# On the Metrics for Evaluating Monocular Depth Estimation

Akhil Gurram
Huawei Munich Research Center, Germany
Univ. Autònoma de Barcelona (UAB), Spain
<EMAIL_ADDRESS>The majority of the work was done while working at
Huawei Munich Research Center Antonio M. López
Computer Vision Center (CVC),
Univ. Autònoma de Barcelona (UAB), Spain
<EMAIL_ADDRESS>
###### Abstract

Monocular Depth Estimation (MDE) is performed to produce 3D information that
can be used in downstream tasks such as those related to on-board perception
for Autonomous Vehicles (AVs) or driver assistance. Therefore, a relevant
arising question is whether the standard metrics for MDE assessment are a good
indicator of the accuracy of future MDE-based driving-related perception
tasks. We address this question in this paper. In particular, we take the task
of 3D object detection on point clouds as proxy of on-board perception. We
train and test state-of-the-art 3D object detectors using 3D point clouds
coming from MDE models. We confront the ranking of object detection results
with the ranking given by the depth estimation metrics of the MDE models. We
conclude that, indeed, MDE evaluation metrics give rise to a ranking of
methods which reflects relatively well the 3D object detection results we may
expect. Among the different metrics, the absolute relative (abs-rel) error
seems to be the best for that purpose.

##  1 Introduction

Monocular depth estimation (MDE) is addressed from different settings
determined by the data available at training time, _e.g_., LiDAR [19] and
virtual-world [36] supervision, stereo [8] and structure-from-motion (SfM)
[37] self-supervision, and combinations of those [11]. MDE results are
compared by using _de facto_ standard metrics (_e.g_., abs-rel, rms, _etc_.)
established by Eigen _et al_. [7]. Reviewing literature results, we can
observe that, in terms of such MDE metrics, the difference among different
proposals is not too large even the way of training the model is quite
different. For instance, we can see it in Table 2. Focusing on the abs-rel
metric by now, we can see that for MonoDEVS-SfM abs-rel=0.090, while for
MonoDELS-SfM abs-rel=0.077. The former is based on virtual-world supervision
and SfM self-supervision [13], while the latter uses LiDAR supervision instead
of the virtual-world one [14]. Thus, in terms of physical sensors, the former
requires a monocular systems at training time, while the latter requires a
relatively dense LiDAR and a camera both calibrated and synchronized. Thus, in
our opinion, a reasonable question is whether the difference of $\sim 0.13$
points justify the use of a LiDAR-based setting. In fact, when performing MDE
on-board an autonomous or assisted vehicle, obtaining depth estimation maps is
just an intermediate step of a perception stack. Thus, one may wonder if those
differences on depth estimation will be consolidated once such MDE models are
used to support the targeted perception task. The aim of this paper is to do a
step forward to answer this question, using 3D object detection as downstream
perception task.

More specifically, we use already trained and publicly available MDE models
from the state-of-the-art to generate depth maps. These depth maps are then
used to generate 3D point clouds, termed as Pseudo-LiDAR [28] in analogy with
LiDAR point clouds. Pseudo-LiDAR is used for training and testing 3D object
detectors. We compare the ranking of MDE models according to their performance
estimating depth, and their performance supporting 3D object detection through
the generation of Pseudo-LiDAR. We consider eight MDE models as well as three
different CNN architectures for 3D object detection (Point R-CNN [27], Voxel
R-CNN [6], CenterPoint [31]). After analysing our experimental results, based
on KITTI benchmark [9], we have seen that the abs-rel metric is well aligned
with 3D object detection results in terms of ranking the MDE methods. What
remains as future work is to investigate if we can predict accuracy
improvements in 3D object detection from the improvements observed in the abs-
rel metric. Otherwise, we recommend to incorporate 3D object detection as part
of the evaluation of MDE models.

Sect. 2 summarizes the related literature. Sect. 3 presents the models and
methods used in our experimental work, namely, the MDE models, the 3D object
detection architectures, and the procedure to generate Pseudo-LiDAR from depth
maps. Sect. 4 describes the quantitative and qualitative results, including
the performance of the MDE models for both estimating depth and supporting 3D
object detection. This allows to compare the rankings of MDE generated by MDE
metrics _vs_. 3D object detection metrics. Finally, Sect. 5 summarizes the
presented work.

##  2 Related Work

We focus on MDE methods and 3D object detectors.

###  2.1 Monocular Depth Estimation (MDE)

As can be seen in the survey by de Queiroz _et al_. [5], deep convolutional
neural networks (CNNs) have significantly boost MDE performance. We can
categorize the different MDE proposal according to the data required at
training time. For instance, many proposals assume the availability of camera
and LiDAR calibrated data, thus, densified LiDAR depth maps are used as
supervision to train CNN-based MDE models [19, 2, 29, 33, 1]. When available,
pixelwise semantic and LiDAR supervision have been used together to improve
the performance of MDE models [15]. To avoid the use of LiDAR data, stereo
rigs have been used to provide depth self-supervision [8, 10], also combined
with semantic supervision [3]. Other works propose stereo self-supervision to
improve upon LiDAR-only supervision [16], assuming a sparse LiDAR setting. In
order to avoid the training-time requirement of having access to either a
calibrated camera-LiDAR suite or a stereo rig, depth self-supervision has also
been based on structure-from-motion (SfM) principles. In other words, only on-
board monocular sequences are required to train the corresponding MDE model
[37, 32, 35]. Since by using SfM supervision alone we can only estimate
relative depth, stereo and SfM self-supervision have also been combined [11],
so still keeping a vision-only setting at training time. Other approaches to
provide absolute depth in SfM self-supervision settings have been the use of
complementary supervision such as the ego-vehicle speed (available as a
readable car signal) [12], or synthetic images with depth and semantic
supervision [13]. On the other hand, SfM self-supervision has been also used
to improve LiDAR-only supervision [14]. Finally, it is also worth to mention
that many approaches [22, 36, 34, 23, 13] explore the use of synthetic images
with automatically generated depth supervision, which implies addressing the
synth-to-real domain gap.

###  2.2 3D Object Detection

While there are proposals on pure image-based 3D object detection [18], for
our study, we are interested in point cloud based object detection. A priori,
the point clouds can come from LiDAR sensors or stereo rigs, however, the
former case is the most usual in the literature. In any case, we can
categorize point cloud based 3D object detection, according to the manner the
point cloud is represented. Essentially, we can find point-based and voxel-
based approaches.

Point-based. These methods work directly on the raw point clouds, so
preserving the geometric information of the world. PointNet [24] processes
point clouds for 3D object recognition and segmentation, while PointNet++ [25]
runs PointNet in a hierarchical manner to capture local structures and
granular patterns of the point cloud. Then, inspired by the popular 2D object
detection methods Faster R-CNN [26] and SSD [20], the 3D object detectors
Point R-CNN [27] and 3D SSD [30] have been proposed, which use PointNet++ as a
backbone feature extractor.

Voxel-based. These methods transform the raw point clouds into a volumetric
representation, _i.e_., a voxel grid. VoxelNet [39] was the first end-to-end
trainable network to learn the informative features by dividing the point
cloud into a voxel grid. The grid is processed using a voxel feature encoding
(VFE) network to extract 3D features. These features are fed to a region
proposal network (RPN) to obtain object probability scores and regress 3D BBs.
In fact, VoxelNet has been used as a backbone network in several 3D object
detection architectures. For instance, Voxel R-CNN [6] uses VoxelNet as part
of a RPN. Then, given the corresponding 3D features and 3D BBs, a voxel
grouping is performed by a process called _voxel query_ (inspired in Ball
query [25]). Finally, a PointNet-inspired process is applied to obtain grid
point features, which are fed to fully connected networks to perform the final
refinement and classification of the BBs. CenterPoint [31], inspired by
CenterNet [38], transforms the point cloud into voxels or pillars, using
either VoxelNet or PointPillars [17], respectively. When using VoxelNet as
backbone network, the extracted 3D features are flatten to obtain 2D features.
Then, a keypoint detector is applied on these 2D features to find the center
of potentially detected objects. After, an anchor-free network produces
heatmaps to extract additional object properties such as 3D size and
orientation. In addition, a light-weighted point network extracts point
features at the center of each side of each of the 3D BBs provided by
VoxelNet. These point features and those from the object center point are
concatenated to pass through MLPs which provide the classification score and
3D BB of each potential object.

##  3 Methods

In order to assess the usefulness of MDE methods for performing 3D object
detection, we consider different MDE approaches and 3D object detectors which
work on point clouds. While LiDAR already provides such point clouds, for MDE
we have to produce the so-called Pseudo-LiDAR point clouds [28] by properly
sampling the respective depth maps. Fig. 1, briefly illustrates the overall
idea of performing 3D object detection from monocular images.

Figure 1: MDE-based 3D object detection in a nutshell. Given an image ($I$) we
apply a MDE model ($\Psi$) to generate its corresponding depth map ($0pt$).
With the intrinsic parameters, $\mathcal{K}$, of the camera capturing the
images, as well as a set of parameters, $\mathcal{L}$, defining a LiDAR-
inspired sampling procedure, the depth map can be converted to a 3D point
cloud, $\hat{\varrho}$. Then, $\hat{\varrho}$ is used to train and test a 3D
object detector ($\Xi$) originally developed to work with LiDAR point clouds.

