T}\rangle=\frac{v_{T}}{\sqrt{2}}(1,0,0)\,,\qquad\qquad\langle\varrho\rangle=\frac{v_{\varrho}}{\sqrt{2}}\,,$
$\displaystyle~{}~{}\langle\chi\rangle=\langle\tilde{\chi}\rangle=\frac{v_{\chi}}{\sqrt{2}}\,,\qquad\quad~{}\langle\eta\rangle=\frac{v_{\eta}}{\sqrt{2}}(1,0)\,.$
(5)

Denoting $\nabla_{\cal F}\equiv v_{\cal F}/(\sqrt{2}M_{5})$ and following the
procedure in Ref.Ahn:2018cau , we obtain

$\displaystyle\nabla_{\Theta}=\nabla_{S}/\kappa=\big{|}\delta^{\rm
G}_{1}/\delta^{\rm G}_{2}\big{|}\sqrt{2/(1+\kappa^{2})}\nabla_{\chi}.$ (6)

The brane-localized Yukawa Lagrangian for up- and down-type quarks and
leptons, which is invariant under $G_{\rm SM}\times G_{F}$, involve the
exchanges of their flavored bulk fermions, but no tree-level interactions in
the SM:

$\displaystyle{\cal
L}_{Y}=\delta(y)\big{\\{}\sum\limits_{i}^{3}Z^{u}_{ii}\Psi^{c}_{ui}Q_{i}H_{u}+Z^{d}_{ii}\Psi^{c}_{di}Q_{i}H_{d}+\sum\limits_{i>j}^{3}Z^{d}_{ij}\Psi^{c}_{di}Q_{j}(\Phi_{S}\Phi_{S})_{j}\frac{H_{d}}{M^{2}_{5}}+\big{(}y_{\nu}\frac{\Theta}{M_{5}}+\tilde{y}_{\nu}\frac{\Phi_{S}}{M_{5}}\big{)}N^{c}LH_{u}$
$\displaystyle+Z_{L}(\Psi^{c}_{\ell}L)_{{\bf
1}}H_{d}+\frac{1}{2}(y_{R}\Phi_{S}+y_{\Theta}\Theta+y_{\tilde{\Theta}}\tilde{\Theta})(N^{c}N^{c})\big{\\}}+\delta(y-L)\big{\\{}\hat{Z}_{t3}t^{c}\Psi_{u3}+Z_{u2}{\cal
U}^{c}\eta\Psi_{u2}$ $\displaystyle+Z_{u1}{\cal
U}^{c}\eta\frac{\Phi_{T}\varrho}{M^{2}_{5}}\Psi_{u1}+(t\rightarrow b,{\cal
U}\rightarrow{\cal D},u\rightarrow
d)+Z_{e}e^{c}\Psi_{\ell}\frac{\Phi_{T}\Phi_{T}}{M_{5}}+\sum_{\omega=\mu,\tau}Z_{\omega}\omega^{c}\Psi_{\ell}\Phi_{T}\frac{\varrho^{2}}{M^{2}_{5}}\big{\\}}\,,$
(7)

where higher order terms combined by ($\eta\eta$, $\Phi_{T}\Phi_{T}$,
$\Phi_{T}\varrho$) are omitted. In Eq.(7),
$\\{Z^{f}_{ij},Z_{L},Z_{fi},Z_{e,\mu,\tau}\\}$ and $\hat{Z}_{fi}$ are
functions of flavons, i.e. $Z=Z[(\frac{\cal F}{M_{5}})^{p}]$ with $p$ being a
combination of $U(1)_{X_{i}}$ charges assigned so as for ${\cal L}_{Y}$
invariant under $U(1)_{X_{i}}$, and their mass dimensions are $-1/2$ and
$+1/2$, respectively. As will be shown later, the exponent $p$ is responsible
for the hierarchies of fermion masses and mixings. The coupling
$y_{\nu}(\tilde{y}_{\nu}$) and $y_{R,(\Theta,\tilde{\Theta})}$ are
respectively a function of ${\cal F}$ and their mass dimensions are all zero.

##  III 4D effective theory

For the effective theory in 4D, we expand the left- and right-handed bulk
fermions in terms of Kaluza-Klein(KK) modes as
$\Psi_{f_{i}L(R)}(x,y)=\frac{e^{-\frac{3}{2}\sigma(y)}}{\sqrt{L}}\sum\limits_{n}\psi^{n}_{f_{i}L(R)}(x)\,f^{n}_{iL(R)}(y)$.
They are chosen to obey the equation of motion coming from the 4D action
$S=\sum\limits_{n}\int
d^{4}x\bar{\psi}^{n}_{i}(f_{i}\gamma_{\mu}D^{\mu}-m_{f}^{n})\psi^{n}_{f_{i}}$
where $m^{n}_{f}$ is the 4D mass of the $n$-th KK mode, with the normalization
condition $\frac{1}{L}\int^{L}_{0}dy\,f^{m}_{iL(R)}f^{n}_{iL(R)}=\delta_{mn}$.
We choose a gauge $A_{5}=0$ such that the KK modes are independent of the
gauge fields. In terms of left- and right-handed spinors $f^{n}_{iL,R}$, we
obtain

$\displaystyle\big{(}\partial_{y}+\frac{1}{2}\sigma^{\prime}-M_{f_{i}}\big{)}f^{n}_{iR}(y)=e^{-\sigma}m_{f}^{n}\,f^{n}_{iL}(y)\,,$
$\displaystyle\big{(}\partial_{y}+\frac{1}{2}\sigma^{\prime}+M_{f_{i}}\big{)}f^{n}_{iL}(y)=-e^{-\sigma}m_{f}^{n}\,f^{n}_{iR}(y)\,,$
(8)

where $\sigma^{\prime}=\partial_{y}\sigma$, and a boundary condition

$\displaystyle\left.\delta f^{n}_{iL}(y)\,f^{n}_{iR}(y)-\delta
f^{n}_{iR}(y)\,f^{n}_{iL}(y)\right|^{y=L}_{y=0}=0\,.$ (9)

Choose $\delta f^{n}_{iL}(0)=\delta f^{n}_{iR}(L)=0$, all the left-handed and
the right-handed KK modes of $\Psi_{fi}$ vanish, respectively, at the $y=0$
brane and at the $y=L$ brane. Therefore, only the right-handed (left-handed)
modes can couple to the SM fields on the $y=0$ ($y=L$) brane, respectively.
The KK fermion equations in Eq.(8), together with the boundary condition,
always allow a massless mode.

