# SINGLE IMAGE CALIBRATION USING KNOWLEDGE DISTILLATION APPROACHES

###### Abstract

Although recent deep learning-based calibration methods can predict extrinsic
and intrinsic camera parameters from a single image, their generalization
remains limited by the number and distribution of training data samples. The
huge computational and space requirement prevents convolutional neural
networks (CNNs) from being implemented in resource-constrained environments.
This challenge motivated us to learn a CNN gradually, by training new data
while maintaining performance on previously learned data. Our approach builds
upon a CNN architecture to automatically estimate camera parameters ($focal$
$length$, $pitch$, and $roll$) using different incremental learning strategies
to preserve knowledge when updating the network for new data distributions.
Precisely, we adapt four common incremental learning, namely: $LwF$, $iCaRL$,
$LUCIR$, and $BiC$ by modifying their loss functions to our regression
problem. We evaluate on two datasets containing $299008$ indoor and outdoor
images. Experiment results were significant and indicated which method was
better for the camera calibration estimation.

Index Terms—  Image calibration, Machine learning, Knowledge distillation.

##  1 Introduction

Single image calibration estimates camera parameters from a monocular RGB
image. This problem is of significant importance in many computer vision
tasks, especially in applications where capturing conditions are not
controlled [1, 2]. Conventional approaches of single image camera calibration
rely on detecting reference objects in the scene, such as a calibration grid
[3] or co-planar circles [4]. Other methods take advantage of vanishing point
properties, by carefully selecting parallel or orthogonal segments in the 3D
scene [5]. However, most of these methods use classic image processing
techniques to detect geometric cues, which makes them inapplicable in
unstructured environments. To overcome such limitations, recent studies
exploited semantic cues learned by deep neural networks to estimate camera
parameters. Existing approaches focus on estimating intrinsic [6, 7] and/or
extrinsic parameters [8]. Yannick et al. [9] estimate the horizon line as a
proxy of camera parameters. This joint estimation results significantly
improve the performance of the model. The latter is partially due to the
generated large-scale synthetic dataset from panoramic images. Despite the
success of these methods, their generalization requires a very large-scale
dataset covering a large range of image distributions, such as scene types,
visible objects, camera types, etc. In practice, the calibration network must
be updated to the application-specific data stream to provide more accurate
estimations of camera parameters. However, legacy data may be unrecorded,
proprietary, or simply too cumbersome to use in training a new task. As a
consequence, it becomes apparent that more flexible strategies are required to
handle the large-scale and dynamic properties of real-world situations.

In this work, we adopt a CNN architecture to recover extrinsic ($pitch$,
$roll$) and intrinsic ($focal$ $length$) camera parameters from a single input
image. Our main contribution is to explore methods to improve the
generalization capability of our network. Specifically, we consider two data
distribution sets (indoor and outdoor scene). We train our network exclusively
on a single dataset and use different incremental learning strategies to
preserve acquired knowledge when training on the second dataset. To our best
knowledge, this is the first study that applies incremental learning
techniques for the camera parameters estimation.

##  2 Camera model

The process of image formation has been studied extensively in computer vision
[10], allowing for very precise calibration of the camera when there are
enough constraints to fit the geometric camera model. Under the pinhole camera
model, homogeneous pixel coordinates $p_{im}$ of a $3D$ point $p_{w}$ are
given by:

$p_{im}=[\lambda_{u},\lambda_{v},\lambda]=[R\mid t][p_{w}\mid 1]^{T}$ (1)

where $K$ is the camera projection matrix (camera intrinsics), $R$ and $t$ are
the camera rotation and translation in the world reference system (camera
extrinsics). By considering square pixels, no skew, and image center at the
principal point, the projection matrix $K$ can be symplified to
$K=diag([f_{px},f_{px},1])$, where $f_{px}$ is the focal length in pixels. The
rotation matrix $R$, can be parameterized by roll $\psi$, pitch $\theta$, and
yaw $\varphi$ angles. There exists no natural reference frame to estimate
$\varphi$ (left vs right) from an arbitrary image. Therefore, the extrinsic
rotation matrix is constrained to only pitch and roll components
$(R=R_{z}(\psi)R_{x}(\theta))$.

For the case of camera calibration, parameters such as focal length, pitch and
roll are difficult to interpret from image content. As revealed by previous
works [8, 9], we can use horizon line as an intuitive representation for these
parameters. We define the horizon line midpoint $b_{p}$ as the y-coordinate of
its intersection with the central vertical axis in the image. It can be
derived from $\theta$ and $f_{px}$ as :

$b_{p}=2f_{p_{x}}tan(\theta)$ (2)

The roll angle $\psi$ represents the angle between the horizon line and the
horizontal axis of the image. In this image unit representation, the top and
bottom of the image have coordinates $1$ and $-1$ respectively.

##  3 Image calibration network

Our goal is to train a CNN to estimate camera parameters from a single image.
To achieve this, we use independent regressors that share a common pretrained
network architecture, which we update using our data. Precisely, we adopt
Residual Neural Networks with $50$ layers (ResNet50) [11], on which the last
layer is replaced with three outputs estimating $focal$, $pitch$ and $roll$
values. In this work, we specifically aim to learn a generalized model, which
can efficiently estimate camera parameters whether for indoor or outdoor
images. To conduct this evaluation, we train our network using exclusively
indoor or outdoor data and exploit a previously trained network by
transferring the learned features to the target network. However, fine-tuning
the weights of a pretrained network usually degrades the performance on the
previously learned task because the shared parameters change without any
guidance for the original task. To avoid this problem, we use incremental
learning techniques to adapt a CNN to different distributions.