###  3.1 Pseudo-LiDAR Generation

####  3.1.1 From a depth map to a 3D point cloud: $\varrho$

In order to generate a Pseudo-LiDAR point cloud, $\varrho$, from a depth map,
$0pt$, estimated from an image, $I$, we need the intrinsic parameters,
$\mathcal{K}$, of the camera that generated this image. More specifically, we
need its optical center $(C_{u},C_{v})$ and focal length111In practice,
calibration software allows to estimate a different focal length parameters
per image axis, _i.e_., $F_{u}$ and $F_{v}$. However, it is expected that
$F_{u}\approx F_{v}$, since this is basically a numerical trick. Thus, for the
sake of simplicity, we keep the idea of using a single focal length parameter.
$F$, which can be obtained by well-established camera calibration methods
[21]. Therefore, we have $\mathcal{K}=\\{C_{u},C_{v},F\\}$. Given this
information, we can assign a 3D point, $(x,y,z)$, to each pixel, $(u,v)$, of
the depth map (an so of the input image) as follows:

$\centering
z\leftarrow{0pt}_{u,v},\,y\leftarrow(z/F)(v-C_{v}),\,x\leftarrow(z/F)(u-C_{u})\@add@centering$
(1)

Thus, $\varrho$ is generated by applying Eq. 1 in all pixels.

####  3.1.2 Sampled Pseudo-LiDAR: $\hat{\varrho}$

As we will see in Sect. 4, directly working with $\varrho$ drives to poor 3D
object detection results. We believe that this is because the design of the
state-of-the-art 3D object detectors is biased towards the typical 3D pattern
distributions present in point clouds captured by actual LiDARs. Therefore, we
introduce a LiDAR-inspired sampling procedure which aims at making Pseudo-
LiDAR point clouds to be more similar to LiDAR ones. Note that here we are not
addressing a domain adaptation problem, since training and testing data will
come from the same domain. Instead, we aim at adjusting our generated point
clouds to be better suited for training and testing models such as Point
R-CNN, Voxel R-CNN, and CenterPoint.

Figure 2: Top: depth map. Mid-Bottom: white pixels would be sampled according
to the LiDAR-inspired procedure for $N_{b}=64$ (mid) and $N_{b}=16$ (bottom),
where $N_{b}$ is the number of beams of the simulated LiDAR.

Let us introduce the parameters, $\mathcal{L}$, required for such LiDAR-
inspired sampling. We assume a rotational LiDAR mounted with the rotation axis
mainly perpendicular with respect to the road plane. The Velodyne LiDAR HDL
64e used in KITTI dataset, is an example. We term as $N_{b}$ the number of
beams of the LiDAR under consideration. We term as $\mathcal{V}$ and
$\mathcal{H}$ the vertical and horizontal field of view (FOV) of the LiDAR,
respectively. The vertical angle resolution is $\mathcal{V}/N_{b}$, while the
horizontal angle resolution, $\theta_{\mathcal{H}}$, depends on the rotation
mechanism. For instance, for the mentioned Velodyne used in KITTI dataset, we
have $N_{b}=64$, $\mathcal{V}\approx 26.9^{\circ}$, $\mathcal{H}=360^{\circ}$,
and $\theta_{\mathcal{H}}\approx 0.08^{\circ}$. Moreover,
$d^{\mbox{{\scriptsize max}}}$ and $h^{\mbox{{\scriptsize max}}}$ denote the
maximum depth and height we want to consider above the camera, respectively.
Finally, it is common to discard the rows of the depth map above a threshold
$r^{\mbox{{\scriptsize min}}}$. Therefore, we have
$\mathcal{L}=\\{N_{b},\mathcal{V},\mathcal{H},\theta_{\mathcal{H}},d^{\mbox{{\scriptsize
max}}},h^{\mbox{{\scriptsize max}}},r^{\mbox{{\scriptsize min}}}\\}$.

Table 1: Details about the selected MDE models.

Model Training Data Backbone Encoder Working Resolution (pixels) Weights
(millions) PackNet-SfM Monocular sequences PackNet $1280\times 384$ 129.88
MonoDepth2-St Stereo pairs ResNet-18 $1024\times 320$ 14.84 MonoDepth2-St+SfM
Stereo pairs & Monocular Seq. ResNet-18 $1024\times 320$ 14.84 MonoDEVS-SfM
Virtual depth & Monocular Seq. HRNet-w48 $1280\times 384$ 93.34 MonoDELS-SfM
LiDAR & Monocular Seq. HRNet-w48 $1280\times 384$ 93.34 MonoDELS-SfM/RN LiDAR
& Monocular Seq. ResNet-18 $1280\times 384$ 14.84 MonoDELS-St LiDAR & Stereo
pairs HRNet-w48 $1280\times 384$ 93.34 AdaBins LiDAR EfficientNet B5
$[1224-1242]\times[370-376]$ 78.25

Now we can think of the sampling procedure as follows. We have a _virtual ray_
originated in the camera optical center $(C_{u},C_{v})$. This ray samples the
image plane (which is at a distance $F$ from the principal point), by
increments of $\theta_{\mathcal{H}}$ in its horizontal-component motion, and
increments of $\mathcal{V}/N_{b}$ in its vertical-component motion. In
addition, $\mathcal{V},\mathcal{H},d^{\mbox{{\scriptsize max}}}$ and
$h^{\mbox{{\scriptsize max}}}$ set bounds in the 3D space to be considered,
while $r^{\mbox{{\scriptsize min}}}$ sets a bound in the image space. For the
research carried out in this paper, we have set $\mathcal{V}$ and
$\mathcal{H}$ so that we consider the full image area, $d^{\mbox{{\scriptsize
max}}}=80$m (points with greater depth are not considered),
$h^{\mbox{{\scriptsize max}}}=1$m (points with higher height above the camera
are not considered), and $r^{\mbox{{\scriptsize min}}}$ is set to discard the
top 40% rows of the depth maps. Note also that the mentioned ray will
intersect the image plane in sub-pixel coordinates, however, we take the
nearest neighborhood approach to select corresponding pixel coordinates. Fig.
2 shows what pixels from a depth map would be sampled to generate the final 3D
point cloud following Eq. 1. We term this Pseudo-LiDAR 3D point cloud as
$\hat{\varrho}$.

###  3.2 MDE models

In order to generate the Pseudo-LiDAR point clouds, we consider well-
established and diverse state-of-the-art methods. Moreover, we prioritize MDE
models publicly available, so ready to produce depth maps. Accordingly, based
on self-supervision we use MonoDepth2 [11], PackNet [12], and MonoDEVSNet
[13]. Two variants of MonoDepth2 are used, with only stereo self-supervision
and with SfM and stereo self-supervision, we will call them MonoDepth2-St and
MonoDepth2-St+SfM, respectively. Analogously, since PackNet relies on SfM
self-supervision we will term it here as PackNet-SfM. MonoDEVSNet relies on
synthetic data supervision and SfM self-supervision, so we will term it here
as MonoDEVSNet-SfM. Based on LiDAR supervision we use AdaBins [1], and
MonoDELSNet [14], which also incorporates SfM self-supervision and, so, we
term it here as MonoDELSNet-SfM. We have developed a MDE model by replacing
SfM self-supervision with stereo self-supervision on MonoDELS-SfM, we term it
as MonoDELS-St. Table 1 summarizes the main details related to the selected
MDE models.

###  3.3 3D object detectors

Regarding 3D object detection, we consider three relatively different
approaches which are Point R-CNN [27], Voxel R-CNN [6], and CenterPoint [31].
In terms of Sect. 3.1.2, any of these 3D object detectors can play the role of
$\Xi$. Beyond differences in their respective CNN architectures, an important
difference for our study is the fact that they rely on different strategies to
represent the 3D point clouds. More specifically, as we have introduced in
Sect. 3, Point R-CNN assumes a point-based representation, while Voxel R-CNN
and CenterPoint rely on a voxel-based representation. Another important point
for our research is that these methods have publicly available code for
training and testing the respective 3D object detectors.

##  4 Experimental Results

###  4.1 Datasets and evaluation metrics

We have downloaded the MDE models reported in Sect. 3.2, except for
MonoDELSNet-St. The downloaded models were trained on the training set of the
Eigen _et al_. [7] split of the KITTI Raw [9] dataset, considering the subset
established by Zhou _et al_. [37] when using SfM self-supervision.
Accordingly, we have used the same training and testing data for our
MonoDELSNet-St. Overall, we can compare all the MDE models since they have
been trained on the same data, and will be tested in the same data too.

Figure 3: Protocol for experiments. DE stands for depth estimation. TS and VS
stands for training and validation set, resp. Here, the DE TS is the training
set of the Eigen _et al_. [7] split, the 3D-OD TS and 3D-OD VS are the
training and validation sets of the Chen _et al_. [4] split. Thus, the
datasets are from KITTI benchmark [9]. Eval DE is based on the standard
metrics to evaluate depth estimation (abs-rel, rms, ${\delta<\tau}$), while
Eval 3D-OD is based on metrics to assess 3D object detection accuracy
($AP_{BEV}$, $AP_{3D}$). Blue paths work at training time, while red paths
work at inference/validation time.

For evaluating 3D object detectors, we consider KITTI object detection
benchmark [9]. Moreover, we use the Chen _et al_. [4] split, consisting of
3,712 images for training and 3,769 for validation. As we have mentioned
before, we use Point R-CNN, Voxel R-CNN, and CenterPoint as 3D object
detectors. In order to make the most of the hyper-parameter tuning done by the
respective authors, we use their corresponding framework settings. This
implies to train Voxel R-CNN only for the class _car_ , while Point R-CNN and
CenterPoint will also include the classes _pedestrian_ and _cyclist_. An
important question for our experiments is that this total of 7,481 images come
with corresponding 3D point clouds based on a 64-beams Velodyne LiDAR. Camera
calibration matrices are also available. Thus, we can obtain the Pseudo-LiDAR
depth (Sect. 3.1) by setting $N_{b}=64$. In the rest of the paper, we term all
these data as KITTI-3D-OD.