Setting $m^{f}_{n=0}=0$ with $M_{f_{i}}$ being constant, we get
$f^{0}_{iR}(y)=f^{0}_{iR}(L)e^{\frac{1}{2}[\sigma(L)-\sigma(y)]-M_{f_{i}}(L-y)},$
and $f^{0}_{iL}(y)=f^{0}_{iL}(0)e^{-\frac{1}{2}\sigma(y)+M_{f_{i}}y}$.
Rescaling dimensionful operators such as $\hat{Z}\rightarrow
e^{\frac{1}{2}\sigma(y)}\hat{Z}$ and $Z\rightarrow
e^{-\frac{1}{2}\sigma(y)}Z$, we get 4D Yukawa interactions given as,

$\displaystyle-{\cal L}_{4D}^{q}$ $\displaystyle=$
$\displaystyle\Big{[}y^{u}_{11}\big{(}{\cal
U}^{c}\eta\frac{\Phi_{T}\sigma}{M^{3}_{5}}\big{)}_{{\bf
1}}+\tilde{y}^{u}_{11}\big{(}{\cal
U}^{c}\eta\frac{\eta\eta}{M^{3}_{5}}\big{)}_{{\bf
1}}\Big{]}Q_{1}H_{u}+\Big{[}y^{u}_{22}({\cal U}^{c}\frac{\eta}{M_{5}})_{{\bf
1}^{\prime\prime}}+\tilde{y}^{u}_{22}\big{(}{\cal
U}^{c}\frac{\eta\Phi_{T}\sigma}{M^{3}_{5}}\big{)}_{{\bf
1}^{\prime\prime}}+\bar{y}^{u}_{22}\big{(}{\cal
U}^{c}\frac{\eta\eta\eta}{M^{3}_{5}}\big{)}_{{\bf
1}^{\prime\prime}}\Big{]}Q_{2}H_{u}$ (10) $\displaystyle+$ $\displaystyle{\bf
y}^{t}_{33}t^{c}Q_{3}H_{u}+{\bf y}^{b}_{33}b^{c}Q_{3}H+\Big{[}y^{d}_{22}({\cal
D}^{c}\frac{\eta}{M_{5}})_{{\bf
1}^{\prime\prime}}+\tilde{y}^{d}_{22}\big{(}{\cal
D}^{c}\frac{\eta\Phi_{T}\sigma}{M^{3}_{5}}\big{)}_{{\bf
1}^{\prime\prime}}+\bar{y}^{d}_{22}\big{(}{\cal
D}^{c}\frac{\eta\eta\eta}{M^{3}_{5}}\big{)}_{{\bf
1}^{\prime\prime}}\Big{]}Q_{2}H_{d}$ $\displaystyle+$
$\displaystyle\Big{[}y^{d}_{11}\big{(}{\cal
D}^{c}\frac{\eta\Phi_{T}\sigma}{M^{3}_{5}}\big{)}_{\bf
1}+\tilde{y}^{d}_{11}\big{(}{\cal
D}^{c}\frac{\eta\eta\eta}{M^{3}_{5}}\big{)}_{\bf 1}\Big{]}Q_{1}H_{d}+{\bf
y}^{b}_{32}b^{c}(\frac{\Phi_{S}\Phi_{S}}{M_{5}^{2}})_{{\bf
1}^{\prime}}Q_{2}H_{d}+{\bf
y}^{b}_{31}b^{c}(\frac{\Phi_{S}\Phi_{S}}{M_{5}^{2}})_{{\bf
1}^{\prime\prime}}Q_{1}H_{d}$ $\displaystyle+$
$\displaystyle\Big{[}y^{d}_{21}({\cal D}^{c}\frac{\eta}{M_{5}})_{{\bf
1}^{\prime\prime}}+\tilde{y}^{d}_{21}\big{(}{\cal
D}^{c}\eta\frac{\Phi_{T}\sigma}{M^{3}_{5}}\big{)}_{{\bf
1}^{\prime\prime}}+\bar{y}^{d}_{21}\big{(}{\cal
D}^{c}\eta\frac{\eta\eta}{M^{3}_{5}}\big{)}_{{\bf
1}^{\prime\prime}}\Big{]}(\frac{\Phi_{S}\Phi_{S}}{M_{5}^{2}})_{{\bf
1}^{\prime}}Q_{1}H_{d}$ $\displaystyle-{\cal L}_{4D}^{l}$ $\displaystyle=$
$\displaystyle\Big{[}y_{e}\big{(}L\frac{\Phi_{T}\Phi_{T}}{M^{2}_{5}}\big{)}_{\bf
1}+\tilde{y}_{e}\big{(}L\frac{\Phi_{T}\Phi_{T}}{M^{2}_{5}}\big{)}_{\bf
3}\frac{\Phi_{T}\sigma}{M^{2}_{5}}\Big{]}e^{c}H_{d}+\Big{[}y_{\mu}(L\Phi_{T})_{{\bf
1}^{\prime}}\frac{\sigma^{2}}{M^{3}_{5}}+\tilde{y}_{\mu}\Big{(}L\frac{\eta\eta}{M^{3}_{5}}\Big{)}_{{\bf
1}^{\prime}}\sigma\Big{]}\mu^{c}H_{d}$ (11) $\displaystyle+$
$\displaystyle\Big{[}y_{\tau}(L\Phi_{T})_{{\bf
1}^{\prime\prime}}\frac{\sigma^{2}}{M^{3}_{5}}+\tilde{y}_{\tau}\Big{(}L\frac{\eta\eta}{M^{3}_{5}}\Big{)}_{{\bf
1}^{\prime\prime}}\sigma\Big{]}\tau^{c}H_{d}+y_{\nu}(N^{c}L)_{\bf
1}\frac{\Theta}{M_{5}}H_{u}+\tilde{y}_{\nu}(N^{c}L)_{\bf
3}\frac{\Phi_{S}}{M_{5}}H_{u}$ $\displaystyle+$
$\displaystyle\frac{1}{2}(y_{\Theta}\Theta+y_{\tilde{\Theta}}\tilde{\Theta})(N^{c}N^{c})_{\bf
1}+\frac{1}{2}y_{R}\Phi_{S}(N^{c}N^{c})_{\bf 3}+{\rm h.c.}$