###  3.1 Incremental learning

In the real world, many computer vision applications require learning new
visual capabilities while maintaining performance on existing ones. However,
CNN-based systems often suffer from the “catastrophic forgetting” of the
previous knowledge. This problem is mainly due to two facts: (1) the updates
can override the knowledge acquired from the previous data, and (2) the model
can not replay the entire previous data to regain the old knowledge. To
overcome these limits, several methods of incremental learning have been
proposed, which are generally divided into three categories: (1)
regularization-based methods: use regularization terms in the loss function to
alleviate forgetting [12, 13, 14, 15, 16]; (2) rehearsal-based methods:
construct a small exemplar set from old data [17, 13] or synthesize samples to
keep the performance for old classes [18, 19]; (3) dynamic architecture
methods: utilize different network parameters for different tasks [20, 21].

In this work, we consider a single incremental learning task: we have a
regression model already trained on old data, and we need to update it to new
data distribution. To solve this problem, we use methods that introduce
regularization terms in their loss functions to consolidate previous knowledge
when learning from new data. Following, we introduce the common loss
distillation-based approaches and their adaptation to our regression problem.

####  3.1.1 Knowledge Distillation

Knowledge distillation is first proposed in [22] to transfer knowledge from a
large pre-trained teacher network (or network assembly) to a smaller student
network for more efficient deployment. Thereafter, knowledge distillation was
introduced in $LwF$ (learning without forgetting) to avoid catastrophic
forgetting by appointing a previous snapshot of the model as a teacher while
new tasks are learned [12]. More precisely, $LwF$ preserves the outputs of the
old model by optimizing a loss function defined as:

$\mathcal{L}_{KD}=\mathcal{L}_{new}(y_{n},\hat{y}_{n})+\lambda_{0}\mathcal{L}_{old}(y_{o},\hat{y}_{o})$
(3)

where: $\mathcal{L}_{new}$ is the common cross-entropy, which encourages new
task predictions $\hat{y}_{n}$ to be consistent with ground truths $y_{n}$;
$\mathcal{L}_{old}$ is a distillation loss used to prevent the actual outputs
of the updated network $\hat{y}_{o}$ from deviating too much from stored
outputs of its older version; ${y}_{o}$ and $\lambda_{0}$ is a loss balance
weight. The original distillation loss function is based on a modified cross-
entropy loss that produces a softer probability distribution over classes
[22]. In our case, we solve our regression problem using a smooth-L1 loss (Eq.
4), which is less sensitive to outliers than the mean square error loss.

$\mathcal{L}(y_{,}\hat{y})=\frac{1}{n}\sum_{i}^{L}\left\\{\begin{array}[]{l
l}0.5(y-\hat{y})^{2}\quad\text{if}&\mid y-\hat{y}\mid<1\\\ \mid
y-\hat{y}\mid-0.5&\text{otherwise,}\end{array}\right.$ (4)

Where $n$ is the number of regression outputs, and $L$ is the batch size.

####  3.1.2 Replay-based methods

$LwF$ is the first work addressing catastrophic forgetting in class
incremental learning. Recent studies show that selecting a few exemplars from
the old classes can alleviate the imbalance problem. Incremental classifier
and representation learning $iCaRl$ [17] employs the nearest mean of exemplars
classification strategy to select the most representative samples of each
previous class and replay them together with the learning data of new tasks.
Learning a unified classifier incrementally via rebalancing $Lucir$ [13] uses
the $iCaRL$ baseline to select exemplars and incorporates a series of
regularization terms to mitigate negative effects caused by data imbalance.
Precisely, they apply a cosine normalization in the last layer of the network
to make the magnitudes of old and new data predictions more comparable and
introduce a less forget constraint to enforce the previous knowledge. They
also incorporate a margin ranking loss to avoid ambiguities between old and
new data.

In this work, we adapt $Lucir$ to resolve our regression problem by (1)
applying cosine normalization in the last layer of ResNet50; (2) incorporating
the less forget constraint into our loss function. Formally, we calculate
$\mathcal{L}_{lucir}$ by summing $\mathcal{L}(y,\hat{y})$ and
$\mathcal{L}_{dist}$, where
$\mathcal{L}_{dist}=1-<(\bar{f^{*}}(x),\bar{f(x)})>$. $\bar{f^{*}}(x)$ and
$\bar{f(x)}$ are respectively, the normalized features extracted by the
original model and those by the current one.

Table 1: $MSE$ errors of calibration parameters obtained using indoor and outdoor model. Evaluation | Indoor model | Outdoor model
---|---|---
| Focal | Roll | Pitch | $\mu MSE$ | Focal | Roll | Pitch | $\mu MSE$
Validation indoor | 0.1 | 1.33 | 0.07 | 0.50 | 1.59 | 2.43 | 0.75 | 1.59
Validation outdoor | 2.17 | 2.84 | 0.58 | 1.86 | 0.08 | 1.40 | 0.08 | 0.52
Test indoor | 0.09 | 0.39 | 0.06 | 0.18 | 1.1 | 2.88 | 0.91 | 1.63
Test outdoor | 2.17 | 2.84 | 0.58 | 1.86 | 0.09 | 1.46 | 0.06 | 0.54

####  3.1.3 Bias Correction (BiC)

Most data replay-based incremental learning methods follow the $iCaRL$
benchmark protocol to select exemplars. Recent studies show that approaches
selecting few exemplars from old classes perform well on small datasets.
However, they suffer from significant performance degradation when the number
of classes becomes large. To solve this issue, a bias correction model called
BiC was introduced in [23]. The training process consists of two stages: (1)
learning the convolution layers of the network, and (2) applying a linear
model to the last FC layer to correct the bias in the validation data, which
approximates the real distribution of both old and new classes.

In this work, we adapt our network to perform bias correction by applying a
BiC linear model ($q_{k}=\alpha o_{k}+\beta$) to the last FC layer of our
network, where $\alpha$ and $\beta$ are the BiC model parameters and $o_{k}$
is the k-th regression output, where $k=3$ for focal, pitch and roll. In the
training phase, we use the distillation function (equation. (3)) for the first
stage. For the second stage, we freeze convolution and FC layers and optimize
$\mathcal{L}(q,y)$ loss function (4)).