In this study, we focus on three _de facto_ standard and complementary metrics
for evaluating depth estimation [7], defined as follows: $\textbf{abs-
rel}\leftarrow(1/n)\sum_{ij}|d_{gt}-d_{p}|/d_{gt}$,
$\textbf{rms}\leftarrow\sqrt{(1/n)\sum_{ij}(d_{gt}-d_{p})^{2}}$, and, for a
threshold $\tau$, $\delta<\tau\leftarrow\text{\% pixels
with}\max\left(|d_{p}|/d_{gt},|d_{gt}|/d_{p}\right)<\tau,$ where $n$ is the
total number of valid pixels (_i.e_., containing depth ground truth) in the
full set of testing images, $d_{gt}$ and $d_{p}$ are the ground truth and
predicted depths at a given pixel of any testing image, respectively. The
absolute relative error (abs-rel) and the accuracy with threshold $\tau$
(${\delta<\tau}$) are percentage measurements, while the root mean square
error (rms) is reported in meters.

For 3D object detection we use KITTI metrics [9]. More specifically, we report
Average Precision (AP) for 3D and bird-eye-view (BEV), _i.e_., AP3D and APBEV.
Specific results for the detection-difficulty categories used with KITTI
metrics are also reported, _i.e_., for _easy_ , _moderate_ , and _hard_. For
computing these APs, we consider an intersection-over-union (IoU) threshold
equal to 0.7 (over 1.0) for the easy case, and 0.5 for the moderate and hard
cases. Moreover, as we have mentioned above, the _car_ class is the only in
common for the default configurations of Point R-CNN, Voxel R-CNN, and
CenterPoint. We also trained Voxel R-CNN for the multi-class task. However,
performance on pedestrians and cyclists is poor. In fact, for Point R-CNN and
CenterPoint neither is so good. Thus, we focus the quantitative analysis on
car detection, while qualitative results are presented for the classes that
each detector considers.

###  4.2 Experiment protocol

The protocol to conduct our experiments is summarized in Fig. 3 with the
following description of the Steps:

  1. 1.

Training depth estimation. MDE models are trained on the training set of the
Eigen et al. split (from KITTI Raw dataset). As mentioned in Sect. 4.1, we use
available trained models (MonoDepth2-Sfm, MonoDepth2-SfM+St, MonoDEVSNet-SfM,
MonoDELSNet-SfM, MonoDELSNet-SfM/RN, PackNet-SfM, AdaBins), and train our
MonoDELSNet-St.

  2. 2.

Testing depth estimation. MDE models are applied to the images of the
validation set of the Chen et al. split. Their performance is assessed with
the LiDAR-based ground truth (depth) of the same validation set. This is done
based on abs-rel, rms, and ${\delta<\tau}$.

  3. 3.

Generating 3D point clouds. MDE models are applied to the images of the
training and validation set of the Chen et al. split (from KITTI-3D-OD
dataset). This generates the corresponding depth maps, which are converted
into Pseudo-LiDAR as explained in Sect. 3.1. Note that KITTI images and LiDAR
data are calibrated, so it is right to assume that we can use the same 3D BBs
for LiDAR and Pseudo-LiDAR data. Note that, in case of using an on-board
vision-based system to fully replace LiDAR, these 3D BBs should be directly
annotated in the Pseudo-LiDAR.

  4. 4.

Training 3D object detection. The Pseudo-LiDAR obtained from the images of the
training set of the Chen et al. split are used to train the 3D objected
detection models summarized in Sect. 3.3. (_i.e_., Point R-CNN, Voxel R-CNN,
CenterPoint).

  5. 5.

Testing 3D object detection. The 3D object detectors are applied to the
Pseudo-LiDAR obtained from the images of the validation set of the Chen et al.
split. Performance is assessed with $AP_{BEV}$, and $AP_{3D}$.

  6. 6.

Depth estimation _vs_. 3D object detection. Select a metric on MDE and rank
the MDE models accordingly. Assess if this ranking matches with the rankings
that would result from the metrics on 3D object detection. The better is the
matching, the better the MDE metric is for comparing depth models regarding
its expected performance when used for 3D object detection.

Table 2: MDE results (up to $80$m) for the images of the validation set of Chen _et al_. [4] split. Best and second best coding is used. Taking Fig. 3 as reference, abs-rel, rms, and $\delta<\tau$ (here $\tau=1.25$) play the role of Eval DE. |  |  |
---|---|---|---
Model | abs-rel | rms | $\delta<1.25$
|  |  |
PackNet-SfM | 0.101 | 4.774 | 0.888
MonoDepth2-St | 0.096 | 4.719 | 0.883
MonoDepth2-St+SfM | 0.095 | 4.609 | 0.892
MonoDEVS-SfM | 0.090 | 4.107 | 0.903
MonoDELS-SfM | 0.077 | 3.837 | 0.911
MonoDELS-SfM/RN | 0.079 | 3.904 | 0.909
MonoDELS-St | 0.073 | 3.761 | 0.918
AdaBins | 0.080 | 3.821 | 0.920
|  |  |
Table 3: Results on 3D car detection using Point R-CNN, Voxel R-CNN and CenterPoint. LiDAR PC refers to training with actual raw LiDAR 3D point clouds. Thus, this case can be seen as the upper-bound for the rest using MDE-based Pseudo-LiDAR. Best and second best coding refers to each 3D object detector block. Taking Fig. 3 as reference, $AP_{BEV}$ and $AP_{3D}$ play the role of Eval 3D-OD. |  |  |  |  |  |
---|---|---|---|---|---|---
| $AP_{BEV}$ | $AP_{3D}$
|  |  |  |  |  |
Model | easy | mod. | hard | easy | mod. | hard
|  |  |  |  |  |
Point R-CNN
|  |  |  |  |  |
LiDAR PC | 90.63 | 89.55 | 89.35 | 90.62 | 89.51 | 89.28
|  |  |  |  |  |
PackNet-SfM | 47.86 | 32.53 | 30.50 | 45.74 | 31.39 | 27.84
MonoDepth2-St | 64.34 | 40.88 | 37.21 | 59.96 | 39.36 | 32.41
MonoDepth2-St+SfM | 58.81 | 37.68 | 31.63 | 53.14 | 35.37 | 30.10
MonoDEVS-SfM | 66.17 | 47.66 | 45.30 | 64.33 | 45.93 | 40.01
MonoDELS-SfM | 75.58 | 56.07 | 48.68 | 71.15 | 53.31 | 45.92
MonoDELS-SfM/RN | 70.50 | 48.97 | 45.41 | 64.89 | 47.15 | 39.87
MonoDELS-St | 74.57 | 54.35 | 47.25 | 68.13 | 47.42 | 42.84
AdaBins | 70.86 | 48.31 | 41.12 | 65.66 | 45.98 | 39.56
|  |  |  |  |  |
Voxel R-CNN
|  |  |  |  |  |
LiDAR PC | 97.33 | 89.71 | 89.35 | 97.29 | 89.70 | 89.33
|  |  |  |  |  |
PackNet-SfM | 55.23 | 37.12 | 35.80 | 52.19 | 34.71 | 33.86
MonoDepth2-St | 65.21 | 45.74 | 42.64 | 62.21 | 42.54 | 37.47
MonoDepth2-St+SfM | 57.98 | 37.72 | 35.09 | 52.87 | 35.40 | 30.54
MonoDEVS-SfM | 65.73 | 46.90 | 45.24 | 63.06 | 44.69 | 42.84
MonoDELS-SfM | 74.91 | 56.07 | 53.17 | 71.51 | 53.11 | 47.06
MonoDELS-SfM/RN | 69.94 | 52.59 | 46.62 | 65.42 | 47.68 | 43.87
MonoDELS-St | 75.90 | 57.39 | 54.23 | 73.97 | 55.64 | 48.73
AdaBins | 70.77 | 52.34 | 46.46 | 66.18 | 47.44 | 43.90
CenterPoint
|  |  |  |  |  |
LiDAR PC | 95.25 | 89.88 | 89.30 | 95.17 | 89.85 | 89.21
|  |  |  |  |  |
PackNet-SfM | 49.89 | 35.10 | 32.10 | 45.44 | 32.33 | 29.10
MonoDepth2-St | 62.14 | 43.22 | 37.59 | 56.43 | 39.82 | 34.96
MonoDepth2-St+SfM | 51.00 | 35.12 | 31.72 | 47.01 | 31.50 | 28.12
MonoDEVS-SfM | 61.13 | 44.63 | 42.16 | 56.66 | 41.63 | 37.10
MonoDELS-SfM | 69.23 | 52.87 | 46.46 | 63.60 | 48.56 | 43.29
MonoDELS-SfM/RN | 66.72 | 49.65 | 44.82 | 62.57 | 45.65 | 41.14
MonoDELS-St | 74.04 | 56.14 | 51.96 | 68.69 | 53.58 | 47.27
AdaBins | 67.92 | 47.21 | 43.04 | 63.62 | 44.85 | 38.63
|  |  |  |  |  |

###  4.3 Testing MDE and 3D object detection

MDE results are shown in Table 2, while Table 3 present 3D car detection
results. The images used to generate the depth maps (and so the Pseudo-LiDAR
$\hat{\varrho}$) are from the training set of the Chen _et al_. [4] split,
while the validation is performed on the validation set of this split too.
Concerning 3D object detection, we can see that MonoDELSNet-SfM and
MonoDELSNet-St are consistently outperforming the rest of MDE models. The
results are still significantly far from the upper-bound based on LiDAR point
clouds. Considering only the detection results corresponding to the use of
actual LiDAR point clouds, voxel-based methods (_i.e_., Voxel R-CNN and
CenterPoint) outperform Point R-CNN. In this case, CenterPoint and Point R-CNN
can be actually compared since they consider the same three classes (although
quantitative results are reported only for car detection here). However, Voxel
R-CNN was tuned for car detection only, so they may outperform the others
because of this. On the other hand, this is not crucial here since we compare
rankings of MDE _vs_. 3D object detection.

In addition to the quantitative analysis, Fig. 4 presents qualitative results
in terms of 3D object detection for the different MDE methods combined with
Point R-CNN. Moreover, Figures 5, 6, and 7, show results for the MonoDELSNet-
St model when using Point R-CNN, Voxel R-CNN, and CenterPoint, respectively.
Despite there are errors (false positives and negatives), we think these are
very promising results taking into account we are able to provide relatively
accurate 3D object BBs (for the true positives) from single images.