with well-defined Yukawa-coupling functions

$\displaystyle{\bf y}^{\alpha}_{ij}=4Z^{\omega}_{ij}\hat{Z}_{\alpha
k}\,e^{-M_{\omega_{k}}L}\cosh\sigma(L)/2\,,$ $\displaystyle
y^{\alpha}_{ij}/M_{5}=4Z^{\omega}_{ij}Z_{\alpha
k}\,e^{-M_{\omega_{k}}L}\cosh\sigma(L)/2\,,$ $\displaystyle
y_{\ell}/M_{5}=4Z_{\ell}Z_{L}\,e^{-M_{\ell}L}\cosh\sigma(L)/2\,,$ (12)

where $\alpha=(d,s,b,u,c,t)$, $\omega=(u,d)$, and $\ell=(e,\mu,\tau)$.

##  IV Hierarchies of fermion masses and mixings

###  IV.1 Charged fermions

After electroweak symmetry breaking, we get the VEVs for two Higgs doublets as
$\langle H_{u,d}\rangle=v_{u,d}$ where $v_{u}=v\sin\beta/\sqrt{2}$ and
$v_{d}=v\cos\beta/\sqrt{2}$ with $v=246$ GeV. Given the specific vacuum
alignment given by Eqs. (4,5), we obtain the up(down)-type quark mass matrices
and diagonal form of charged-lepton mass matrix, in which each entries is
proportional to $e^{-M_{f_{i}}L}\cosh\sigma(L)/2$, see Eq.(33).

From the diagonalization of the quark mass matrices, we finally obtain the
quark mixings $\theta^{q}_{ij}$ in the standard formPDG  corresponding to the
rotations in the $ij=12,23,13$ planes, as follows;

$\displaystyle\theta^{q}_{ij}$ $\displaystyle\simeq$ $\displaystyle
3\kappa^{2}f(\hat{\bf y},\hat{y})\Big{|}\frac{\delta^{\rm G}_{1}}{\delta^{\rm
G}_{2}}\Big{(}\frac{2}{1+\kappa^{2}}\Big{)}^{\frac{1}{2}}\nabla_{\chi}\Big{|}^{[{\cal
Q}_{X_{1}}(Q_{j})-{\cal Q}_{X_{1}}(Q_{i})]}\,,$
$\displaystyle\delta^{q}_{\scriptscriptstyle{CP}}$ $\displaystyle=$
$\displaystyle\arg(f(\hat{\bf y},\hat{y}))\,,$ (13)

where $i\neq j=1,2,3$ and ${\cal Q}_{X_{i}}(q)$ represents $U(1)_{X_{i}}$
quantum number ${\cal Q}$ of field $q$, and $f_{i}(\hat{\bf y},\hat{y})$ are
functions of the associated hat Yukawa couplings being of order unities (cf.
Eqs.(1,12, 33)). Note that in the limit of $\kappa\rightarrow 0$ there is no
quark mixing. Clearly, it shows the quark mixings only depend on the local
effects of $G_{F}$. For the charged lepton mass ratios are given by

$\displaystyle\frac{m_{e}}{m_{\mu}}\simeq\hat{y}_{e}\nabla^{2}_{T}\,\nabla^{10}_{\chi}/(\hat{y}_{\mu}\nabla_{T}\nabla_{\varrho}+\hat{\tilde{y}}_{\mu}\nabla^{2}_{\eta})\nabla_{\varrho}\,,$
$\displaystyle\frac{m_{\mu}}{m_{\tau}}=(\hat{y}_{\mu}\nabla_{T}\nabla_{\varrho}+\hat{\tilde{y}}_{\mu}\nabla^{2}_{\eta})\nabla^{4}_{\chi}/(\hat{y}_{\tau}\nabla_{T}\nabla_{\varrho}+\hat{\tilde{y}}_{\tau}\nabla^{2}_{\eta})\,.$
(14)

In the limit, $\nabla_{T}\sim\nabla_{\varrho}\sim\nabla_{\eta}$, and for order
one $\hat{y}_{\alpha=e,\mu,\tau}$ and $\hat{\tilde{y}}_{\alpha}$,
$\frac{m_{e}}{m_{\mu}}\simeq\nabla^{10}_{\chi}\nabla_{\varrho}$ and
$\frac{m_{\mu}}{m_{\tau}}\simeq\nabla^{4}_{\chi}$.

###  IV.2 Neutrinos

In this model, the low energy effective neutrino masses are generated by the
usual seesaw mechanismMinkowski:1977sc  with the inclusion of right-handed
$SU(2)$-singlet Majorana neutrinos $N^{c}$. The seesaw scale $M$ should be
larger than the electroweak scale but smaller than $M_{5}$. The fundamental
gravity scale $M_{5}$ can be derived in terms of $F_{A}$ or $M$:

$\displaystyle M_{5}=\frac{|\delta^{\rm
G}_{2}|}{\sqrt{2}}\frac{F_{A}}{\nabla_{\chi}}=\frac{M}{\nabla_{\Theta}}\quad\text{with}~{}M\equiv|\delta^{\rm
G}_{1}|\frac{F_{A}}{\sqrt{1+\kappa^{2}}}\,,$ (15)

where $F_{A}$ is the QCD axion decay constant, given by
$F_{A}\equiv\frac{f_{A}}{N_{C}}=\frac{f_{a_{i}}}{\sqrt{2}\,\delta^{\rm
G}_{i}}$ with $f_{a_{1}}=\sqrt{2}v_{\chi}$ and
$f_{a_{2}}=v_{\Theta}\sqrt{1+\kappa^{2}}$. Contrary to the quark and charged-
lepton sectors, the neutrino Yukawa couplings do not depend on the non-local
effects of the extra-dimension. After seesawing ${\cal
M}_{\nu}\simeq-m^{T}_{D}M^{-1}_{R}m_{D}$ where $M_{R}$ and $m_{D}$ are
respectively right-handed Majorana and Dirac neutrino mass matrices, in a
basis where charged lepton masses are real and diagonal, we obtain the
effective light neutrino mass matrix