##  4 Experimentation

In this section, we begin by explaining the generation of the datasets, then
we evaluate and compare the methods introduced in section 3. Our experiments
are divided into two main parts: (1) cross-evaluation of indoor and outdoor
models; (2) network generalization.

###  4.1 Dataset generation

We took inspiration from [9] to synthetically generate images and their ground
truth camera parameters. In this work, we divided the data into two
categories: indoor and outdoor datasets (Fig.1). To construct the indoor
dataset, we choose $292$ panoramic images from the publicly available SUN360
database [24]. Furthermore, we collect $292$ public panoramic images from the
internet to construct the outdoor dataset (we particularly chose images of the
same resolution as the indoor SUN360 images $[1024\times 512px]$). On one
hand, panoramic images are used to emulate any amount of $360^{\circ}$ field
of view. On the other hand, we can point the virtual camera to different
orientations to observe different parts of the scene and mimic tilted cameras.

Fig. 1: Examples of panoramas used to generate synthetic training datasets :
(left) indoor image, (right) outdoor image.

In this work, we randomly generate focal lengths in a range ($50$,$500$
pixel). For rotation angles, we randomly generate pitch $\theta$ in range
$(-90^{\circ},0^{\circ})$ with a roll angle $\psi$ comprised between
$(-45^{\circ},45^{\circ})$. We divided ground truths by the max absolute value
of the corresponding parameter to obtain a comparable scales:
$focal\in[0,1],pitch\in[-1,0],roll\in[-1,1]$. By following this approach, we
automatically generated two databases (indoor and outdoor) each containing
$149504$ images, 80% were used for training and 20% for validation. Similarly,
we generate two test sets with $3584$ images each, where the size of images is
$299$ $\times$ $299$ pixels.

###  4.2 Cross-evaluation

We conduct our experiments by a cross-evaluation of indoor and outdoor models
(as stated in section 3) using exclusively indoor images and we test the
resulting network on indoor and outdoor datasets. Conversely, we train our
network on outdoor images and test the resulting model on the two datasets.
ResNet50 is trained to minimize the L1-smooth loss between ground truths and
predictions. The model is trained on a single GPU with a batch size of $16$.
We use the SGD optimizer with an initial learning rate of $0.003$, which is
reduced by a factor of $0.1$ if no improvement is seen over $2$ epochs.

The results of cross-testing are reported in Table.1. We specifically show the
mean squared error ($MSE$) of the camera parameters ($focal$, $pitch$, $roll$)
and compare results according to the mean $MSE$ of the three parameters that
we call $\mu MSE$.

Fig. 2: Example of image calibration predictions on indoor (Top row) and
outdoor (Bottom row) scene.

We note that the two models perform well when tested on the same training data
distribution. This result can be observed in Fig.2 where we draw the horizon
lines according to the predicted parameters [9]. We can see that, for indoor
images, horizon lines are closer to ground truths than outdoor model lines.
Conversely, for outdoor images, we obtain more precise horizon lines with the
outdoor model.

###  4.3 Network generalization

The purpose of this work is to estimate the camera parameters from a single
image whether for indoor or outdoor scenes. In this section, we compare the
performance of several incremental learning techniques (see section.3.1). The
aim is to determine which learning model is more suitable to preserve
knowledge from different data distributions. Fig. 3, compares $\mu MSE$ errors
over indoor and outdoor validation sets.

Fig. 3:  $\mu MSE$ errors over balanced indoor and outdoor validation sets.

As shown in the Fig. 3, fine-tuning shows a much larger error on indoor data
than outdoor data since it learns new data parameters without taking any
measures to prevent catastrophic forgetting of indoor data (equivalent to
$\lambda=0$ in Eq. (3)). In contrast, the indoor error is reduced by $LwF$,
which can be seen as a hybrid of fine-tuning and knowledge distillation.
Indeed, this technique prevents the deviation of the prediction of the new
updated model from the previous one. Further experiments using replay-based
methods were performed. These methods store exemplars from previously learned
distribution and interleave them with the current learning data. To estimate
the number of exemplars, we conducted extensive tests of the $iCaRl$ baseline,
by combining the outdoor training set with different percentages of indoor
exemplars. Note that, $iCaRL$ with $0\%$ of exemplars is equivalent to $LwF$.

Fig. 4: $\mu MSE$ errors over indoor and outdoor using $(\%)$ of exemplars.

The obtained result from Fig. 4 show that $iCaRL$ reduces $LwF$ indoor error
from $0.82$ to $0.57$ using $20\%$ of exemplars. Thereafter, the indoor error
decreases slightly as the number of exemplars increases. When we kept $60\%$
of exemplars, iCaRL shows a good performance for both indoor and outdoor
images. We can explain this by that the indoor images, added during training,
contain features shared between indoor and outdoor (edges, contours, etc.).
However, we show that uMsE increases when we kept $80\%$ of exemplars, whereas
the performance for indoor images keeps increasing as more indoor data are
involved during training ($uMsE=0.15$ at $100\%$). The issue can be related to
the selected portion of images, which may be largely biased towards indoor
images and have no features related to outdoor images. Therefore, we kept
$20\%$ the best trade-off between indoor and outdoor error.

Fig.3 compares $iCaR$L with $Lucir$ and $BiC$ methods. All of them utilize
knowledge distillation to prevent catastrophic forgetting. As we can see in
this figure, $BiC$ performs better than $Lucir$ and $iCaRL$ both on indoor and
outdoor scenes. This improvement is achieved using the bias correction layer
in the validation stage. In contrast, $Lucir$ degrades the performances of
outdoor data ($\mu MSE=0.95$) because it uses a distillation loss on the
feature space, which tends to focus on preserving previous knowledge from old
data at the expense of learning new inputs. We also note that replay methods
perform slightly better on indoor data than outdoor data. This can be
justified by the complexity and variety of outdoor images compared to the
indoor scenes, which have closer distribution. To confirm our preliminary
results, we evaluated the three methods on the new test data (Fig. 5).

