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# High-energy neutrino signals from supernova explosions:
a new window into dark photon parameter space

Vsevolod Syvolap  Instituut-Lorentz for Theoretical Physics, Leiden
University, 2333 CA Leiden, Netherlands  Oleg Ruchayskiy  Niels Bohr
Institute, Copenhagen University, Jagtvej 155A, Copenhagen, DK-2100, Denmark

###### Abstract

Dark photons, hypothetical feebly interacting massive vector bosons, appear in
many extensions of the Standard Model. This study investigates their
production and subsequent decay during supernova explosions. We demonstrate
that the decay of dark photons, with masses ranging from 200 to 400 MeV, can
lead to the emission of neutrinos with energies surpassing those emitted by
supernovae. These neutrinos therefore serve as a distinct signal of new
physics, allowing for the exploration of previously uncharted regions of the
dark photon parameter space and complementing both accelerator-based searches
and other astrophysical constraints. The signal is largely unaffected by the
specifics of the supernova’s temperature and density radial profiles outside
the SN core, rendering the prediction both robust and model-independent. Our
results indicate that searching for high-energy neutrinos accompanying
supernova explosions provides a novel approach to probe physics beyond the
Standard Model, including dark photons, heavy neutral leptons, and other
feebly interacting particles with masses in the hundreds of MeV range.

##  I Introduction

The cores of exploding supernovae (SNe) can reach high matter densities
($\rho\sim 10^{14}$ g/cm3) and temperatures
($T\sim$30\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$), see e.g. [1, 2]. These
astrophysical environments serve therefore as unique sites of copious
productions of hypothetical particles that interact with matter “weaker than
neutrinos”. Such particles are collectively known as _feebly interacting
particles_ or _FIPs_ , see [3, 4, 5, 6] for the recent overviews of their
properties and of models that predict them. A commonly used approach to
constrain feebly interacting particles with supernovae utilizes the fact, that
SN right after the explosion is cooled primarily via the active neutrino
emission which lasts for approximately 10 seconds. In the presence of
additional cooling channels, like the FIPs emission, this duration might be
shortened significantly [7] which would contradict the SN1987a observations
[8, 9, 10]. Using the energy-loss argument different FIPs have been studied in
the past [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. However, improving existing
bounds necessitates detailed SN explosion models that incorporate how neutrino
emissions are influenced by any additional cooling channels - [21] and is
highly model-dependent, given the current level of understanding of details of
SN explosions, c.f. [22].

Given the advancement of neutrino experiments, one expects a plethora of
observational data, should a nearby supernova explode. This warrants new
studies about more detailed constraints on the properties of FIPs, based on
the detailed measurements of SN neutrino signal. This work has already started
[23, 24, 25].

This work extends the supernova constraints on the properties of _dark
photons_ (DP) — massive vector bosons that interact with the Standard Model
sector exclusively through kinetic mixing with photons. The Lagrangian of the
model is given by

$\mathcal{L}_{\text{DP}}=\mathcal{L}_{\text{SM}}-\frac{1}{4}F^{\prime}_{\mu\nu}{F^{\prime}}^{\mu\nu}+\frac{m_{A^{\prime}}^{2}}{2}A^{\prime}_{\mu}{A^{\prime}}^{\mu}-\frac{e}{2}UF^{\mu\nu}F^{\prime}_{\mu\nu}$
(1)

where $\mathcal{L}_{\text{SM}}$ is the Lagrangian of the Standard Model, $e$
is the electric charge, $U$ is the dimensionless kinetic mixing
parameter,111Often called $\varepsilon$ in the literature $F^{\mu\nu}$ and
$F^{\prime\mu\nu}$ are field strength tensors, associated with the SM photon
field and the dark photon respectively (original references can be found in
[26, 27], and the current model status and experimental searches are reviewed
in [28, 29]). Transitioning to the mass basis means that electrically charged
Standard Model particles inherit an interaction with the dark photon, carrying
a “charge” of $U\times e$. This mechanism enables the potential production and
detection of dark photons. Previously the production of dark photons in SN
interiors and their impact on the SN cooling have been studied in [17, 18, 19,
20]. These works considered the additional SN cooling channel via dark photons
and derived the limits based on the energy-loss arguments. An alternative test
for the presence of dark photons can be the search for a detectable gamma-ray
signal that dark photons would produce when decaying to the Standard model
photons [30].