$\displaystyle{\cal M}_{\nu}$ $\displaystyle=$ $\displaystyle
m_{0}\frac{e^{i\pi}}{3}{\left(\begin{array}[]{ccc}1&1&1\\\ 1&1&1\\\
1&1&1\end{array}\right)}+\frac{m_{0}\,e^{i\pi}}{1-e^{2i\phi}\tilde{\kappa}^{2}}{\left(\begin{array}[]{ccc}a_{\nu}&-\frac{1}{2}a_{\nu}&-\frac{1}{2}a_{\nu}\\\
-\frac{1}{2}a_{\nu}&a_{\nu}-1&1-\frac{1}{2}a_{\nu}\\\
-\frac{1}{2}a_{\nu}&1-\frac{1}{2}a_{\nu}&a_{\nu}-1\end{array}\right)}$ (22)
$\displaystyle+$ $\displaystyle
m_{0}\,e^{i\pi}\frac{\kappa^{2}}{1-e^{2i\phi}\tilde{\kappa}^{2}}{\left(\begin{array}[]{ccc}-b_{\nu}-c_{\nu}\,\tilde{\kappa}&\frac{1}{2}(b_{\nu}+c_{\nu}\tilde{\kappa})+d_{\nu}&\frac{1}{2}(b_{\nu}+c_{\nu}\tilde{\kappa})-d_{\nu}\\\
\frac{1}{2}(b_{\nu}+c_{\nu}\tilde{\kappa})+d_{\nu}&\frac{1}{2}(b_{\nu}-c_{\nu}\tilde{\kappa})-d_{\nu}&\frac{1}{2}c_{\nu}\tilde{\kappa}-b_{\nu}\\\
\frac{1}{2}(b_{\nu}+c_{\nu}\tilde{\kappa})-d_{\nu}&\frac{1}{2}c_{\nu}\tilde{\kappa}-b_{\nu}&\frac{1}{2}(b_{\nu}-c_{\nu}\tilde{\kappa})+d_{\nu}\end{array}\right)},$
(26)

where
$m_{0}=|\hat{y}_{\nu}|^{2}\frac{v^{2}_{u}}{M}\left|\frac{X_{2}\delta^{\rm
G}_{1}}{X_{1}\delta^{\rm
G}_{2}}\nabla_{\chi}\right|^{16}\left(\frac{2}{1+\kappa^{2}}\right)^{8}$,
$\tilde{\kappa}=\kappa|\hat{y}_{R}/\hat{y}_{\Theta}|$ with
$\phi=\arg(\hat{y}_{R}/\hat{y}_{\Theta})$, and each component is given by
$a_{\nu}$, $b_{\nu}$, $c_{\nu}$, $d_{\nu}$ that are functions of $\kappa$,
$\tilde{\kappa}$, $\phi$, and other phases.

As a result, the neutrino masses $m_{\nu_{i}}$ ($i=1,2,3$) are obtained by the
transformation, $U^{T}_{\rm PMNS}\,{\cal M}_{\nu}\,U_{\rm PMNS}={\rm
diag}(m_{\nu_{1}},m_{\nu_{2}},m_{\nu_{3}})$, where $U_{\rm PMNS}$ is the
mixing matrix of three mixing angles, $\theta_{12}$ (solar), $\theta_{13}$
(reactor), $\theta_{23}$ (atmospheric), and three CP-odd phases (one
$\delta_{CP}$ for the Dirac neutrinos and two $\varphi_{1,2}$ for the Majorana
neutrinos)PDG . In the limit $\kappa\rightarrow 0$ (leading to
$\tilde{\kappa}\rightarrow 0$ and $a_{\nu}\rightarrow\frac{2}{3}$), the light
neutrino masses generated via Eq.(26) become degenerate with no neutrino
mixings (and also no quark mixings, see Eq.(13)). Hence, it is reasonable to
take a nonzero $\kappa$ for generating the observed neutrino mixing angles.
Then, the neutrino mass eigenvalues of Eq.(26) can be expanded in terms of
$\kappa$: for normal mass ordering (NO),
$m^{2}_{\nu_{3}}=m^{2}_{\nu+}>m^{2}_{\nu_{2}}\equiv
m^{2}_{0}>m^{2}_{\nu_{1}}=m^{2}_{\nu-}$ with
$m^{2}_{\nu\pm}=m^{2}_{0}\big{(}Q_{\nu}\pm\sqrt{J_{\nu}}\big{)}/P_{\nu}$, and
for inverted mass ordering (IO), $m^{2}_{\nu_{2}}\equiv
m^{2}_{0}>m^{2}_{\nu_{1}}=m^{2}_{\nu+}>m^{2}_{\nu_{3}}=m^{2}_{\nu-}$, where
$Q_{\nu}$, $J_{\nu}$, and $P_{\nu}$ are functions of the parameters in
Eq.(26). The limit $\tilde{y}_{\nu}\rightarrow 0$ in Eq.(11) is equivalent to
$a_{\nu}\rightarrow\frac{2}{3}(1-e^{i\phi}\tilde{\kappa})$ and
$\\{b_{\nu},c_{\nu},d_{\nu}\\}\rightarrow 0$, and it gives rise to the so-
called tri-bimaximal mixing of neutrinos. Then, the deviation from the tri-
bimaximal mixingTBM  can be presented in terms of $\kappa$:

$\displaystyle|\theta_{23}-\pi/4|\propto|d_{\nu}|\kappa^{2}\,,\qquad\theta_{13}\propto|d_{\nu}|\kappa^{2}\,,$
$\displaystyle\theta_{12}-\frac{1}{2}\tan^{-1}(2\sqrt{2})\simeq\frac{2\sqrt{2}|3|a_{\nu}|(\kappa^{2}-\tilde{\kappa}^{2})\,S_{\nu}+{\cal
O}(\kappa^{3},\tilde{\kappa}^{4})|}{R_{\nu}+3|a_{\nu}|\big{(}\tilde{\kappa}^{2}-\frac{\kappa^{2}}{2}\big{)}\,S_{\nu}+{\cal
O}(\kappa^{3},\tilde{\kappa}^{4})}\,,$ (27)

where
$R_{\nu}=\frac{1}{3}-\frac{3}{4}|a_{\nu}|^{2}-\frac{2}{3}\tilde{\kappa}^{2}\cos^{2}\phi$,
and $S_{\nu}$ is functions of $|b_{\nu}|,|c_{\nu}|,\tilde{\kappa}$ and
relevant phases (especially, $S_{\nu}\rightarrow 0$ and ${\cal
O}(\kappa^{3},\tilde{\kappa}^{4})\rightarrow 0$ for
$\\{b_{\nu},c_{\nu}\\}\rightarrow 0$).