Fig. 5: $\mu MSE$ errors over balanced indoor and outdoor test sets ($3584$
images).

Results demonstrate that the $BiC$ method outperforms $iCaRl$ and $Lucir$,
even without using a bias correction layer in the inference.

##  5 Conclusion

In this study, we address the problem of adapting a single image calibration
network to a new distribution while preserving its previous knowledge.
Particularly, we adapted the current incremental learning methods $LwF$,
$iCarl$, $Lucir$, and $BiC$ to predict camera parameters whether for indoor or
outdoor scenes. The comparison results show that $BiC$ performs remarkably
well when distilling knowledge, outperforming the evaluated state-of-the-art
methods. In future work, we will explore dynamically expandable networks which
only change the relevant part of the previously trained network, while still
allowing the expansion of its capacity when necessary.

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# Dark Matter Decay to Neutrinos

Carlos A. Argüelles  Department of Physics & Laboratory for Particle Physics
and Cosmology, Harvard University, Cambridge, MA 02138, USA  Diyaselis Delgado
<EMAIL_ADDRESS>Department of Physics & Laboratory for Particle Physics
and Cosmology, Harvard University, Cambridge, MA 02138, USA  Avi Friedlander
Department of Physics, Engineering Physics and Astronomy, Queen’s University,
Kingston ON K7L 3N6, Canada  Arthur B. McDonald Canadian Astroparticle Physics
Research Institute, Kingston ON K7L 3N6, Canada
Ali Kheirandish  Department of Physics & Astronomy, University of Nevada, Las
Vegas, NV, 89154, USA  Nevada Center for Astrophysics, University of Nevada,
Las Vegas, NV 89154, USA  Ibrahim Safa  Department of Physics, Columbia
University, New York, NY, 10027, USA  Department of Physics & Wisconsin
IceCube Particle Astrophysics Center, University of Wisconsin, Madison, WI
53706, USA  Department of Physics & Laboratory for Particle Physics and
Cosmology, Harvard University, Cambridge, MA 02138, USA  Aaron C. Vincent
Department of Physics, Engineering Physics and Astronomy, Queen’s University,
Kingston ON K7L 3N6, Canada  Arthur B. McDonald Canadian Astroparticle Physics
Research Institute, Kingston ON K7L 3N6, Canada  Perimeter Institute for
Theoretical Physics, Waterloo ON N2L 2Y5, Canada  Henry White  Department of
Physics, Engineering Physics and Astronomy, Queen’s University, Kingston ON
K7L 3N6, Canada  Arthur B. McDonald Canadian Astroparticle Physics Research
Institute, Kingston ON K7L 3N6, Canada

###### Abstract

It is possible that the strongest interactions between dark matter and the
Standard Model occur via the neutrino sector. Unlike gamma rays and charged
particles, neutrinos provide a unique avenue to probe for astrophysical
sources of dark matter, since they arrive unimpeded and undeflected from their
sources. Previously, we reported on annihilations of dark matter to neutrinos;
here, we review constraints on the decay of dark matter into neutrinos over a
range of dark matter masses from MeV to ZeV, compiling previously reported
limits, exploring new electroweak corrections and computing constraints where
none have been computed before. We examine the expected contributions to the
neutrino flux at current and upcoming neutrino experiments as well as photons
from electroweak emission expected at gamma-ray telescopes, leading to
constraints on the dark matter decay lifetime, which ranges from $\tau\sim
1.2\times 10^{21}$ s at 10 MeV to $1.5\times 10^{29}$ s at 1 PeV.

##  I Introduction

The angular power spectrum of the cosmic microwave background (CMB) as well as
the matter distribution on large scales, the clustering of galaxies, and the
measured kinematics of stars and gas within those galaxies all point to a
large component of weakly-interacting dark matter (DM), constituting 85% of
all matter in the Universe [1, 2]. While these observations imply an equation
of state consistent with a cold, collisionless fluid, no microphysical
connection has yet been found between DM and the Standard Model (SM) of
particle physics. Numerical coincidences such as the one-to-five ratio of
dark-to-ordinary matter sustain our hope that DM decoupled late enough in the
history of the Universe to require a coupling well below the Planck scale and
thus be describable in the language of particle physics.

The parameter space of such nongravitational interactions is immense, and
myriad portals are potentially available. Traditional searches for electroweak
and supersymmetry-inspired WIMPs in the GeV-TeV mass range that scatter with
or annihilate to quarks have expanded in the past decades to encompass light
axionlike [3] and minicharged particles [4, 5, 6, 7, 8], sub-GeV nonthermal DM
candidates [9, 10], primordial black holes [11], and other exotic objects. In
some of these scenarios, dark matter can be unstable and decay to Standard
Model particles.

Direct searches for such DM rely on elastic scattering with electrons or
nuclei, while indirect searches look for signatures of decay or annihilation
into SM particles. Products of DM decay (or annihilation) into SM particles
eventually create a flux of stable particles, i.e., protons, electrons,
photons, or neutrinos. Here, we focus on the latter. A direct neutrino portal
would render direct detection impracticable, and indirect detection very
difficult, owing to the minuscule cross section of neutrinos at low energies.
However, at high energies, the neutrino cross section grows and is no longer
suppressed by the mass of the heavy bosons but by the momentum transfer as is
the case of photon-nucleon interactions [12, 13, 14]. Additionally, high-
energy gamma rays can be attenuated as they travel from their sources of
production to Earth, while neutrinos voyage unimpeded. Therefore, the study of
neutrinos represents a final frontier in the search for indirect signatures of
DM. The study of this channel is further motivated by connections between the
dark sector and neutrinos. These have been proposed in a variety of different
contexts, including the scotogenic scenarios where neutrinos gain their mass
via interacting with the dark sector [15, 16, 17, 18, 19, 20, 21], the Majoron
scenario [22] or see-saw models [23]. In many of these models, UV physics can
destabilize the DM, leading to a decay to $\bar{\nu}\nu$, which may dominate
over other channels. These models have motivated numerous dedicated studies,
mainly in the context of discovering heavy DM using neutrino line searches
[24, 25, 26, 27, 28, 29], and many neutrino experiments have hunted for DM
signatures in their observations [30, 31, 32, 33, 34, 35], so far yielding
null results.