The present study focuses on the potential for detecting signals induced by
dark photons using neutrino detectors, should a nearby Galactic supernova
explode. We propose a method similar to our previous work [25], see also [31,
24, 23] – the detection of high-energy neutrinos accompanying the SN neutrino
signal. In contrast to the case of heavy neutral leptons explored in [31, 25],
dark photons do not have a sizeable direct decay into neutrinos. However, dark
photons with masses $m_{A}>2m_{\mu}$ ($2m_{\pi}$), produced within the SN
core, can decay into pairs of muons (correspondingly, charged pions), which in
turn will decay into neutrinos. These decays will occur in less dense regions
of the supernova, allowing the resulting neutrinos to escape and be detected
by current and future neutrino detectors, including the Hyper-Kamiokande
(_Hyper-K_) neutrino telescope [32], JUNO [33], and DUNE [34]. Given that dark
photons are primarily produced in the core, where temperatures are
significantly higher than that of the neutrinosphere [1], the energy of
neutrinos resulting from dark photon decays will be noticeably higher than
that of the typical neutrino signal, making these signals distinguishable.

  * The paper is organized as follows.

  * –

In the section II, the production of the dark photons are described and their
spectra are calculated.

  * –

In Section III we compute the distribution function of the DP’s decay products
– muons and pions, the sources of the secondary neutrinos. Here we also
analyze their thermalization happening before decay, which modifies the
particles’ spectra.

  * –

In Section V, the decays of remaining muons and pions leading to the neutrinos
as secondary particles are discussed and the neutrino spectra are computed.

  * –

The propagation and flavor content of the secondary neutrino flux is described
in Section V.2.

  * –

Section V.3 discusses detection by the secondary neutrinos by Hyper-K and
other neutrino detectors. Our main result is then summarized in Figure 6.

  * –

In the final Section VII, we discuss the results and future prospects.

##  II Dark photons in the supernova core

In the supernova medium the production of the dark photon occurs via
scatterings of charged particles. The main production channel is the proton-
nucleon bremsstrahlung:

$p+N\to p+N+A^{\prime}\;.$ (2)

(here $N$ is neutron or another proton). DP can also be produced in the
scatterings, involving charged leptons:

$e^{\pm}+X\to e^{\pm}+X+A^{\prime}$ (3)

where $X$ is either a proton or another charged lepton. Since the nucleon
number density is higher than that of the leptons and the cross-section of
strong interactions is higher, than the electromagnetic ones, the channels (3)
are subdominant and will be neglected in our calculations.

The period after the core bounce, when the temperature and density in the core
are the highest, leads to the most efficient DP production. A proper
analytical description of such a process is missing. To simplify the
calculation (and to make our results readily reproducible) we will employ a
simplified model of the SN core – a sphere with a uniform baryon density
$\rho_{B}=$3\text{\times}{10}^{14}\text{\,}\mathrm{g}\mathrm{/}\mathrm{c}\mathrm{m}^{3}$$
and the temperature $T=$30\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$. We take
both quantities as constant for the time duration $t_{\text{emm}}=10$ sec. The
most intense production of the dark photons photons occurs inside the core and
our result is not sensitive to the temperature/density profile outside (see
also discussion below, in Section V).