Figure 4: 3D object detection for different MDE methods combined with Point
R-CNN. Red BBs are  ground truth, green ones are  detections. Figure 5: 3D
object detection based on MonoDELSNet-St combined with Point R-CNN. Figure 6:
3D object detection based on MonoDELSNet-St combined with Voxel R-CNN.

Figure 7: 3D object detection based on MonoDELSNet-St combined with
CenterPoint.

###  4.4 Depth estimation _vs_. 3D object detection

Despite examining the results of Table 2 and Table 3 has a great interest in
itself, in this paper, we focus on confronting them. Concerning the 3D object
detection metrics, it is common practice to focus on the moderate setting
(mod.). Moreover, we see in Table 2 that $AP_{BEV}$ and $AP_{3D}$ give rise to
analogous rankings in terms of 3D object detection under the moderate setting.
Thus, for the shake of simplicity, we are going to focus only on $AP_{BEV}$.
Accordingly, Fig. 8 compares the $AP_{BEV}-mod$ ranking with the abs-rel, rms,
and $\delta<\tau$ $(\tau=1.25)$ rankings, for all the considered 3D object
detection models. Briefly, the correspondence between MDE and object detection
rankings can be visually observed by looking at the arrows in these figures. A
perfect correspondence between rankings shows as parallel arrows, while the
lower the correspondence, the more arrows crossing each other. Taking this
into account, it is clear that the MDE metric abs-rel predicts the 3D object
detection results better than the others. The $\delta<\tau$ metric is more
messy, and rms metric is performing in between the other two.

Figure 8: Comparing rankings: abs-rel, rms, $\delta<1.25$ (MDE) _vs_.
$AP_{BEV}-mod$ (3D OD). In the mid column, we have ordered the MDE models from
best (top/1) to worse (bottom/8). Then, we have replicated the mid columns as
left (voxel-based detectors, which produce the same ranking) and right (Point
R-CNN) columns. Afterwards, we have connected the models in left and right
columns to its ranking number according to $AP_{BEV}-mod$. Thus, a perfect
correspondence between rankings shows as parallel arrows, while the lower the
correspondence, the more arrows crossing each other. We abbreviate MonoDepth2
as MD2, MonoDEVSNet as MDEVS, MonoDELSNet as MDELS.

##  5 Conclusion

When performing MDE on-board an autonomous or assisted vehicle, obtaining
depth estimation maps is just an intermediate step of a perception stack. For
instance, the perception goal may be to perform semantic segmentation or/and
object detection with attached depth information to allow for actual vehicle
navigation. On the other hand, reviewing the literature of MDE, it is common
to see relatively small quantitative differences among the evaluated MDE
models. Thus, one may wonder if those differences will be consolidated once
such MDE models are used in the targeted perception task. We have addressed
this question by using 3D object detection as target perception task. Depth
maps based on different MDE models have been converted to Pseudo-LiDAR 3D
point clouds, where 3D object detectors can be trained and tested. We have
considered eight MDE models, as well as three different CNN architectures for
3D object detection (Point R-CNN, Voxel R-CNN, CenterPoint). Using KITTI
benchmark data, we have seen that, indeed, the abs-rel metric commonly used in
MDE assessment, is well aligned with 3D object detection results in terms of
ranking the MDE methods. What remains as future work is to investigate if we
can predict accuracy improvements in 3D object detection (in absolute terms),
from the improvements observed in the abs-rel metric. In case this is not
possible, we recommend to incorporate 3D object detection as part of the
evaluation of MDE models. It is worth to mention that we have also seen that
the way depth maps are sampled to produce Pseudo-LiDAR matters because of the
dependency that 3D object detectors have on this.

## Acknowledgement

Antonio acknowledges the financial support received for this research from the
Spanish TIN2017-88709-R (MINECO/AEI/FEDER, UE) project. Antonio acknowledges
the financial support to his general research activities given by ICREA under
the ICREA Academia Program. Antonio acknowledges the support of the
Generalitat de Catalunya CERCA Program as well as its ACCIO agency to CVC’s
general activities.

We thank Antoni Bigata Casademunt for assistance with sampling Pseudo-LiDAR.
We thank Gabriel Villalonga and Onay Urfalioglu for helpful discussions.

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# A hidden gauged $U(1)$ addressing radiative neutrino mass,
dark matter, $(g-2)_{\mu}$, and $H_{0}$ tension

Ujjal Kumar Dey<EMAIL_ADDRESS>Department of Physical Sciences, Indian
Institute of Science Education and Research Berhampur, Transit Campus,
Government ITI, Berhampur 760010, Odisha, India  Hiroshi Okada
<EMAIL_ADDRESS>Asia Pacific Center for Theoretical Physics (APCTP) -
Headquarters San 31, Hyoja-dong, Nam-gu, Pohang 790-784, Korea  Department of
Physics, Pohang University of Science and Technology, Pohang 37673, Republic
of Korea

###### Abstract

We propose a radiative seesaw model with a hidden $U(1)$ gauge symmetry. In
order to have anomaly cancellations, we need to introduce several new fermions
that contribute to muon anomalous magnetic dipole moment $(g-2)_{\mu}$ as well
as explain the neutrino oscillation data. We also consider a fermionic dark
matter candidate that correlates with $(g-2)_{\mu}$ and neutrino mass matrix
at the same time. We show allowed regions in our input parameters satisfying
several constraints. Finally, we also briefly discuss the possibility of the
resolution of the Hubble tension via neutrino self-interactions mediated by a
lighter hidden gauge boson without spoiling our model.

##  I Introduction

Explaining nonzero neutrino masses is one of the important issues to be
resolved beyond the standard model (SM) of particle physics. In addition,
there are certain experimental observations, namely existence of dark matter
(DM), anomalous magnetic dipole moment of muon $(g-2)_{\mu}$ etc., which can
also not be explained by the SM. In the standard paradigm of cosmology, i.e,
the $\Lambda$CDM model there exists a statistically significant discrepancy in
the measurements of the Hubble constant $H_{0}$ which is crucial in the
estimation of the expansion rate of the Universe, see Di Valentino et al.
(2021) for a review. This is famously known as Hubble tension. Even though
there exist a lot of mechanisms to generate the non-vanishing neutrino masses,
a radiative seesaw scenario is attractive due potentially to involving the DM
candidate. For most of the cases, an additional symmetry is imposed to
maintain the neutrino loop as well as assuring the stability of DM. If one
introduces a hidden $U(1)$ gauge symmetry to realize a radiative neutrino mass
model, there is a possibility that Hubble tension can also be explained Berbig
et al. (2020). In addition, the DM problem can be addressed by the lightest
Majorana neutrino present in the model, the DM stability is ensured by a
remnant symmetry such as $\mathbb{Z}_{2}$ after spontaneous breaking of the
hidden $U(1)$ symmetry.

In this paper, we study a radiative seesaw model introducing a hidden $U(1)$
gauge symmetry. In order to have the gauge anomaly cancellations among
fermions, several fields are introduced in which a DM candidate naturally
emerges. In addition, we can explain sizable $(g-2)_{\mu}$ as a bonus of this
model due to appropriate charge assignments. A new gauge boson arising from
this model can, in principle, give rise to a self-interaction of neutrinos
which can potentially explain the $H_{0}$ tension as well.

This paper is organized as follows. In Section II, we present our model
formulating the renormalizable Lagrangian, scalar sector, neutrino sector,
gauge sector, dark matter, lepton flavor violations (LFVs), and $(g-2)_{\mu}$.
In Section III, we carry out numerical analysis showing our allowed input
parameters for normal and inverted hierarchies of the neutrino masses as well
as $(g-2)_{\mu}$ and LFVs. In Section IV, we discuss the possibility of
explaining the $H_{0}$ tension in this model. Finally we summarise and
conclude in Section V.

##  II Model setup and constraints

|  $U_{R},\ U_{L}$ |  $D_{R},\ D_{L}$ |  $E_{R},\ E_{L}$ |  $N_{R},\ N_{L}$ |  $\eta$ |  $\chi$ |  $\varphi$ |  $\varphi^{\prime}$
---|---|---|---|---|---|---|---|---
$SU(3)_{C}$ | $\bm{3}$ | $\bm{3}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$
$SU(2)_{L}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{2}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$
$U(1)_{Y}$ | $\frac{2}{3}$ | $-\frac{1}{3}$ | $-1$ | $0$ | $\frac{1}{2}$ | $0$ | $0$ | $0$
$U(1)_{H}$ | $4,\ 1$ | $-4,\ -1$ | $-4,\ -1$ | $4,\ 1$ | $4$ | $1$ | $3$ | $2$
Table 1:  Charge assignments of the our fields under $SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}\times U(1)_{H}$, where all the SM fields are zero
charges under the $U(1)_{H}$ symmetry. We suppose that all the exotic fermions
have three families.