Previously, we presented an updated compendium of constraints on particle DM
annihilation to neutrinos [36]. Here, we turn our attention to the production
of neutrinos from DM decay, directing special attention to the higher-mass
region. At masses greater than $\sim$ TeV, electroweak (EW) corrections can
“decloak” the DM, producing high-energy photons, even when directly decaying
to neutrinos that can be detectable at current and future gamma-ray
observatories [37]. We thus present limits from current measurements and
sensitivities for upcoming experiments covering a DM mass range starting at 20
MeV and spanning well into the ultraheavy domain up to $10^{11}$ GeV. Although
recent LHAASO results are on par with IceCube’s recent analyses, we will find
that current and future neutrino telescopes retain superior sensitivity across
nearly the entire range of masses that we consider. This is due to three
factors: the loop suppression of the gamma-ray production rate, the growth of
the electroweak cross section with energy, and the loss of high-energy gamma
rays to interactions in the interstellar and intergalactic medium.

We begin by briefly describing the signature of DM decaying to neutrinos at
both neutrino telescopes and gamma-ray observatories. We present our new
results in Section III and offer parting words of wisdom in Section IV.

##  II Dark Matter decays to neutrinos

The expected flux per neutrino flavor at Earth from decay of DM with mass
$m_{\chi}$ and lifetime $\tau_{\chi}$ is

$\frac{d\Phi_{\nu+\bar{\nu}}}{dE_{\nu}}=\frac{1}{4\pi}\frac{1}{\tau_{\chi}m_{\chi}}\frac{1}{3}\frac{dN_{\nu}}{dE_{\nu}}D(\Omega).$
(1)

Below the electroweak scale, the neutrino spectrum per annihilation is
${dN_{\nu}}/{dE_{\nu}}=2\delta(1-2E/m_{\chi})m_{\chi}/2E^{2}$; at higher
masses a low-energy tail arises as discussed in Ref. [37]. Because relevant
backgrounds follow a power-law distribution, only the delta contribution is
relevant for neutrino constraints. The so-called $D$-factor, $D(\Omega)$, is
an integral of the DM distribution $\rho(x)$ along the line of sight and solid
angle $\Delta\Omega$:

$D\equiv\int d\Omega\int_{\mathrm{l.o.s.}}\rho_{\chi}(x)dx.$ (2)

We assume the Galactic DM spatial distribution is modeled by an Navarro-Frenk-
White (NFW) profile with a slope parameter $\gamma=1.2$ and a scale radius
$r_{s}=20~{}{\rm kpc}$, and we set the local DM density to
$\rho_{0}=0.4~{}{\rm GeV~{}cm}^{-3}$. These parameters are consistent with the
results of, e.g., Ref. [38], which point out a strong dependence on how the
baryonic potential is modeled. We take the distance to the galactic centre to
be $R_{0}=8.127~{}{\rm kpc}$ [39]. Here, we mainly strive to make results as
self-consistent as possible by using common halo parameters in all of our
analyses. We have assumed equal production of each flavor, which leads to
equal flavors at Earth. Due to neutrino oscillation, this will remain
approximately true regardless of the initial flavor composition.

Table 1: Neutrino (top) and gamma-ray (bottom) observatories considered in this work. Here, “All Flavors” denotes both neutrinos and antineutrinos of electron, muon, and tau flavor. The experiments given in italic font are upcoming or proposed detectors. Energy (GeV) |  Experiment | Dir. | Particles
---|---|---|---
$\left(2.5-15\right)\times 10^{-3}$ |  Borexino [40] | $\boldsymbol{\times}$ | $\bar{\nu}_{e}$
$\left(8.3-18.3\right)\times 10^{-3}$ |  KamLAND [41] |  | $\bar{\nu}_{e}$
$\left(10-40\right)\times 10^{-3}$ |  JUNO [42] |  | $\bar{\nu}_{e}$
$\left(1.5-100\right)\times 10^{-2}$ |  SK [43] | $\boldsymbol{\times}$ | $\bar{\nu}_{e}$
$0.1-30$ |  DUNE [44] | $\boldsymbol{\times}$ | $\nu_{e},\bar{\nu}_{e},\nu_{\tau},\bar{\nu}_{\tau}$
$0.1-30$ |  HK [44] | $\boldsymbol{\times}$ | $\nu_{e},\bar{\nu}_{e},\nu_{\tau},\bar{\nu}_{\tau}$
$1-10^{4}$ |  SK [45, 46] |  | All Flavors
$20-10^{4}$ |  IceCube [47] |  | All Flavors
$50-10^{5}$ |  ANTARES [48] |  | $\nu_{\mu},\,\bar{\nu}_{\mu}$
$10^{3}-10^{7}$ |  P-ONE [49] |  | All Flavors
$10^{4}-10^{7}$ |  KM3NeT [50] |  | All Flavors
$10^{6}-10^{8}$ |  TAMBO [51] |  | $\nu_{\tau},\,\bar{\nu}_{\tau}$
$>10^{7}$ |  IceCube-Gen2 [52] |  | All Flavors
$>10^{8}$ |  RNO-G [53] |  | All Flavors
$>10^{8}$ |  GRAND [54] |  | $\nu_{\tau},\,\bar{\nu}_{\tau}$
$10^{-1}-10^{2}$ |  Fermi-LAT [55] |  | $\gamma$
$10^{3}-10^{9}$ |  CTA [56] |  | $\gamma$
$10^{4}-10^{9}$ |  HAWC [57] |  | $\gamma$
$10^{5}-10^{9}$ |  LHAASO [58] |  | $\gamma$
$10^{6}-10^{9}$ |  IceTop [59] |  | $\gamma$
$10^{7}-2\times 10^{9}$ |  KASCADE [60] |  | $\gamma$
$10^{8}-2\times 10^{10}$ |  CASA-MIA [61] |  | $\gamma$
$10^{9}-2\times 10^{12}$ |  EAS-MSU [62] |  | $\gamma$
$10^{11.5}-10^{14}$ |  TA-SD [63] |  | $\gamma$