To find the dark photon spectra $f_{A^{\prime}}$, we use the Boltzmann
equation:

$\frac{df_{A^{\prime}}(p_{A^{\prime}})}{dt}=\int\prod_{i=1,2,3,4}\frac{d^{3}p_{i}}{(2\pi)^{3}2E_{i}}S|\mathcal{M}|^{2}(2\pi)^{5}\delta(p_{3}+p_{4}+p_{A^{\prime}}-p_{1}-p_{2})f_{1}f_{2}(1-f_{3})(1-f_{4})(1+f_{A^{\prime}}).$
(4)

where $p_{i}$ are the momenta of initial and final nucleons;
$S|\mathcal{M}|^{2}$ – matrix element of the processes (2) with the symmetry
factor $S$ included in it; and $f_{i}$’s are the phase-space distribution
functions of the nucleons. We solve Eq. (4) following the approach of [20,
17]. For scattering energies, that are small compared to the nucleon mass, the
integration over the 12-dimensional space can be considerably reduced, leading
to the 3-dimensional integral:

$\frac{dN_{A^{\prime}}}{dE}=e^{2}U^{2}\alpha_{\pi}^{2}n_{i}n_{j}\sqrt{\frac{T^{3}}{8\pi
m_{N}^{3}}}\int
dudvdz\sqrt{uv}e^{-u}\sqrt{1-\frac{q^{2}}{x^{2}}}\delta(u-v-E/T)\mathcal{I}_{ij}.$
(5)

where $\alpha_{\pi}\approx(2m_{N}/m_{\pi})^{2}/4\pi$ is the pion-nucleon
coupling [35]; $n_{i},n_{j}$ are the number densities of nucleons, $u,v,z,q$ –
dimensionless variables, related to momenta and angles in the scattering
process. Finally the quantities $\mathcal{I}_{ij}$ are the expression,
different for pp and pn processes, and discussed further in Appendix B.

A low energy fraction of these dark protons will be gravitationally trapped
inside the SN core, where they will decay and all their decay products will
eventually thermalize, not affecting the SN neutrino spectra. The relevant
part of the energy spectrum of dark photons is therefore given by:

$\frac{dN_{A^{\prime}}}{dE_{A^{\prime}}}\to\frac{dN_{A^{\prime}}}{dE_{A^{\prime}}}\cdot\theta(E_{A^{\prime}}-E_{g}-m_{A^{\prime}})$
(6)

where the gravitational binding energy is defined as

$E_{g}=\frac{r_{g}}{R_{c}}m_{A^{\prime}}$ (7)

with $R_{c}\simeq 10$ km is the typical radius of the SN core, and $r_{g}$ –
its Schwartzhield radius. The examples of the dark photon spectra
$dN_{A^{\prime}}/dE_{A^{\prime}}$ produced during
$t_{\text{emm}}\sim$10\text{\,}\mathrm{s}\mathrm{e}\mathrm{c}$$ are presented
in Fig. 1.

Figure 1: The energy spectra of dark photons, generated in the supernova core
during the initial 10 sec post-bounce. Parts of the spectra represented by
dashed lines are excluded due to gravitational trapping (see the text for
details).

##  III Dark photon decays

Our study concentrates on the parts of the dark photon parameter space where
these particles decay _inside_ the supernova (but of course, outside of the
supernova core). Along with their decay, dark photons can get re-absorbed in
the supernova environments. It has been shown in [20] that for dark photons
with $m_{A^{\prime}}>2m_{\mu}$ their decays are dominant for the depletion of
the dark photon flux. For masses
$m_{A^{\prime}}\lesssim$600\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$ there are
three main decay channels [28]:

$\displaystyle A^{\prime}$ $\displaystyle\rightarrow$ $\displaystyle
e^{-}+e^{+}$ $\displaystyle A^{\prime}$ $\displaystyle\rightarrow$
$\displaystyle\mu^{-}+\mu^{+}$ (8) $\displaystyle A^{\prime}$
$\displaystyle\rightarrow$ $\displaystyle\pi^{-}+\pi^{+}$

The decay width into the lepton pair is given by

$\Gamma_{A^{\prime}\rightarrow
l^{+}l^{-}}=\frac{\alpha}{3}U^{2}m_{A^{\prime}}\sqrt{\left(1-\frac{4m_{l}^{2}}{m_{A^{\prime}}^{2}}\right)}\left(1+\frac{2m_{l}^{2}}{m_{A^{\prime}}^{2}}\right)$
(9)

where $m_{l}$ is the mass of the relevant lepton. The decay width into hadrons
is related to it as

$\Gamma_{A^{\prime}\rightarrow\text{hadrons}}=\Gamma_{A^{\prime}\rightarrow\mu^{+}\mu-}\left(\frac{\sigma_{e^{+}e^{-}\rightarrow\text{hadrons}}}{\sigma_{e^{+}e^{-}\to\mu^{+}\mu^{-}}}\right)\biggr{|}_{\sqrt{s}=m_{A^{\prime}}}$
(10)

where $\sigma_{e^{+}e^{-}\rightarrow\text{hadrons}}$ is the inclusive
inelastic cross-section. The branching ratios for these channels are shown in
Fig. 2.