In this section we review our model. We introduce isospin singlet exotic
quarks $U_{R(L)}$ and $D_{R(L)}$, and exotic leptons $E_{R(L)}$ and
$N_{R(L)}$, where we impose nonzero charges under a hidden $U(1)$ gauge
symmetry, $U(1)_{H}$. In Table 1, we depict the relevant charges of the fields
present in the model. The $U(1)_{H}$ chiral anomalies are canceled among these
fields Cai et al. (2019), and we need at least two families of $N_{R(L)}$ in
order to reproduce the neutrino oscillation data. Thus, all the exotic
fermions have to have two or more than two families in order to realize the
radiative seesaw model. Here, we fix three families for our setup. As for the
bosonic sector, we introduce two types of inert scalars, namely, $\eta$ and
$\chi$, where $\eta$ is an isospin doublet and $\chi$ is an isospin singlet.
Furthermore, we introduce two singlets $\varphi$ and $\varphi^{\prime}$ both
of which have nonzero VEVs that contribute to the spontaneous breaking of
$U(1)_{H}$. The new scalar contents and their charge assignments are also
summarized in Table 1. The SM Higgs is denoted by $H$ and its VEV is defined
by $\langle H\rangle\equiv[0,v/\sqrt{2}]^{T}$. Then the valid renormalizable
Yukawa Lagrangian and Higgs potential under these symmetries are given by,

$\displaystyle-{\cal L}_{Y}$
$\displaystyle=y_{\ell_{ii}}\bar{L}_{L_{i}}He_{R_{i}}+f_{ai}\bar{E}_{R_{a}}\eta^{*}L_{L_{i}}+g_{ia}\bar{e}_{R_{i}}E_{L_{a}}\chi$
$\displaystyle~{}~{}+y_{\eta_{ia}}\bar{L}_{L_{i}}\tilde{\eta}N_{R_{a}}+y_{E_{aa}}\varphi\bar{E}_{L_{a}}E_{R_{a}}+y_{{N}_{aa}}\bar{N}_{L_{a}}N_{R_{a}}\varphi^{*}+y_{{N^{\prime}}_{ab}}\bar{N}^{C}_{L_{a}}N_{L_{b}}\varphi^{\prime}+{\rm
h.c.},$ (1) $\displaystyle V$
$\displaystyle=\sum_{\phi}^{H,\eta,\varphi,\varphi^{\prime},\chi}\left(\mu^{2}_{\phi}\phi^{\dagger}\phi+\lambda_{\phi}|\phi^{\dagger}\phi|^{2}\right)+\sum_{\phi\neq\phi^{\prime}}^{H,\eta,\varphi,\varphi^{\prime},\chi}\lambda_{\phi\phi^{\prime}}|\phi|^{2}|\phi^{\prime}|^{2}+\lambda^{\prime}_{H\eta}(H^{\dagger}\eta)(\eta^{\dagger}H)$
$\displaystyle~{}~{}+\mu_{0}\chi\varphi^{*}\varphi^{\prime}+\mu_{1}\chi^{2}\varphi^{\prime*}+\lambda_{0}\chi^{*}\varphi^{*}\varphi^{\prime
2}+\lambda_{1}(H^{\dagger}\eta)\varphi\chi+{\rm h.c.},$ (2)

where $\tilde{\eta}\equiv i\sigma_{2}\eta^{*}$, $\sigma_{2}$ being the second
Pauli matrix, $\lambda_{\phi\phi^{\prime}}\equiv\lambda_{\phi^{\prime}\phi}$,
lower indices $(a,b,i)=1,2,3$ are the number of families. The Yukawa couplings
$y_{\ell},\ y_{E},\ y_{N}$ can be taken to be diagonal by field-phase
redefinitions without loss of generality.

###  II.1 Scalar sector

At first, we define each scalar as follows,

$\displaystyle H=\left[\begin{array}[]{c}w^{+}\\\
\frac{v+h+iz}{\sqrt{2}}\end{array}\right],\quad\eta=\left[\begin{array}[]{c}\eta^{+}\\\
\frac{\eta_{R}+i\eta_{I}}{\sqrt{2}}\end{array}\right],$ (7)
$\displaystyle\varphi\equiv\frac{v_{\varphi}+r+z_{\varphi}}{\sqrt{2}},\
\varphi^{\prime}\equiv\frac{v_{\varphi^{\prime}}+r^{\prime}+z_{\varphi^{\prime}}}{\sqrt{2}},\
\chi\equiv\frac{\chi_{R}+\chi_{I}}{\sqrt{2}},$ (8)

where $w^{\pm}$ is absorbed by the SM singly-charged gauge boson $W^{\pm}$,
and one degree of freedom in the CP-odd scalar sector $z$ and the Nambu-
Goldstone boson $G$ after diagonalizing $z_{\varphi}$ and
$z_{\varphi^{\prime}}$ are respectively eaten by the neutral SM gauge boson
$Z$ and the hidden gauge boson $Z^{\prime}$. 111In our model, there is one
physical Goldston boson and its phenomenology was discussed in ref. Cai et al.
(2019). Here, we define the relevant mass eigenstates and their mixing
matrices $O_{R(I)}$ for inert bosons $\chi$ and $\eta$. Due to non-trivial
terms $\mu$ and $\lambda_{0}$, they mix with each other.

$\displaystyle\left(\begin{array}[]{c}\chi_{R}\\\
\eta_{R}\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta_{R}&-\sin\theta_{R}\\\
\sin\theta_{R}&\cos\theta_{R}\end{array}\right)\left(\begin{array}[]{c}H_{1}\\\
H_{2}\end{array}\right),$ (15)
$\displaystyle\left(\begin{array}[]{c}\chi_{I}\\\
\eta_{I}\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta_{I}&-\sin\theta_{I}\\\
\sin\theta_{I}&\cos\theta_{I}\end{array}\right)\left(\begin{array}[]{c}A_{1}\\\
A_{2}\end{array}\right),$ (22)

where $O_{R}(O_{I})$ is the matrix parameterized by $\theta_{R}(\theta_{I})$
above. Notice here that we neglect the mixing among $h,r,r^{\prime}$ for
simplicity that would be favored by experiments.

###  II.2 Neutrino sector

After spontaneous symmetry breaking, we have a neutral fermion mass matrix of
6$\times$6 in the basis of $\left(N_{R}^{C},N_{L}\right)$. This is given by,

$\displaystyle M_{N}$ $\displaystyle=\left[\begin{array}[]{cc}0&m_{D}\\\
m_{D}&M_{N_{L}}\\\ \end{array}\right],$ (25)

where $m_{D}\equiv y_{N}v_{\varphi}/\sqrt{2}$, $M_{N_{L}}\equiv
y_{N^{\prime}}v_{\varphi^{\prime}}/\sqrt{2}$. $M_{N}$ is diagonalized by a
unitary matrix $V_{N}$ as $D_{N}\equiv V_{N}^{T}M_{N}V_{N}$,
$N_{R_{i}}\equiv\sum_{a=1}^{6}V_{N_{ia}}\psi_{R_{a}}$ and
$N^{C}_{L_{i}}\equiv\sum_{a=1}^{6}V_{N_{i+3,a}}\psi_{R_{a}}$ ($i=1,2,3$). The
entries of the diagonal matrix $D_{N}$ represent the mass eigenvalues and
$\psi_{R_{a}}$ are the mass eigenstates. The relevant Lagrangian in terms of
mass eigenstate is found to be

$\displaystyle-{\cal
L}=\frac{1}{\sqrt{2}}\bar{\nu}_{L_{i}}y_{\eta_{ia}}V_{N_{ab}}\psi_{R_{b}}(s_{R}H_{1}+c_{R}H_{2})-\frac{i}{\sqrt{2}}\bar{\nu}_{L_{i}}y_{\eta_{ia}}V_{N_{ab}}\psi_{R_{b}}(s_{I}A_{1}+c_{I}A_{2})+{\rm
h.c.},$ (26)

where we have used short-hand notations $s_{R(I)}$ $c_{R(I)}$ for
$\sin\theta_{R(I)}$ and $\cos\theta_{R(I)}$, respectively. Then, the neutrino
mass matrix is given by Cai et al. (2019)

$\displaystyle(m_{\nu})_{ij}$
$\displaystyle=\frac{(y_{\eta}V_{N})_{ia}D_{N_{a}}(y_{\eta}V_{N})^{T}_{aj}}{2(4\pi)^{2}}\left[-2F_{I}^{a}+F_{II}^{a}\right],$
(27) $\displaystyle F_{I}^{a}$
$\displaystyle=\int[dx]_{3}\ln\left[\frac{xD^{2}_{N_{a}}+ym^{2}_{H_{1}}+zm^{2}_{H_{2}}}{xD^{2}_{N_{a}}+ym^{2}_{A_{1}}+zm^{2}_{A_{2}}}\right],$
(28) $\displaystyle F_{II}^{a}$
$\displaystyle=\int[dx]_{3}\left[\frac{c^{2}_{R}m^{2}_{H_{1}}+s^{2}_{R}m^{2}_{H_{2}}}{xD^{2}_{N_{a}}+ym^{2}_{H_{1}}+zm^{2}_{H_{2}}}-\frac{c^{2}_{I}m^{2}_{A_{1}}+s^{2}_{I}m^{2}_{A_{2}}}{xD^{2}_{N_{a}}+ym^{2}_{A_{1}}+zm^{2}_{A_{2}}}\right],$
(29)

where $\int[dx]_{3}\equiv\int_{0}^{1}dx\int_{0}^{1-x}dy|_{z=1-x-y}$.

The neutrino mass matrix is diagonalized by unitary matrix $U_{MNS}$;
$D_{\nu}=U_{MNS}^{T}m_{\nu}U_{MNS}$, where $D_{\nu}\equiv{\rm
diag}(m_{1},m_{2},m_{3})$. Then, we parametrize $y_{\eta}$ in terms of
neutrino experiments and some model parameters as follows Nomura and Okada
(2019),

$\displaystyle y_{\eta}=U_{MNS}^{*}\sqrt{D_{\nu}}O_{\rm
mix}R_{N}^{-1}\lesssim\sqrt{4\pi},$ (30)

where the last inequality suggests the dimensionless couplings $y_{\eta}$ has
to be perturbative. Here $O_{\rm mix}$ is an arbitrary $3\times 3$ orthogonal
matrix where this matrix can be parametrized by three angles
$\alpha,\beta,\gamma$ as a standard parametrization, and $R_{N}$ is a lower
unit triangular matrix Nomura et al. (2017), which is uniquely decomposed to
be $(V_{N}D_{N}V_{N}^{T})_{3\times 3}=(R_{N}R^{T}_{N})_{3\times 3}$, since it
is symmetric.