Based on Eq. 1, we produce limits on the DM decay rate, using results from
different analyses of existing data [36, 64, 65, 54, 47, 66, 67, 68, 69, 28,
70, 71, 72, 43, 73, 74] or forecasted sensitivities [75, 76, 77, 78, 79, 80].
The full list of neutrino experiments is given in the top section of Table 1.
We also list the neutrino energy range covered by each experiment, spanning
from 10 MeV at Borexino to $>10^{11}$ GeV at IceCube and AUGER, as well as
each experiment’s neutrino flavor sensitivity. For a detailed description of
each experiment and its sensitivity, we point the reader to Ref. [36]. The
$D$-factors for each experiment are computed by integrating the exposure of
each telescope over 24 hours. The resulting exposures and $D$-factors are
tabulated in Table 2 in the Supplementary Material.

Our forecasts assume five years of exposure for each of the following
experiments: JUNO [42], DUNE [44], Hyper-Kamiokande (HK) [44], RNO-G [53],
IceCube-Gen2 [52], KM3NeT [50], P-ONE [49], TAMBO [51], and GRAND [54].
Constraints that are not derived by us, but are reported by experiments or
other groups, are rescaled to match the $D$-factors used in this work. This
enables a fair comparison between different experimental constraints.

###  II.1 Gamma rays from electroweak corrections

Above the TeV scale, electroweak corrections can lead to the production of
photons. These result in two distinct gamma-ray fluxes. First, “prompt” high-
energy flux consisting of primary photons emitted during the dark matter decay
to neutrinos. Second, lower-energy (GeV–TeV) photon signal due to scattering
of primary photons with the CMB and extragalactic background light (EBL). The
prompt gamma-ray spectra can be obtained via the HDMSpectra package [37],
which solves the Dokshitzer-Gribo-Lipatov-Altarelli-Parisi (DGLAP) equations
above the electroweak scale with initial conditions given by the DM decay
channel. At the edge of this scale, HDMSpectra matches its solution with
Pythia-8.2 [81], which calculates the effects of particle showers,
hadronization, and light particle decays.

We use the gamma-ray distribution from this package to derive constraints
using gamma-ray data sets in a similar way to how we proceed with neutrinos.
One important difference is the inclusion of an additional factor of
$\exp(-\tau_{\gamma\gamma})$ in Eq. 1. This factor accounts for attenuation
due to pair-production from scattering of high-energy gamma-rays with CMB
photons, the dominant attenuation channel. For the galactic component, we
conservatively include this as a constant factor by taking the average
attenuation rate over a distance of 10 kpc. We will present limits from Fermi-
LAT [55], HAWC [57], LHAASO [58], IceTop [59], KASCADE-Grande [60], CASA-MIA
[61], EAS-MSU [62], and TA-SD [63] observations as well as a projected
sensitivity for CTA [56]. For CTA, we consider the differential sensitivity
from [82] and convert it to an upper limit on the total flux per decade of
energy, which is defined as the minimum flux required to obtain a $5\sigma$
point source detection from CTA Southern array for a total observation time of
50 hours.

In the case of IceTop, we use the differential upper limits at 90% C.L.
reported in [83]. These are then converted to total integrated emission per
energy decade. For KASCADE, KASCADE-Grande, CASA-MIA, EAS-MSU, and TA-SD, we
use the integral gamma-ray flux upper limits reported in Refs. [84, 85, 86,
87]. For HAWC, we follow the same procedure for the flux upper limit in each
declination band [88] and then further select the most stringent constraint
among all bands. All integrated gamma-ray fluxes are then compared to the
expected total flux from DM decay to neutrinos with the photon spectrum from
electroweak corrections. This comparison then yields our constraints on the DM
decay lifetime. Limits for decaying DM to all neutrino flavors from LHAASO
were taken from [89].

Figure 1: Expected integral gamma-ray fluxes produced by electroweak
corrections to dark matter decay to neutrinos overlaid on the observed gamma-
ray distributions. Integral fluxes, defined as the integral of the flux from
$E_{\gamma}$ to infinity, for four different dark matter masses and lifetime
of $\tau_{\chi}=10^{27}{\rm s}$, are shown as solid lines. Colored symbols
indicate observations detailed in the bottom half of Table 1.

At sufficiently large masses, gamma rays produced from decays outside our
galaxy can scatter down to produce a signal that is observable at lower
energies in experiments such as Fermi-LAT. High-energy gamma rays traversing
the intergalactic medium (IGM) are absorbed and scattered by photons from the
CMB and EBL, attenuating the signal [90]; see Ref. [91] for a recent detailed
discussion. Scattering and absorption of gamma rays result in cascades that
transform any sufficiently high-energy gamma-ray source into a universal
spectrum [92] that peaks within the Fermi telescope’s sensitivity range. In
what follows we take advantage of this universality to extend gamma-ray limits
to higher dark matter masses, and convert them to limits on decay to
neutrinos.

Ref. [93] sets constraints on the lifetime of DM decay to SM particles using
Fermi observations of the isotropic gamma-ray background. We use the limits
presented there for DM decays to neutrino pairs that extend up to
$m_{\chi}=10^{7}$ GeV. The limits presented in Ref. [93] constrain the channel
$\chi\rightarrow\bar{b}b$ up to $10^{10}$ GeV. We use these to obtain
corresponding limits in the channel of interest, DM decay to neutrino pairs.
The idea of the universal spectrum means that regardless of the initial gamma-
ray spectral shape, the spectrum arriving at Earth is universal. Therefore
limits on dark matter decay to neutrinos are related to the
$\chi\rightarrow\bar{b}b$ limits by a factor of
$F^{\gamma}_{\chi\rightarrow\bar{\nu}\nu}/F^{\gamma}_{\chi\rightarrow\bar{b}b}=0.06$,
where $F^{\gamma}_{\chi\rightarrow\bar{X}X}$ is the fraction of energy per
decay to species $X$ going into photons. Suppl. Fig. 1 shows that this
rescaling yields the published $\bar{\nu}\nu$ results below $10^{7}$ to a
reasonable accuracy, allowing us to confidently extend these limits up to
$10^{10}$ GeV. We find that the Fermi-LAT constraints remain subdominant over
the full range of masses considered.