Figure 2: The branching ratios for the dominant dark photon decay modes. For
masses of interest, no other decay channels with the branching fractions $\geq
1\%$ level exist.

The electron decay channel is not interesting to us as they are stable, and
can not lead to the production of neutrinos, but muons and pions are long-
lived and have neutrinos as their decay products. The decay of these particles
can therefore lead to delayed neutrino emission – the subject of this study.

The energy spectra of the decay products have been calculated before [36] and
for the 2-particle decay scenario has the form:

$f_{x}(E)=\frac{m_{A^{\prime}}}{2\bar{E}}\text{Br}_{A\to\mu/\pi}\int\limits_{\text{E}_{\text{min}}(E)}\frac{dE_{A^{\prime}}}{p_{A^{\prime}}}\frac{dN_{A^{\prime}}}{dE_{A^{\prime}}}$
(11)

The subscript $x$ stands for $\mu/\pi$, $\bar{E}\equiv m_{A^{\prime}}/2$ is
the energy of the decay product in the rest frame of the parent particle,
$p_{A^{\prime}}$ is the absolute value of the DP’s 3-momentum; and the
quantity $\text{E}_{\text{min}}$ shows the minimum values of the DP energy,
that can produce a pion/muon with the energy $E$ in the laboratory frame:

$\text{E}_{\text{min}}(E)=\max\left(E_{g},E+\frac{m_{A^{\prime}}^{2}}{4E}\right)$
(12)

In both decay processes the kinematics are the same, therefore the calculation
will be the same up to the change in the product’s mass.

An example of the resulting spectra of the pions and muons is given in Fig.3

Figure 3: Energy spectra of pions and muons from dark photon decay with mass
$m_{A^{\prime}}=340$ MeV (solid lines). The smaller quantity of pions results
from a lower branching ratio for this decay channel. The distinction between
dashed and solid lines is due to gravitational trapping.

The suppression of the pion spectra is due to the small BR for the presented
mass (BR${}_{A^{\prime}\rightarrow\pi^{+}\pi^{-}}\approx 3.7\%$ for
$m_{A^{\prime}}=340$ MeV). For higher masses, the importance of pions will
increase. The distinctive shape of the "flat" spectra is due to gravitational
trapping, implemented as a low-energy cut-off for the spectra of dark photons.

##  IV Muons and pions inside the supernova

Muons and charged pions participate in the EM interactions with the SN plasma
– mostly with electrons, protons, and photons. The temperature outside the SN
core is significantly lower than the mass of pion or muon. Therefore, if such
interactions occur at a rate higher than the decay of the corresponding
particles, they will cool down, losing their momenta and becoming non-
relatistic. We will call such a process thermalization, implying that it is
only a kinetic energy equilibration with plasma.

We start our analysis with the case of muons. Three main thermalization
channels are

  1. 1.

Scattering off photons $\mu^{\pm}+\gamma\rightarrow\mu^{\pm}+\gamma$

  2. 2.

Scattering off free electrons in plasma222the abundance of positrons at $T\ll
1$ MeV is negligible. $\mu^{\pm}+e^{-}\rightarrow\mu^{\pm}+e^{-}$

  3. 3.

Scattering on free protons333At large radii, there are also He nuclei, but as
we will see, this process is subdominant $\mu^{\pm}+p\rightarrow\mu^{\pm}+p$

###  IV.1 Scattering of muons on free electrons.

The stopping power and energy loss of $E_{\mu}\sim 100$ MeV muons in regular
matter is dominated by the ionization processes [37], which in the case of
muon kinetic energy $T_{\mu}\sim$\mathrm{M}\mathrm{e}\mathrm{V}$$ is
equivalent to scattering on free electrons. We can expect, that in the case of
the SN plasma, the result will be similar.