###  II.3 Gauge sector

Since we have an additional $U(1)_{H}$ gauge symmetry, the most general $U(1)$
gauge Lagrangian including the kinetic mixing with the SM $U(1)_{Y}$ is given
as follows,

$\mathcal{L}_{\text{gauge}}=-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}-\frac{1}{4}B^{\prime}_{\mu\nu}B^{\prime\mu\nu}-\frac{1}{2}\xi
B_{\mu\nu}B^{\prime\mu\nu},$ (31)

where $B_{\mu\nu}$ and $B^{\prime}_{\mu\nu}$ are the field strength tensors of
$U(1)_{Y}$ and $U(1)_{H}$ gauge symmetries, respectively. We then diagonalize
Eq. (31) by the following transformation:

$\displaystyle\left(\begin{array}[]{c}\tilde{B}^{\prime}_{\mu}\\\
\tilde{B}_{\mu}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\sqrt{1-\xi^{2}}&0\\\ \xi&1\\\
\end{array}\right)\left(\begin{array}[]{c}B^{\prime}_{\mu}\\\ B_{\mu}\\\
\end{array}\right)$ (38)

where $\xi$ is a dimensionless quantity ($\xi\ll 1$) and we parameterize
$\rho=-\xi/\sqrt{1-\xi^{2}}$. Under the transformation Eq. (38), the gauge
Lagrangian can be rewritten as follows,

$\displaystyle\mathcal{L}_{\text{gauge}}=-\frac{1}{4}\tilde{B}_{\mu\nu}\tilde{B}^{\mu\nu}-\frac{1}{4}\tilde{B}^{\prime}_{\mu\nu}\tilde{B}^{\prime\mu\nu}$
(39)

where
$\tilde{B}_{\mu\nu}=\partial_{\mu}\tilde{B}_{\nu}-\partial_{\nu}\tilde{B}_{\mu}$
and
$\tilde{B}^{\prime}_{\mu\nu}=\partial_{\mu}\tilde{B}^{\prime}_{\nu}-\partial_{\nu}\tilde{B}^{\prime}_{\mu}$.

The kinetic term of the scalar fields with nonzero VEVs is

$\displaystyle\mathcal{L}_{\text{kin}}=(D_{\mu}H)^{\dagger}(D^{\mu}H)+(D_{\mu}\varphi)^{\dagger}(D^{\mu}\varphi)+(D_{\mu}\varphi^{\prime})^{\dagger}(D^{\mu}\varphi^{\prime}).$
(40)

The covariant derivatives of scalar fields are written by

$\displaystyle D_{\mu}H$ $\displaystyle=$
$\displaystyle\Big{(}\partial_{\mu}+ig_{1}\frac{\tau^{a}}{2}W_{\mu}^{a}+i\frac{g_{2}}{2}\tilde{B}_{\mu}+i\frac{g_{H}}{2}\rho\tilde{B^{\prime}}_{\mu}\Big{)}H,$
$\displaystyle D_{\mu}\varphi$ $\displaystyle=$
$\displaystyle\Big{(}\partial_{\mu}-3ig_{H}\frac{\rho}{\xi}\tilde{B}^{\prime}_{\mu}\Big{)}\varphi,$
$\displaystyle D_{\mu}\varphi^{\prime}$ $\displaystyle=$
$\displaystyle\Big{(}\partial_{\mu}-2ig_{H}\frac{\rho}{\xi}\tilde{B}^{\prime}_{\mu}\Big{)}\varphi^{\prime},$
(41)

where $g_{1}$, $g_{2}$ and $g_{H}$ are respectively gauge couplings of
$SU(2)_{L}$, $U(1)_{Y}$ and $U(1)_{H}$, $W^{a}$ are the $SU(2)_{L}$ gauge
fields, and $\tau^{a}$ are the Pauli matrices.

The masses of the gauge bosons come from Eqs. (40) and (41). The mass matrix
written in the basis of neutral gauge fields
$(W_{\mu}^{3},\tilde{B}_{\mu},\tilde{B}^{\prime}_{\mu})$ is

$\displaystyle\mathcal{L}_{\text{gauge}}^{\text{mass}}=\frac{1}{2}\left(\begin{array}[]{c}W_{\mu}^{3}\\\
\tilde{B}_{\mu}\\\
\tilde{B}^{\prime}_{\mu}\end{array}\right)^{T}M^{2}_{\text{gauge}}\left(\begin{array}[]{c}W_{\mu}^{3}\\\
\tilde{B}_{\mu}\\\ \tilde{B}^{\prime}_{\mu}\end{array}\right),$ (48)

where

$\displaystyle
M^{2}_{\text{gauge}}=\frac{1}{4}\left(\begin{array}[]{ccc}g_{1}^{2}v^{2}&-g_{1}g_{2}v^{2}&-g_{1}g_{2}v^{2}\rho\\\
-g_{1}g_{2}v^{2}&g_{2}^{2}v^{2}&g_{2}^{2}v^{2}\rho\\\
-g_{1}g_{2}v^{2}\rho&g_{2}^{2}v^{2}\rho&g_{2}^{2}v^{2}\rho+4g_{H}^{2}(9v^{2}_{\varphi}+4v^{2}_{\varphi^{\prime}})\frac{\rho^{2}}{\xi^{2}}\\\
\end{array}\right).$ (52)

Here we parameterize

$\displaystyle
M_{Z^{\prime}}^{2}=\frac{1}{4}g_{2}^{2}v^{2}\rho^{2}+g_{H}^{2}(9v^{2}_{\varphi}+4v^{2}_{\varphi^{\prime}})\frac{\rho^{2}}{\xi^{2}}.$
(53)

We rotate the fields $(W_{\mu}^{3},\tilde{B}_{\mu})$ by Weinberg angle
$\theta_{W}$ to obtain the massless photon field $A_{\mu}$ 222In general,
Weinberg angle is defined to be the mixing between $W^{3}_{\mu}$ and $B_{\mu}$
in the SM, this can be realized by $\rho\to 0$. In fact, we expect $\rho$ be
so small.

$\displaystyle\left(\begin{array}[]{c}W^{3}_{\mu}\\\ \tilde{B}_{\mu}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta_{W}&\sin\theta_{W}\\\
-\sin\theta_{W}&\cos\theta_{W}\\\
\end{array}\right)\left(\begin{array}[]{c}\tilde{Z}_{\mu}\\\ A_{\mu}\\\
\end{array}\right).$ (60)

And the mass matrix for the massive neutral gauge bosons is given as follows:

$\displaystyle\mathcal{L}^{\text{mass}}_{\text{gauge}}=\frac{1}{2}\left(\begin{array}[]{c}\tilde{Z}_{\mu}\\\
\tilde{B}^{\prime}_{\mu}\\\
\end{array}\right)^{T}\left(\begin{array}[]{cc}M_{Z,SM}^{2}&-\Delta^{2}\\\
-\Delta^{2}&M_{Z^{\prime}}^{2}\end{array}\right)\left(\begin{array}[]{c}\tilde{Z}_{\mu}\\\
\tilde{B}^{\prime}_{\mu}\\\ \end{array}\right),$ (67)

where $\Delta^{2}=\frac{1}{4}g_{2}v^{2}\rho\sqrt{g_{1}^{2}+g_{2}^{2}}$ and
$M_{Z,SM}^{2}=\frac{1}{4}v^{2}(g_{1}^{2}+g_{2}^{2})$. The physical masses are

$\displaystyle m_{Z}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}M_{Z,SM}^{2}+M_{Z}^{\prime
2}+\sqrt{(M_{Z,SM}^{2}-M_{Z^{\prime}}^{2})^{2}+4\Delta^{4}}\Big{]},$
$\displaystyle m_{Z^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}M_{Z,SM}^{2}+M_{Z}^{\prime
2}-\sqrt{(M_{Z,SM}^{2}-M_{Z^{\prime}}^{2})^{2}+4\Delta^{4}}\Big{]},$ (68)

where we expect $m_{Z}\gg m_{Z^{\prime}}$ to address the Hubble tension well.
The mass matrix in Eq. (67) can be diagonalized by rotation matrix

$\displaystyle\left(\begin{array}[]{c}\tilde{Z}_{\mu}\\\
\tilde{B}^{\prime}_{\mu}\\\ \end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}\cos\epsilon&\sin\epsilon\\\
-\sin\epsilon&\cos\epsilon\\\
\end{array}\right)\left(\begin{array}[]{c}Z_{\mu}\\\ Z^{\prime}_{\mu}\\\
\end{array}\right),$ (75) $\displaystyle\tan 2\epsilon$
$\displaystyle=\frac{2\Delta^{2}}{M_{Z,SM}^{2}-M_{Z^{\prime}}^{2}}\approx
2\frac{g_{2}\rho}{\sqrt{g_{1}^{2}+g_{2}^{2}}}=2\rho\sin\theta_{W}\approx\rho,$
(76)

where $\sin\theta_{W}^{2}\approx 0.2229$, $Z_{\mu}$ and $Z^{\prime}_{\mu}$ are
the two physical gauge bosons which respectively correspond to the SM $Z$
boson and extra gauge boson. The above relation suggests
$\epsilon\approx\rho/2$ if $\epsilon$ is small enough. The quantity $\rho$ is
restricted by the absolute discrepancy between $m_{Z}$ and $M_{Z,SM}$. It is
given by $|\Delta m_{Z}|\approx M_{Z,SM}(\sqrt{1+g_{2}^{4}\rho^{2}}-1)$ and
$|\Delta m_{Z}|$ has to be within the experimental uncertainty as follows:

$\displaystyle|\Delta m_{Z}|\lesssim 0.0021{\rm GeV}.$ (77)

Finally, we obtain

$\displaystyle\epsilon\lesssim 0.0312,$ (78)

where we have used the following input values; $M_{Z,SM}=91.1876$ GeV,
$g_{2}=0.33$. However, the kinetic mixing receives several stringent
constraints from astrophysical experiments in light mass scale of hidden gauge
boson; 1 eV - 100 eV. These experiments suggest $\epsilon\ll{\cal O}(10^{-9})$
Jaeckel and Ringwald (2010) that is far from the constraint from electroweak
precision test of $Z$. Thus, we need to rely on mixing among neutral fermions
via Yukawa terms. Therefore, $\eta$ has to have tiny VEV. In this case, the
active neutrino mass is induced at tree level via inverse seesaw in general.
But, we assume that this tree level neutrino mass is negligibly small compared
to the one-loop contribution. Theoretically, it is easy to realize by
controlling the parameter of VEV of $\eta$. Then, the corresponding mixing
paramter is found to be $\tan\tilde{\epsilon}\sim fv_{\eta}/(\sqrt{2}m_{D})$
Berbig et al. (2020).