Figure 1:  Plot of $|g_{a\gamma\gamma}|$ versus $m_{a}$ for KSVZKSVZ  (blue-
dotted line) and the model (“localized” red line) in terms of $E_{A}/N_{C}=$
$0$ and $-761/99$, respectively.

Figure 2:  Left-plot for predictions of $\delta_{CP}$ as a function of
$\theta_{23}$, for NO (red crosses) and IO (blue asters). Right-plot for
$0\nu\beta\beta$-decay rate as a function of the lightest neutrino mass,
$m_{\rm lightest}$, for NO ($m_{\rm lightest}=m_{1}$ red crosses) and IO
($m_{\rm lightest}=m_{3}$ blue asters), where the most stringent limit ($90\%$
CL) is given by KamLAND-ZenKamLAND-Zen:2016pfg .

##  V Numerical results

Using the quark and charged-lepton masses, we can determine the warp factor
$\sigma(L)=kL$ with the help of Eq.(3) via flavor dynamics. In the end, the
compactification length, $L[m]$, is given by $L[m]\simeq 5.82\times
10^{-13}\,\cdot\frac{e^{-\sigma(L)}\sigma(L)}{\sinh\sigma(L)}\big{(}\frac{10^{11}{\rm
GeV}}{M_{5}}\big{)}^{3}$. For $\hat{y}(\hat{\bf y})={\cal O}(1)$, there are
fourteen degrees of freedom left due to the two constraints in Eq.(6) among 16
parameters (8 local parameters
$\\{\nabla_{\chi},\nabla_{\Theta},\nabla_{S},\kappa,\nabla_{\eta},\nabla_{T},\nabla_{\varrho},\tan\beta\\}$
plus 8 non-local parameters $\\{\sigma(L),M^{u(d)}_{1,2,3},M_{\ell}\\}$).
Thirteen observables in the charged-fermion sectors can be determined once one
of non-local parameters is fixed.

We perform a numerical simulation by using the linear algebra
toolsAntusch:2005gp . With inputs $\sigma(L)=18.25$, $\tan\beta=3.1$,
$\kappa=0.4$, $\nabla_{\chi}=0.52$, $\nabla_{\eta}=0.51$, $\nabla_{T}=0.35$,
$\nabla_{\varrho}=0.34$, the empirical results of the SM fermion mixing angles
and massesPDG  are obtained with the appropriate Yukawa couplings $\hat{y}$ of
order of unity of Eq.(1). Plugging the inputs into Eq.(15), we can derive the
scale of $M_{5}$ as $(1-2)\times 10^{11}\,{\rm GeV}\lesssim M_{5}\lesssim
2.8\times 10^{11}\,{\rm GeV}$, leading to the QCD axion mass,
$m_{a}=5.44^{+2.33}_{-2.58}\times 10^{-4}\,\text{eV}$. Then, we obtain the
axion photon coupling $|g_{a\gamma\gamma}|=1.13^{+0.48}_{-0.53}\times
10^{-12}\,\text{GeV}^{-1}$ for $m_{u}/m_{d}=0.47$, as shown in red line in
FIG. 1. Moreover, we can predict $\delta_{CP}$ precisely by $\theta_{23}$.
Namely, $\theta_{23}$ would favor $\sim 51^{\circ}$ for $\delta_{CP}\sim
180^{\circ}$ and $\theta_{23}\sim 45^{\circ}$ for $\delta_{CP}\sim 90^{\circ}$
and $270^{\circ}$, as shown in the left plot of FIG. 2. We note that the
pattern of the light neutrino mass spectrum, NO
($m_{\nu_{3}}>m_{\nu_{2}}>m_{\nu_{1}}$) or IO
($m_{\nu_{2}}>m_{\nu_{1}}>m_{\nu_{3}}$), can be distinguished by the
measurement of $0\nu\beta\beta$-decay rate, as can be seen in the right plot
of FIG. 2.

##  VI Flavored axion

The flavored-axion $A_{1}$ produces the flavor-changing neutral Yukawa
interaction, $-{\cal L}^{aq}\supset
i\frac{A_{1}}{f_{a1}}\lambda(1-\frac{\lambda^{2}}{2})(m_{d}-m_{s})\bar{d}s+{\rm
h.c.}$ where $\lambda$ is the Cabbibo anglePDG , which gives the strongest
bound on the QCD axion decay constant $F_{A}$. From the present experimental
upper bound, ${\rm Br}(K^{+}\rightarrow\pi^{+}A_{i})<(3-6)\times 10^{-11}$ at
$90\%$ CLNA62:2021zjw , we obtain $F_{A}\gtrsim(0.7-1.5)\times 10^{10}\,{\rm
GeV}$.