Naively, the dimensional estimate for the cross-section would be similar to
the Thomson scattering $\sigma_{T}\sim\alpha^{2}/E^{2}$ with typical energy in
reaction $E\sim m_{\mu}$, but instead, due to small electron mass and, as a
result, small transferred momentum, this cross-section will be significantly
higher as well as energy-loss rate - see App C.

The temperature $T_{e,th}$, where such thermalization occurs can be estimated
as

$N_{e,th}^{-1}\cdot\sigma_{\mu e}\cdot
n_{e}(T_{e,th})=\Gamma_{\text{dec}}^{\mu}$ (13)

here, $N_{e,th}$ \- number of reactions needed to thermaliza muon,
$\sigma_{e\mu}$ \- electron-muon scattering cross-section, $n_{e}$ \- electron
number density. Such an estimate will lead to the thermalization temperature

$T_{e,th}\sim 10^{-5}~{}\text{MeV}$

corresponding to radius $R\gtrsim 10^{5}-10^{6}$ km. As will be seen from the
results, such a large radius would mean that the major part of the parameter
space of the dark photons’ parameters that are potentially interesting for us,
would correspond to the thermalized muons.

###  IV.2 Muon-photon scattering

In a limit, where the energy of the photon is small $\omega\sim T\ll m_{\mu}$,
the Thomson cross-section is applicable with the rate:

$\Gamma_{\mu\gamma}(T)=\sigma^{\mu}_{T}\cdot
n_{\gamma}(T),~{}~{}~{}\sigma^{\mu}_{T}=\left(\frac{m_{e}}{m_{\mu}}\right)^{2}~{}0.665~{}\text{b},$
(14) $n_{\gamma}(T)=2\frac{\zeta(3)}{\pi^{2}}T^{3}$

here $\sigma^{\mu}_{T}$ \- Thomson scattering cross-section for muons. The
energy carried by photons after scattering is

$\omega^{\prime}\approx\gamma^{2}\omega(1+\beta\cos\theta^{\prime})(1-\beta\cos\theta)$
(15)

where $\gamma,\beta$ \- are Lorentz factors for incoming muon,
$\theta,\theta^{\prime}$ \- angular directions before and after scattering,
$\omega,\omega^{\prime}$ \- energy of the photon before and after scattering.
In our scenario, the energy of the muon will not be ultra-relativistic and
$\gamma\sim\mathcal{O}(1)$. To thermalize the muon
$N_{\gamma,th}\sim\frac{E_{\mu}}{\gamma^{2}\cdot\braket{\omega}}$ reactions
are needed. So, using the similar estimate as for the case of the electron
scattering:

$N_{\gamma,th}^{-1}\sigma^{\mu}_{T}\cdot
n_{\gamma}(T_{\gamma,th})=\Gamma_{\text{dec}}^{\mu}$ (16)

will give a typical value of thermalization temperature

$T_{\gamma,th}\approx 0.03\text{MeV}$

which corresponds to $R\sim 10^{3}$ km. Since this radius is much smaller than
in the case of electrons,

###  IV.3 Scattering on protons

The calculations for this process in the approximation of the point-like
proton can be done with the replacement of the electron mass with the proton
mass in the $e-\mu$ scattering. As a result, it can be seen (see C), that due
to the large proton mass, the rate of the energy loss due to muon-proton
scattering is significantly lower, than in the case of electron-muon scenario.

Therefore, the $e+\mu\to e+\mu$ process will be dominant in the thermalization
of muons. If the decay of the dark photon occurs at a higher temperature, the
resulting muon spectra will become non-relativistic, and for further
calculations, they are considered to be at rest at the moment of decay. On the
other hand, if the temperature, where the decay of the DP occurs is lower,
$\mu$’s spectra will still be given by Eqn.(11).