Figure 2: Constraints on the lifetime of dark matter decaying to neutrinos
$\boldsymbol{\chi\rightarrow\bar{\nu}\nu}$. Solid lines bordering shaded
regions represent limits from existing neutrino telescope data, solid lines
without shading correspond to limits from existing gamma-ray observatories (as
shown in Fig. 3), and dashed lines show the reach of future experiments.
Labels with a heart symbol ($\heartsuit$) correspond to limits derived for
this work.

##  III Results

Using the methods outlined above, we present constraints on the DM decay
lifetime in Fig. 2. We label the results derived for this work with a heart
($\heartsuit$).

Constraints from neutrino telescopes are shown as shaded regions bordered by
solid lines. The overlap in experimental sensitivities yields continuous
constraints on the DM lifetime that are much greater than the age of the
Universe, ranging from $\tau>10^{19}$ s at $m_{\chi}\sim 50$ MeV to
$\tau>10^{27}$ s for $m_{\chi}\sim 10^{11}$ GeV. The expected neutrino flux at
Earth from DM decay is independent of mass. Below $\sim 10^{7}$ GeV, this is
reflected by sensitivity closely following the growth of the electroweak cross
section with energy, with some scaling between experiments owing to
differences in effective volumes. Above $\sim 10^{7}$ GeV energies, the Earth
becomes opaque to neutrinos, and detection technologies become sensitive to a
much smaller solid angle, usually restricted to an area just around the
horizon.

Estimated sensitivities of future observatories are shown as dashed lines;
these assume five years of data taking. JUNO, Hyper-K, DUNE, KM3NeT, P-ONE,
and IceCube-Gen2 should each lead to an improvement of one to two orders of
magnitude over current bounds, mainly owing to much larger effective detector
volumes. Projected improvements from future radio (GRAND, RNO-G) and modular
Cherenkov arrays (TAMBO) are more modest, which we mainly attribute to
restricted fields of view.

Limits from gamma-ray observatories are marked with a $\gamma$ superscript.
These are also shown separately in Fig. 3. Four experiments dominate the
constraints at three different energy ranges. At masses below $\sim 10^{5}$
GeV, the flux of extragalactic gamma-rays produced by interactions with the
IGM is probed by Fermi-LAT, yielding the dominant source of gamma-ray
constraints in this mass range. At masses between $10^{6}$ and $10^{7}$ GeV,
recent measurements by LHAASO supersede prior experiments and improve
constraints by nearly four orders of magnitude compared to HAWC. At masses
above $10^{7.5}$ GeV, KASCADE-Grande measurements establish the most
competitive constraints on the DM decay lifetime limits, outperforming
existing neutrino telescopes; at $m_{\chi}\gtrsim 10^{10}$ GeV, TA supersedes
all other experiments thanks to is monumental effective area. Other
experiments considered, such as HAWC or IceTop, remain subdominant over the
entire mass range probed here.

Figure 3: Gamma-ray constraints on dark matter decay lifetime
$\boldsymbol{\chi\rightarrow\bar{\nu}\nu}$ due to $\boldsymbol{\gamma}$
emission from electroweak processes. Solid lines correspond to existing
constraints, while the dashed line is a projection for a future experiment.
Hearts indicate the new constraints derived in this work. Gamma-ray emission
below the electroweak scale is suppressed by powers of $M_{W}$ [37].

##  IV Future Prospects & Conclusions

As shown in Fig. 2, existing neutrino telescopes are able to constrain the
lifetime of DM decay to neutrino pairs to values ranging from $10^{3}$ to
$10^{12}$ times the age of the Universe. Upcoming neutrino telescopes will
make improvements of one to two orders of magnitude: DUNE and Hyper-Kamiokande
will fill in the gap around $m_{\chi}\sim$ GeV, while the strongest
improvements will take place for the next generation of large-volume water and
ice Cherenkov telescopes: KM3NeT, P-ONE, and IceCube-Gen2. Though not included
here, the ongoing scintillator phase of SNO+ [94] may also shore up
constraints on the low-mass end, depending on the timeline for tellurium
filling; the inclusion of directional information in, e.g., Borexino [95] or
KamLAND analyses could also yield a modest improvement in reach [96, 97].

Above the $\sim$ TeV range, the electroweak emission of gamma rays opens a new
opportunity for discovery, and above $10^{8}$ GeV, gamma rays become the
dominant source of information, thanks to the large telescope areas and
unsuppressed electroweak emission of photons. What’s more the observation of
an electromagnetic counterpart will be key in the event of a discovery.
Intriguingly, the Square Kilometre Array (SKA) will be sensitive to
$\chi\rightarrow\bar{\nu}\nu$ [29] in nearby dwarf galaxies for DM masses
above a few hundred GeV. Taken together, these observations highlight the
importance of multimessenger observations when it comes to elucidating the
nature of dark matter.