Under the $U(1)_{\rm PQ}$ transformation, the QCD axion field $A$ changes
$A\rightarrow A+\frac{f_{A}}{N_{C}}\alpha$, where $f_{A}=\sqrt{2}\,\delta^{\rm
G}_{2}f_{a_{1}}=\sqrt{2}\,\delta^{\rm G}_{1}f_{a_{2}}$. From $-{\cal
L}^{a\ell}\supset(\frac{\partial_{\mu}A_{1}}{2f_{a_{1}}}-41\frac{\partial_{\mu}A_{2}}{2f_{a_{2}}}+\frac{\partial_{\mu}A_{T}}{v_{t}})\bar{e}\gamma^{\mu}\gamma_{5}e$,
we can identify the effective axion couplings as
$g_{1ee}=m_{e}/(\sqrt{2}|\delta^{\rm G}_{1}|F_{A})$ and
$g_{2ee}=41m_{e}/(\sqrt{2}|\delta^{\rm G}_{2}|F_{A})$. Thus, the flavored-
axions $A_{1,2,T}$ interactions with leptons can be searched for in stellar
evolutions in astroparticle physicsRaffelt:1985nj . Some stellar cooling hints
can be interpreted as axion-electron couplings $7.2\times
10^{-14}\lesssim|g_{iee}|\lesssim 2.2\times 10^{-13}$Rowell:2011wp ,
determining the favored values of $F_{A}$ via the $A_{i}$ couplings to
electrons. Otherwise, the lower limit on $F_{A}$ is set to about $10^{8}\,{\rm
GeV}$ by SN1987A while the upper bound on $F_{A}$ is about $10^{12}\,{\rm
GeV}$ from the dark matter abundance. Combining the stellar cooling hints
(from astro-physics) with the constraint from $K^{+}\rightarrow\pi^{+}+A_{1}$
process (from particle physics), we obtain the consistent axion decay constant
as $(0.7-1.5)\times 10^{10}\lesssim F_{A}[{\rm GeV}]\lesssim 1.9\times
10^{10}$.

##  VII Conclusion

We have proposed an extra-dimension scenario for understanding the origin of
the fermion mass and mixing hierarchies by introducing the localized flavon
fields and imposing the flavor symmetry $G_{F}(=$non-Abelian$\times$Abelian)
through the bulk. We fixed the charges of the extra gauged $U(1)$ symmetries
by the U(1) gravitation anomaly-free condition and found that they play a
crucial role in achieving the desirable fermion mass and mixing hierarchies
and protecting the PQ symmetry from quantum gravity corrections.

When the bulk flavor symmetry $G_{F}$ is broken due to the flavor fields
localized at the 3-branes, we showed that the $SU(2)$ singlet flavored
fermions in the bulk are integrated out to provide the effective Yukawa
couplings for quarks and leptons, inherited with the $G_{F}$ breaking in the
two sectors. We also showed that there is a viable parameter space for the
flavored axion in our model, which is consistent with the solution to the
strong CP problem, the dark matter relic abundance with misalignment mechanism
as well as the bounds from rare meson decays.

##  Appendix A Superpotential

We present the vacuum structure for the flavon fields in our model.

In order to obtain the relevant vacuum configuration for the desirable lepton
and quark mixings, several SM singlet fields are introduced in Table-1: flavon
fields ${\cal F}$ has a zero $U(1)_{R}$ charge and transforms nontrivially
only under $G_{F}$ and are responsible for the symmetry breaking of $G_{F}$;
driving fields has $U(1)_{R}$ charge $+2$ and breaks the flavor group along
the required VEV directions. The vacuum configuration for flavons is
simplified in a supersymmetric theory. Thus, we consider the brane-localized
superpotential with the driving fields at the leading order, which is
invariant under $SL_{2}(F_{3})\times U(1)_{X_{i}}$ ($i=1,2,T$), as follows,

$\displaystyle
W_{v}=\delta(y)\big{\\{}\Phi^{S}_{0}(g_{s_{1}}\Phi_{S}\Phi_{S}+g_{s_{2}}\tilde{\Theta}\Phi_{S})$
$\displaystyle+\Theta_{0}(g_{\Theta_{1}}\Phi_{S}\Phi_{S}+g_{\Theta_{2}}\Theta\Theta+g_{\Theta_{3}}\Theta\tilde{\Theta}+g_{\Theta_{4}}\tilde{\Theta}\tilde{\Theta})\big{\\}}$
$\displaystyle+\delta(y-L)\big{\\{}\Phi^{T}_{0}\big{(}\mu_{T}\Phi_{T}+g_{T}\Phi_{T}\Phi_{T}\frac{\sigma}{M_{5}}+g_{T\chi}\Phi_{T}\frac{\chi\tilde{\chi}}{M_{5}}\big{)}$
$\displaystyle+\sigma_{0}\big{(}\mu_{\sigma}\sigma+g_{\sigma\chi}\chi\tilde{\chi}\frac{\sigma}{M_{5}}\big{)}$
$\displaystyle+\eta_{0}\big{(}\mu_{\eta}\eta+g_{\eta}\eta\Phi_{T}\frac{\sigma}{M_{5}}\big{)}+\chi_{0}\big{(}g_{\chi}\chi\tilde{\chi}-\mu^{2}_{\chi}\big{)}\big{\\}},$
(28)

where $\\{\Phi_{0},\Theta_{0},\eta_{0},\sigma_{0},\chi_{0}\\}$ denote the
driving fields, and $\mu_{i}$ are dimensionful parameters and $g_{i}$ are
dimensionless coupling constants. At this level there is no fundamental
distinction between the $\Theta$ and $\tilde{\Theta}$ in $W_{v}$ so that the
field $\tilde{\Theta}$ is defined as the combination that couples to
$\Phi^{S}_{0}\Phi_{S}$Altarelli:2005yx . Higher-dimensional brane interactions
via the one-loop exchange of the flavored bulk fermions are allowed but
absorbed into the leading order terms of Eq.(28) by redefinition of
coefficients.

##  Appendix B Fermion mass matrices

We present the details of quark and lepton mass matrices obtained in 4D
effective theory.