###  II.4 Dark matter

Here, we suppose our DM to be the lightest Majorana fermion;
$X_{R}\equiv\psi_{R_{1}}$. Here, we define the mass to be $M_{X}$. Then, the
relevant interaction Lagrangian is given by,

$-\mathcal{L}_{DM}=\frac{1}{\sqrt{2}}\bar{\nu}_{L_{i}}G_{i1}X_{R}(s_{R}H_{1}+c_{R}H_{2})-\frac{i}{\sqrt{2}}\bar{\nu}_{L_{i}}G_{i1}X_{R}(s_{I}A_{1}+c_{I}A_{2})+\bar{\ell}_{L_{i}}G_{i1}X_{R}\eta^{-}+{\rm
h.c.},$ (79)

where $G\equiv U_{MNS}^{*}\sqrt{D_{\nu}}O_{\rm mix}R_{N}^{-1}V_{N}$. Then, the
thermally averaged annihilation cross section to explain the DM is $s$-wave
dominant and given by

$\displaystyle\langle\sigma
v_{rel}\rangle\simeq\frac{M_{X}^{2}|G_{i1}G_{1j}^{\dagger}|^{2}}{64\pi}$
$\displaystyle\left[\frac{4}{(M_{X}^{2}+m_{\eta^{-}}^{2})^{2}}\right.$
$\displaystyle+\left.\left(\frac{s_{R}^{2}}{M_{X}^{2}+m_{H_{1}}^{2}}+\frac{c_{R}^{2}}{M_{X}^{2}+m_{H_{2}}^{2}}+\frac{s_{I}^{2}}{M_{X}^{2}+m_{A_{1}}^{2}}+\frac{c_{I}^{2}}{M_{X}^{2}+m_{A_{2}}^{2}}\right)^{2}\right],$
(80)

where we assume $m_{\eta^{-}}\approx m_{A_{2}}$ simply in order to evade the
constraint from oblique parameters under the small mixing of $\theta_{I}$.

The resulting relic density is given by Baek et al. (2016),

$\displaystyle\Omega h^{2}\approx\frac{1.07\times
10^{9}x_{f}}{\sqrt{g_{*}(x_{f})}M_{\rm Pl}\langle\sigma
v_{rel}\rangle}\approx\frac{2.2\times 10^{-10}}{\langle\sigma
v_{rel}\rangle}~{},$ (81)

where the present relic density is $0.1199\pm 0.0054$ at 2$\sigma$ Ade et al.
(2014), $g_{*}(x_{f}\approx 25)\approx 100$ counts the degrees of freedom for
relativistic particles, and $M_{\rm Pl}\approx 1.22\times 10^{19}$ GeV is the
Planck mass. In our numerical analysis below, however, we will use relaxed
observable $0.11\leq\Omega h^{2}\leq 0.13$ which is at around 3$\sigma$.

###  II.5 Lepton flavor violations and muon anomalous magnetic dipole moment
$(g-2)_{\mu}$

Lepton flavor violations (LFV) as well as $(g-2)_{\mu}$ are arisen at one-loop
level, and LFVs form is given by Baek et al. (2016)

$\displaystyle{\rm BR}(\ell_{i}\to\ell_{j}\gamma)=\frac{48\pi^{3}\alpha_{\rm
em}C_{ij}}{{\rm
G_{F}^{2}}m_{\ell_{i}}^{2}}\left(|a_{R_{ij}}|^{2}+|a_{L_{ij}}|^{2}\right),$
(82) $\displaystyle a_{R_{ij}}$
$\displaystyle\approx-\sum_{a=1,2,3}\frac{g_{ja}M_{E_{a}}f_{ai}}{(4\pi)^{2}}$
$\displaystyle\times\left[s_{R}c_{R}(F[M_{E_{a}},m_{H_{1}}]-F[M_{E_{a}},m_{H_{2}}])+s_{I}c_{I}(F[M_{E_{a}},m_{A_{1}}]-F[M_{E_{a}},m_{A_{2}}])\right],$
(83) $\displaystyle\quad a_{L_{ij}}$
$\displaystyle\approx-\sum_{a=1,2,3}\frac{f^{\dagger}_{ja}M_{E_{a}}g^{\dagger}_{ai}}{(4\pi)^{2}}$
$\displaystyle\times\left[s_{R}c_{R}(F[M_{E_{a}},m_{H_{1}}]-F[M_{E_{a}},m_{H_{2}}])+s_{I}c_{I}(F[M_{E_{a}},m_{A_{1}}]-F[M_{E_{a}},m_{A_{2}}])\right],$
(84) $\displaystyle F[m_{1},m_{2}]$
$\displaystyle\approx\frac{m_{1}^{2}-m_{2}^{2}+m_{2}^{2}\ln\left[\frac{m_{2}^{2}}{m_{1}^{2}}\right]}{(m_{1}^{2}-m_{2}^{2})^{2}},$
(85)

where $M_{E}\equiv y_{E}v_{\varphi}/\sqrt{2}$ ${\rm G_{F}}\approx 1.17\times
10^{-5}$[GeV]-2 is the Fermi constant, $\alpha_{\rm em}\approx 1/137$ is the
fine structure constant, $C_{21}\approx 1$, $C_{31}\approx 0.1784$, and
$C_{32}\approx 0.1736$. Experimental upper bounds are respectively given by
${\rm BR}(\mu\to e\gamma)\lesssim 4.2\times 10^{-13}$, ${\rm BR}(\tau\to
e\gamma)\lesssim 3.3\times 10^{-8}$, and ${\rm BR}(\tau\to\mu\gamma)\lesssim
4.4\times 10^{-8}$ Adam et al. (2013); Baldini et al. (2016).
New contribution to $(g-2)_{\mu}$ also arises from the same term as in LFVs,
and it is given by333For a comprehensive review on new physics models for the
$(g-2)_{\mu}$ anomaly as well as lepton flavour violation, please see Ref.
Lindner et al. (2018).

$\displaystyle\Delta a_{\mu}=-m_{\mu}[a_{R}+a_{L}]_{22}.$ (86)

Combined results with the previous BNL, suggests that $(g-2)_{\mu}$ deviates
from the SM prediction by 4.2$\sigma$ level Aoyama et al. (2012, 2019);
Czarnecki et al. (2003); Gnendiger et al. (2013); Davier et al. (2017);
Keshavarzi et al. (2018); Colangelo et al. (2019); Hoferichter et al. (2019);
Davier et al. (2020); Keshavarzi et al. (2020); Kurz et al. (2014); Melnikov
and Vainshtein (2004); Masjuan and Sanchez-Puertas (2017); Colangelo et al.
(2017); Hoferichter et al. (2018); Gérardin et al. (2019); Bijnens et al.
(2019); Colangelo et al. (2020); Blum et al. (2020); Colangelo et al. (2014);
Hagiwara et al. (2011),

$\displaystyle\Delta a_{\mu}=(25.1\pm 5.9)\times 10^{-10}~{}.$ (87)

##  III Numerical analysis

In this numerical analysis, we randomly select absolute values of input
parameters within the following ranges,

$\displaystyle\\{|\alpha|,\ |\beta|,\ |\gamma|\\}\ {\text{in}}\ O_{\rm mix}$
$\displaystyle\in[0,\pi],$ (88a) $\displaystyle\\{s_{R},\ s_{I},\ f,\ g\\}$
$\displaystyle\in[0,0.3],$ (88b) $\displaystyle\\{m_{H_{1}},\ m_{H_{2}},\
m_{A_{1}},\ m_{A_{2}}\\}$ $\displaystyle\in[10^{2},10^{5}]~{}\text{GeV},$
(88c) $\displaystyle\\{m_{D},\ M_{N_{L}}\\}$
$\displaystyle\in[0.1,10^{5}]~{}\text{GeV},$ (88d)

where $f$ and $g$ are relevant Yukawa couplings as described in Eqs. (83) and
(84). Then, we also impose the perturbative limit for
$y_{\eta}\lesssim\sqrt{4\pi}$ as discussed in Eq. (30) and the neutrino
observables are accommodated by Nufit 5.1 as the best fit values in case of
“without SK atmospheric data” nuf  vanishing Majorana phases, observed relic
density, $0.11\lesssim\Omega h^{2}\lesssim 0.13$; and $(g-2)_{\mu}$ within
1$\sigma$, $19.2\times 10^{-10}\lesssim\Delta a_{\mu}\lesssim 31.0\times
10^{-10}$ in Eq. (87).

###  III.1 Normal Hierarchy

Figure 1: Allowed regions of $m_{H_{1}}$(blue) and $m_{A_{1}}$(red) in the
left panel and the one of $m_{H_{2}}$(black) and $m_{A_{2}}$(green) in the
right panel in terms of $M_{X}$ in GeV unit. Here, these points satisfy
$0.11\leq\Omega h^{2}\leq 0.13$.