###### Acknowledgements.

We thank Matheus Hostert, Alejandro Diaz, Nicholas Rodd, and Damiano Fiorillo
for useful discussions. We thank Barbara (Basia) Skrzypek for providing the
code for gamma-ray attenuation. We thank Thomas Hambye for useful discussion
on the model building of dark matter and its connections to neutrinos. We
thank Jean DeMerit for carefully proofreading this manuscript. CAA, IS, and DD
are supported by the Faculty of Arts and Sciences of Harvard University and
the Alfred P. Sloan Foundation. AK work is partially supported by the Nevada
Center for Astrophysics. IS is supported by NSF under grants PLR-1600823 and
PHY-1607644 and by the University of Wisconsin Research Council with funds
granted by the Wisconsin Alumni Research Foundation. AF is supported by an
Ontario Graduate Scholarship. ACV is supported by the Arthur B. McDonald
Canadian Astroparticle Physics Research Institute and NSERC, with equipment
funded by the Canada Foundation for Innovation and the Ontario Government.
Research at Perimeter Institute is supported by the Government of Canada
through the Department of Innovation, Science, and Economic Development, and
by the Province of Ontario.

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##  Appendix A Supplementary Figures and Tables

Here, we include the following tables and figures:

  * •

SUPPL. TABLE 1 shows the full list of references to data sets or prior
analyses used to produce the results of Fig. 2.

  * •

SUPPL. TABLE 2 shows the $D$-factors relevant for each experiment for which we
recast diffuse limit fluxes into limits on dark matter decay.

  * •

SUPPL. FIG. 1 illustrates the rescaling of Fermi-LAT limits on decays to
$\bar{b}b$ pairs to limits on $\bar{\nu}\nu$.

  * •

SUPPL. FIG. 2 is a larger version of the main results Fig. 2.

SUPPL. FIG. 1: Demonstration of the scaling from dark matter decay to
$b\bar{b}$ constraints to neutrinos. Solid lines show lifetime limits on dark
matter decay to $b\bar{b}$ (red) and $\nu_{\tau}\bar{\nu}_{\tau}$ (black) as
determined by Cohen et al. [93] using gamma-ray observation from Fermi-LAT.
Dashed line shows an approximation for extending the
$\chi\to\nu_{\tau}\bar{\nu_{\tau}}$ limits by rescaling the decay to quark
limits, by a factor of 0.06, the ratio between total photon production
fractions in each channel.

Analyses Data Used to Derive Lifetime Limit ANTARES Annihilation Cross Section
Limit [98] Auger Diffuse Neutrino Flux [64] Borexino Upper Limit of Neutrino
Flux [65] DUNE Annihilation Cross Section Sensitivity [36] GRAND-200k Diffuse
Neutrino Flux Sensitivity [54] Hyper-Kamiokande Annihilation Cross Section
Limit [75] IceCube Annihilation Cross Section Limits [47, 66] IC-DeepCore
Annihilation Cross Section Limit [67] IC (atm.) Diffuse Neutrino Flux [68]
IceCube-HE Diffuse Neutrino Flux [99, 100] IceCube-EHE Diffuse Neutrino Flux
[69] IceCube (Bhattacharya) Lifetime Limit [28] IceCube-Gen2 Diffuse Neutrino
Flux [101] JUNO Lifetime Sensitivity [76] KamLand Upper Limit of Neutrino Flux
[70] [65] KM3NET Annihilation Cross Section Sensitivity [77] P-ONE Projected
Effective Areas [79] RNO-G Diffuse Neutrino Flux Sensitivity [78].
SK-$\bar{\nu_{e}}$ Upper Limit of Neutrino Flux [72] SK (Olivares)
Annihilation Cross Section [43] SK Annihilation Cross Section [73] SK (atm.)
Diffuse Neutrino Flux [74] TAMBO Projected Effective Areas [80] IceTop Upper
Limit of Gamma-Ray Flux [83] CTA (Queiroz et al.) Annihilation Cross Section
Projection [102] CTA Projected $\gamma$ Sensitivity [82] HAWC Upper Limit of
Gamma-Ray Flux [88] KASCADE Upper Limit of Gamma-Ray Flux [84] KASCADE-Grande
Upper Limit of Gamma-Ray Flux [84] CASA-MIA Upper Limit of Gamma-Ray Flux [85]
EAS-MSU Upper Limit of Gamma-Ray Flux [86] TA-SD Upper Limit of Gamma-Ray Flux
[87]

SUPPL. TABLE 1: List of data sets used in this work and their references. Left
column names the experiment, while the right one provides a short description
of the data set.

Experiment Exposure $D/{10^{23}}$ All-sky All-sky $2.65$ GRAND Elevation-
dependent exposure, Fig. 24 of [54] $0.298$ ANITA dec =
$[1.5^{\circ},4^{\circ}]$ $0.052$ TAMBO Elevation-dependent exposure, Fig. 3 &
4 of [80] 0.001 Auger ${\rm zenith}=[90^{\circ},95^{\circ}]$ ${\rm
zenith}=[75^{\circ},90^{\circ}]$ ${\rm zenith}=[60^{\circ},75^{\circ}]$ 0.11
0.35 0.33 P-ONE $\cos({\rm zenith})=[-1,-0.5]$ $\cos({\rm zenith})=[-0.5,0.5]$
$\cos({\rm zenith})=[0.5,1]$ 0.83 1.35 0.47 IceTop All-sky $2.63$ CTA Galactic
Centre [102] $0.003$ HAWC dec = [$-25^{\circ}$ \- $5^{\circ}$] $0.0295$
KASCADE dec = [$14^{\circ}$ \- $84^{\circ}$] $0.76$ CASA-MIA dec =
[$-20^{\circ}$ \- $90^{\circ}$] $1.58$ EAS-MSU dec = [$7^{\circ}$ \-
$78^{\circ}$] $0.92$ TA-SD zen = [$-90^{\circ}$ \- $45^{\circ}$] $1.85$

SUPPL. TABLE 2:  Exposures and D-factors for experiments discussed in this
work. Exposures are converted to RA-dec and averaged over 24 hour period to
obtain the corresponding $D$-factors, given in units of GeV cm-2 sr, and
computed according to Eq. (2).

SUPPL. FIG. 2: Larger version of Fig. 2. Solid lines bordering shaded regions
represent limits from existing neutrino telescope data, solid lines without
shading correspond to limits from existing gamma-ray observatories (as shown
in Fig. 3), and dashed lines show the reach of future experiments.