From Eqs.(10) and (11) the up(down)-type quark ${\cal M}_{u(d)}$ and charged-
lepton ${\cal M}_{\ell}$ mass matrices are expressed as

$\displaystyle{\cal
M}_{d}={\left(\begin{array}[]{ccc}m^{d}_{11}\,e^{i(3\frac{A_{1}}{v_{\cal
F}}+14\frac{A_{2}}{v_{g}})}\,e^{-M_{d_{1}}L}&0&0\\\
m^{d}_{21}\,e^{i(3\frac{A_{1}}{v_{\cal
F}}+14\frac{A_{2}}{v_{g}})}\,e^{-M_{d_{2}}L}&m^{d}_{22}\,e^{i(\frac{A_{1}}{v_{\cal
F}}+14\frac{A_{2}}{v_{g}})}\,e^{-M_{d_{2}}L}&0\\\
m^{d}_{31}\,e^{i(11\frac{A_{1}}{v_{\cal
F}}-18\frac{A_{2}}{v_{g}})}\,e^{-M_{d_{3}}L}&m^{d}_{32}\,e^{i(9\frac{A_{1}}{v_{\cal
F}}-18\frac{A_{2}}{v_{g}})}&m^{d}_{33}\,e^{i(3\frac{A_{1}}{v_{\cal
F}}-18\frac{A_{2}}{v_{g}})}\,e^{-M_{d_{3}}L}\end{array}\right)}\cosh\frac{\sigma(L)}{2}\,,$
(32) $\displaystyle{\cal M}_{u}={\rm
diag}\big{(}m^{u}_{11}\,e^{i(2\frac{A_{1}}{v_{\cal
F}}+6\frac{A_{2}}{v_{g}})}\,e^{-M_{u_{1}}L},m^{u}_{22}e^{6i\frac{A_{2}}{v_{g}}}\,e^{-M_{u_{2}}L},m^{u}_{33}\,e^{-11i\frac{A_{2}}{v_{g}}}\,e^{-M_{u_{3}}L}\big{)}\cosh\frac{\sigma(L)}{2},$
$\displaystyle{\cal M}_{\ell}={\rm
diag}\big{(}m^{e}_{11}\,e^{i(\frac{A_{1}}{v_{\cal
F}}-41\frac{A_{2}}{v_{g}}+2\frac{A_{T}}{v_{t}})},m^{\mu}_{22}\,e^{i(\frac{A_{1}}{v_{\cal
F}}-31\frac{A_{2}}{v_{g}}-\frac{A_{T}}{v_{t}})},m^{\tau}_{33}\,e^{i(\frac{A_{1}}{v_{\cal
F}}-27\frac{A_{2}}{v_{g}}-\frac{A_{T}}{v_{t}})}\big{)}e^{-M_{\ell}L}\cosh\frac{\sigma(L)}{2},$
(33)

with the massless modes $A_{1(2,T)}$ and
$v_{t}=\sqrt{v^{2}_{T}+v^{2}_{\varrho}}$, where

$\displaystyle
m^{d}_{11}=(i\hat{y}^{d}_{11}\nabla_{T}\nabla_{\varrho}-\hat{\tilde{y}}^{d}_{11}\nabla^{2}_{\eta})\nabla_{\eta}|\frac{\delta^{\rm
G}_{1}}{\delta^{\rm
G}_{2}}|^{3}(\frac{2}{1+\kappa^{2}})^{\frac{3}{2}}\,\nabla^{17}_{\chi}\,v_{d},$
$\displaystyle
m^{d}_{21}\simeq-3\kappa^{2}\hat{y}^{d}_{21}\nabla_{\eta}|\frac{\delta^{\rm
G}_{1}}{\delta^{\rm
G}_{2}}|^{3}(\frac{2}{1+\kappa^{2}})^{\frac{3}{2}}\,\nabla^{17}_{\chi}\,v_{d},$
$\displaystyle
m^{d}_{22}\simeq-\hat{y}^{d}_{22}\nabla_{\eta}|\frac{\delta^{\rm
G}_{1}}{\delta^{\rm
G}_{2}}|(\frac{2}{1+\kappa^{2}})^{\frac{1}{2}}\,\nabla^{15}_{\chi}\,v_{d}$
$\displaystyle m^{d}_{31}=3\kappa^{2}\hat{\bf y}^{b}_{31}|\frac{\delta^{\rm
G}_{1}}{\delta^{\rm
G}_{2}}|^{11}(\frac{2}{1+\kappa^{2}})^{\frac{11}{2}}\,\nabla^{29}_{\chi}\,v_{d},$
$\displaystyle m^{d}_{32}=3\kappa^{2}\hat{\bf y}^{b}_{32}|\frac{\delta^{\rm
G}_{1}}{\delta^{\rm
G}_{2}}|^{9}(\frac{2}{1+\kappa^{2}})^{\frac{9}{2}}\,\nabla^{27}_{\chi}\,v_{d},$
$\displaystyle m^{d}_{33}=\hat{\bf y}^{b}_{33}|\frac{\delta^{\rm
G}_{1}}{\delta^{\rm
G}_{2}}|^{3}(\frac{2}{1+\kappa^{2}})^{\frac{3}{2}}\,\nabla^{21}_{\chi}\,v_{d},$
(34) $\displaystyle
m^{u}_{11}=(i\hat{y}^{u}_{11}\nabla_{T}\nabla_{\varrho}-\hat{\tilde{y}}^{u}_{11}\nabla^{2}_{\eta})\nabla_{\eta}|\frac{\delta^{\rm
G}_{1}}{\delta^{\rm
G}_{2}}|^{2}\frac{2}{1+\kappa^{2}}\,\nabla^{8}_{\chi}\,v_{u},$ $\displaystyle
m^{u}_{22}\simeq-\hat{y}^{u}_{22}\nabla_{\eta}\,\nabla^{6}_{\chi}\,v_{u},$
$\displaystyle m^{u}_{33}=\hat{\bf y}^{t}_{33}\,\nabla^{11}_{\chi}\,v_{u},$
(35) $\displaystyle
m^{e}_{11}\simeq\hat{y}_{e}\nabla^{2}_{T}\,\nabla^{42}_{\chi}|\frac{\delta^{\rm
G}_{1}}{\delta^{\rm G}_{2}}|(\frac{2}{1+\kappa^{2}})^{\frac{1}{2}}\,v_{d},$
$\displaystyle
m^{\mu}_{22}=(\hat{y}_{\mu}\nabla_{T}\nabla_{\varrho}+\hat{\tilde{y}}_{\mu}\nabla^{2}_{\eta})\nabla^{32}_{\chi}\nabla_{\varrho}|\frac{\delta^{\rm
G}_{1}}{\delta^{\rm G}_{2}}|(\frac{2}{1+\kappa^{2}})^{\frac{1}{2}}\,v_{d},$
$\displaystyle
m^{\tau}_{33}=(\hat{y}_{\tau}\nabla_{T}\nabla_{\varrho}+\hat{\tilde{y}}_{\tau}\nabla^{2}_{\eta})\nabla_{\varrho}\nabla^{28}_{\chi}|\frac{\delta^{\rm
G}_{1}}{\delta^{\rm G}_{2}}|(\frac{2}{1+\kappa^{2}})^{\frac{1}{2}}\,v_{d}.$
(36)