In Fig. 1, we show the allowed regions of $m_{H_{1}}$(blue) and
$m_{A_{1}}$(red) in the left panel and the one of $m_{H_{2}}$(black) and
$m_{A_{2}}$(green) in the right panel of Fig. 1 in terms of $M_{X}$. Here,
these points satisfy $0.11\leq\Omega h^{2}\leq 0.13$. These figures suggest 50
GeV$\lesssim M_{X}\lesssim$140 GeV, 1000 GeV$\lesssim m_{H_{1}}\lesssim$1400
GeV, 3800 GeV$\lesssim m_{A_{1}}\lesssim$5600 GeV, 117 GeV$\lesssim
m_{H_{2}}\lesssim$175 GeV, 12.8 TeV$\lesssim m_{A_{2}}\lesssim$19.23 TeV.

Figure 2: $(g-2)_{\mu}$ allowed region in $M_{E_{1}}$-$M_{E_{2}}$ plane (left
panel); and $f_{22}$-$g_{22}$ plane (right panel). The blue color represents
$(g-2)_{\mu}$ within 1$\sigma$ level, yellow one 2$\sigma$ level, and red one
3$\sigma$ level. Here, we have fixed $\Omega h^{2}=0.12299$.

In Fig. 2, we show the allowed regions to satisfy $(g-2)_{\mu}$ in terms of
$M_{E_{1}}$ and $M_{E_{2}}$ in the left figure, and $f_{22}$ and $g_{22}$ in
the right figure. The blue color represents $(g-2)_{\mu}$ within 1$\sigma$
level, yellow one 2$\sigma$ level, and red one 3$\sigma$ level. Here, we have
subtracted a benchmark point from Fig. 1 so that we have fixed $\Omega
h^{2}=0.12299$. These figures suggest that 100 GeV$\lesssim M_{E_{1}}\lesssim
10^{5}$ GeV, and 100 GeV$\lesssim M_{E_{2}}\lesssim$800 GeV and
0.5$\lesssim(|f_{22}|,\ |g_{22}|)\lesssim$1.5 within 3$\sigma$ level.

###  III.2 Inverted Hierarchy

Figure 3: Allowed regions of $m_{H_{1}}$(blue) and $m_{A_{1}}$(red) [left
panel] and the one of $m_{H_{2}}$(black) and $m_{A_{2}}$(green) [right panel]
in terms of $M_{X}$ in GeV unit. Here, these points satisfy $0.11\leq\Omega
h^{2}\leq 0.13$.

In Fig. 3, we show the allowed regions of $m_{H_{1}}$(blue) and
$m_{A_{1}}$(red) in the left panel and the one of $m_{H_{2}}$(black) and
$m_{A_{2}}$(green) in the right panel in terms of $M_{X}$ in GeV unit. Here,
these points satisfy $0.11\leq\Omega h^{2}\leq 0.13$. These figures suggest
0.2 GeV$\lesssim M_{X}\lesssim$0.5 GeV, 42 TeV$\lesssim m_{H_{1}}\lesssim$62
TeV, 48 TeV$\lesssim m_{A_{1}}\lesssim$72 TeV, 300 GeV$\lesssim
m_{H_{2}}\lesssim$480 GeV, 120 GeV$\lesssim m_{A_{2}}\lesssim$180 GeV.

Figure 4: Muon $(g-2)$ allowed region in the $M_{E_{1}}$-$M_{E_{2}}$ plane
(left panel), and in the $f_{22}$-$g_{22}$ plane (right panel). The blue color
represents the muon $g-2$ within 1$\sigma$ level, yellow one 2$\sigma$ level,
and red one 3$\sigma$ level. Here, we have fixed $\Omega h^{2}=0.118805$.

In Fig. 4, we show the allowed region that satisfies $(g-2)_{\mu}$ constraint
in terms of $M_{E_{1}}$-$M_{E_{2}}$ plane (left) and $f_{22}$-$g_{22}$ plane
(right). The blue color represents $(g-2)_{\mu}$ within 1$\sigma$ level,
yellow one 2$\sigma$ level, and red one 3$\sigma$ level. Here, we have
subtracted a benchmark point from Fig. 3 and fixed $\Omega h^{2}=0.118805$.
These figures suggest that 100 GeV$\lesssim M_{E_{1}}\lesssim 10^{5}$ GeV, and
100 GeV$\lesssim M_{E_{2}}\lesssim$4200 GeV and 0.2$\lesssim(|f_{22}|,\
|g_{22}|)\lesssim$1.5 within 3$\sigma$ level.

##  IV Hubble tension

There has been a widely persisting and statistically significant
($4\sigma$-$6\sigma$) discrepancy in the measurements of the Hubble constant
$H_{0}$ from late time and early time observations. There are attempts to
tweak the standard $\Lambda$CDM to address this Schöneberg et al. (2019); Knox
and Millea (2020). Another attractive possible resolution comes from
considering the scenario of self-interacting neutrinos Blinov et al. (2019);
He et al. (2020); Berbig et al. (2020); Lyu et al. (2021); Das and Ghosh
(2021); Brinckmann et al. (2021). Such an effective interaction is given by,

$\displaystyle\mathcal{L}_{\rm eff}\supset G_{\rm
eff}(\bar{\nu}\nu)(\bar{\nu}\nu),$ (89)

where $G_{\rm eff}$ is the dimensionful effective coupling. Using the Planck
observation, a fit to the CMB data identifies two regimes, namely “strongly
interacting” (SI) and “moderately interacting” (MI), for the effective
coupling $G_{\rm eff}$ Kreisch et al. (2020); Park et al. (2019),

$\displaystyle G_{\rm
eff}=\begin{cases}\left(4.73^{+0.37}_{-0.61}~{}\text{MeV}\right)^{-2}~{}~{}\text{(SI)}\\\
\left(89.12^{+170.89}_{-60.94}~{}\text{MeV}\right)^{-2}~{}~{}\text{(MI)}\end{cases}\approx\begin{cases}\left(5~{}\text{MeV}\right)^{-2}~{}~{}\text{(SI)}\\\
\left(100~{}\text{MeV}\right)^{-2}~{}~{}\text{(MI)}\end{cases}.$ (90)

Note that this $G_{\rm eff}$ is much larger than the Fermi constant $G_{\rm
F}$. It is also shown in Kreisch et al. (2020); Park et al. (2019) that the SI
case is preferable in ameliorating the $H_{0}$ tension as well as remaining
consistent with the local astronomical observations.

In our case, the self-interaction of neutrinos are mediated by the additional
$Z^{\prime}$ arising due to the hidden $U(1)_{H}$ gauge symmetry. It is to be
noted that there are other scalars in the model but since in our model
neutrinos do not self-interact via scalar fields, they will have no effect in
the explanation of the Hubble tension. Clearly, the effective coupling $G_{\rm
eff}$ can be represented in terms of the present model parameters as,

$\displaystyle G_{\rm
eff}=\frac{g_{H}^{2}\tilde{\epsilon}^{4}}{m_{Z^{\prime}}^{2}},$ (91)

where $\tilde{\epsilon}$ represents the mixing between the neutral fermions as
defined in the last part of Sec. II.3. It has been shown in Berbig et al.
(2020) that to maintain BBN and other constraints,
$g_{H}\tilde{\epsilon}^{2}\in[2\times 10^{-7},5\times 10^{-6}]$ and this,
using Eqs. (90) and (91), translates to the constraint on the $Z^{\prime}$
mass as,

$\displaystyle m_{Z^{\prime}}\in\begin{cases}[1,25]~{}\text{eV~{}~{}(SI)}\\\
[20,500]~{}\text{eV~{}~{}(MI)}\end{cases}.$ (92)

Using Eq. (68) one can set VEVs $v_{\varphi}$, $v_{\varphi^{\prime}}$, and the
dark sector gauge coupling $g_{H}$ to obtain $m_{Z^{\prime}}$ in the above-
mentioned ranges. Below, we show two benchmark points of (i)
$(g_{H}\tilde{\epsilon}^{2},m_{Z^{\prime}})=(2\times 10^{-7},1~{}{\rm eV})$,
and (ii) $(g_{H}\tilde{\epsilon}^{2},m_{Z^{\prime}})=(5\times
10^{-6},25~{}{\rm eV})$ satisfying all the constraints that we have discussed
before,

$\displaystyle(i)\quad$ $\displaystyle\tilde{\epsilon}=0.0144,$ (93)
$\displaystyle(ii)\quad$ $\displaystyle\tilde{\epsilon}=0.00288,$ (94)

where we fixed $v_{\varphi}=v_{\varphi^{\prime}}=10^{-2}{\rm GeV},\
\rho=10^{-9}$ and then the following values are commonly obtained
$\Delta=3.70\times 10^{-6}\ {\rm GeV},\ M_{Z^{\prime}}=7.59\times 10^{-7}\
{\rm GeV},\ g_{H}=0.0242$.

##  V Summary and Conclusions

We have proposed a radiative seesaw model in a hidden gauge $U(1)$ symmetry.
In order to have anomaly cancellations, we need to introduce several new
fermions that contribute to $(g-2)_{\mu}$ as well as the neutrino oscillation
data. We have also considered a fermionic DM candidate that correlates with
$(g-2)_{\mu}$ and neutrino mass matrix at the same time. We have demonstrated
allowed regions in our input parameters satisfying several constraints.
Finally, we have briefly discussed the Hubble tension via a lighter hidden
gauge boson. In order to achieve it, we have found that we need to have
nonzero VEV of $\eta$ to mix with neutral fermions that leads to tree-level
neutrino mass matrix. But, we have confirmed that this mass matrix is
negligible compared to the radiative seesaw model by choosing small VEV of
$\eta$. In the process we can generate requisite neutrino self-interaction
mediated by the additional gauge boson to address the Hubble tension.

Acknowledgments

UKD acknowledges the hospitality of APCTP Pohang where this work was
initiated. This research of HO was supported by an appointment to the JRG
Program at the APCTP through the Science and Technology Promotion Fund and
Lottery Fund of the Korean Government. This was also supported by the Korean
Local Governments - Gyeongsangbuk-do Province and Pohang City. HO is sincerely
grateful for the KIAS membership.

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