###### Acknowledgements.

YHA was supported by the National Research Foundation of Korea(NRF) grant
funded by the Korea government(MSIT) (No.2020R1A2C1010617). SKK was supported
by the National Research Foundation of Korea(NRF) grant funded by the Korea
government(MSIT) (No.2019R1A2C1088953). HML was supported in part by Basic
Science Research Program through the National Research Foundation of Korea
(NRF) funded by the Ministry of Education, Science and Technology
(NRF-2019R1A2C2003738 and NRF-2021R1A4A2001897).

## References

  * (1) Y. H. Ahn, Phys. Rev. D 93, no.8, 085026 (2016) [arXiv:1604.01255 [hep-ph]].
  * (2) P. Horava and E. Witten, Nucl. Phys. B 460 (1996), 506-524; N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B 429 (1998), 263-272.
  * (3) N. Arkani-Hamed and M. Schmaltz, Phys. Rev. D 61 (2000), 033005; N. Arkani-Hamed, L. J. Hall, D. Tucker-Smith and N. Weiner, Phys. Rev. D 61 (2000), 116003.
  * (4) T. Gherghetta and A. Pomarol, Nucl. Phys. B 586, 141-162 (2000) doi:10.1016/S0550-3213(00)00392-8 [arXiv:hep-ph/0003129 [hep-ph]].
  * (5) D. E. Kaplan and T. M. P. Tait, JHEP 06, 020 (2000) [arXiv:hep-ph/0004200 [hep-ph]].
  * (6) S. J. Huber and Q. Shafi, Phys. Lett. B 498, 256-262 (2001) [arXiv:hep-ph/0010195 [hep-ph]].
  * (7) K. Agashe, G. Perez and A. Soni, Phys. Rev. D 71, 016002 (2005) [arXiv:hep-ph/0408134 [hep-ph]].
  * (8) S. Casagrande, F. Goertz, U. Haisch, M. Neubert and T. Pfoh, JHEP 10, 094 (2008) [arXiv:0807.4937 [hep-ph]].
  * (9) K. Agashe, T. Okui and R. Sundrum, Phys. Rev. Lett. 102, 101801 (2009) [arXiv:0810.1277 [hep-ph]].
  * (10) P. R. Archer, JHEP 09, 095 (2012) [arXiv:1204.4730 [hep-ph]].
  * (11) M. Frank, C. Hamzaoui, N. Pourtolami and M. Toharia, Phys. Rev. D 91, 116001 (2015) [arXiv:1504.02780 [hep-ph]].
  * (12) A. Ahmed, A. Carmona, J. Castellano Ruiz, Y. Chung and M. Neubert, JHEP 08, 045 (2019) [arXiv:1905.09833 [hep-ph]].
  * (13) G. N. Wojcik and T. G. Rizzo, JHEP 02, 179 (2020) [arXiv:1909.13153 [hep-ph]].
  * (14) Y. J. Kang, S. Kim and H. M. Lee, JHEP 09 (2020), 005 doi:10.1007/JHEP09(2020)005 [arXiv:2006.03043 [hep-ph]].
  * (15) H. M. Lee, J. Song and K. Yamashita, doi:10.1007/s40042-021-00339-0 [arXiv:2110.09942 [hep-ph]].
  * (16) Y. H. Ahn, Phys. Rev. D 98, no.3, 035047 (2018) [arXiv:1804.06988 [hep-ph]].
  * (17) Y. H. Ahn, Phys. Rev. D 91 (2015), 056005.
  * (18) M. Kamionkowski and J. March-Russell, Phys. Lett. B 282, 137 (1992); R. Kallosh, A. D. Linde, D. A. Linde and L. Susskind, Phys. Rev. D 52, 912 (1995).
  * (19) L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999), 3370-3373.
  * (20) P. H. Frampton and T. W. Kephart, Int. J. Mod. Phys. A 10, 4689 (1995); A. Aranda, C. D. Carone and R. F. Lebed, Phys. Rev. D 62, 016009 (2000).
  * (21) W. D. Goldberger and M. B. Wise, Phys. Rev. Lett. 83 (1999), 4922-4925.
  * (22) M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019 update.
  * (23) P. Minkowski, Phys. Lett. B 67, 421 (1977).
  * (24) P. F. Harrison, D. H. Perkins and W. G. Scott, Phys. Lett. B 530, 167 (2002).
  * (25) S. Antusch, J. Kersten, M. Lindner, M. Ratz and M. A. Schmidt, JHEP 0503, 024 (2005).
  * (26) J. E. Kim, Phys. Rev. Lett. 43, 103 (1979); M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 166, 493 (1980).
  * (27) A. Gando et al. [KamLAND-Zen Collaboration], Phys. Rev. Lett. 117, no. 8, 082503 (2016) Addendum: [Phys. Rev. Lett. 117, no. 10, 109903 (2016)] [arXiv:1605.02889 [hep-ex]].
  * (28) E. Cortina Gil et al. [NA62], JHEP 06 (2021), 093.
  * (29) G. G. Raffelt, Phys. Lett. B 166, 402 (1986); S. I. Blinnikov and N. V. Dunina-Barkovskaya, Mon. Not. Roy. Astron. Soc. 266, 289 (1994).
  * (30) S. DeGennaro, T. von Hippel, D. E. Winget, S. O. Kepler, A. Nitta, D. Koester and L. Althaus, Astron. J. 135 (2008), 1-9 [arXiv:0709.2190 [astro-ph]]; N. Rowell and N. Hambly, [arXiv:1102.3193 [astro-ph.GA]].
  * (31) G. Altarelli and F. Feruglio, Nucl. Phys. B 741 (2006), 215-235.