The above-written can be similarly applied for pions with a replacement of
muon mass with the pion mass and corresponding lifetimes. Besides, they can
participate in strong interactions in elastic pion-proton or pion-neutron
scatterings. But pions, as it will be seen, will give a subdominant
contribution to the final result.

As was mentioned previously, to calculate the production of the dark photon
the toy model of the uniform tsphere is used. It doesn’t have any information
about the temperature profile of the media outside the core. But to take into
account the thermalization and its potential effect no simple model can be
used. Therefore, a model of Electron-Capture Supernovae - the 8.8 $M_{\odot}$
spherical symmetry simulation SN [38, 39] has been used to obtain the radial
temperature profiles. The examples of such hydrodynamical profiles are given
in App D.

##  V Neutrinos from dark photon decays

In this Section we will analyze the spectra and flavor composition of the
neutrinos, produced from dark photon decays. The neutrinos are produced in the
decay of secondary particles (Section V.1), change their flavor composition
while propagating (Section V.2) and, finally, get detected at terrestial
detectors (Section V.3).

###  V.1 Spectra of secondary neutrinos at production

Muons decay primarily via

$\mu^{-}\rightarrow e^{-}+\bar{\nu}_{e}+\nu_{\mu}$ (17)

(plus a CP-conjugated process). This decay produces neutrinos of both muon and
electron flavors. In the muon’s rest frame, the distribution of these
neutrinos in energies and direction is given by:

$\frac{d^{2}N}{dE_{\nu}d\cos\theta}\equiv
f_{I}(E_{\nu})=\frac{256}{54}\frac{E_{\nu}^{2}}{m_{\mu}^{3}}\left(3-4\frac{E_{\nu}}{m_{\mu}}\right)$
(18)

(for $\bar{\nu}_{\mu}$ and $\nu_{e}$ in the case of CP-conjugated process)

$f_{II}(E_{\nu})=48\frac{E_{\nu}^{2}}{m_{\mu}^{3}}\left(1-2\frac{E_{\nu}}{m_{\mu}}\right)$
(19)

(for $\bar{\nu}_{e}$ and $\nu_{\mu}$ in the case of CP-conjugated process).
This leads to the following expression for the neutrino spectra in the lab
frame, c.f. [31]:

$f_{\text{3-body}}(E)=\int
dE_{N}~{}d\cos\theta\,\frac{dN_{\mu}}{dE_{\mu}}f_{a}\left(\frac{E}{\gamma(1+\beta\cos\theta)}\right)$
(20)

here $\beta=p_{\mu}/E_{\mu},\gamma=E_{\mu}/m_{\mu}$ are the Lorenz factors for
muons, $f_{a}$ \- spectra of secondary neutrinos/antineutrinos Eqs. (18)–(19),
the subscript "a" stays for $I,II$.

Next, consider the pion decay. The dominant decay channel for pions is [40]:

$\pi^{-}\rightarrow\mu^{-}+\bar{\nu}_{\mu}$ (21)

and other channels can be ignored due to the large helicity suppression. For
this 2-body decay, the expression for the spectra of finalneutrinos is almost
the same as in (11). The only difference is the negligible masses of
neutrinos, for which there is no upper limit in the integration:

$f_{\nu}(E)=\frac{m_{\pi}}{2\bar{E_{\nu}}}\int_{\text{E}_{\text{min},\nu}}\frac{dE_{\pi}}{p_{\pi}}\frac{dN_{\pi}}{dE_{\pi}}$
(22)

where $\bar{E_{\nu}}=\frac{m_{\pi}^{2}-m_{\mu}^{2}}{2m_{\pi}}$, and lower
bound $\text{E}_{\text{min},\nu}$ is a solution for

$m_{\pi}\frac{E}{\bar{E_{\nu}}}=\text{E}_{\text{min},\nu}+\sqrt{\text{E}_{\text{min},\nu}^{2}-m_{\pi}^{2}}$
(23)

####  V.1.1 Thermalization of muons and pions.

In the case of $\mu$ and $\pi$ losing their energy during thermalization and
becoming non-relativistic, the results for the neutrino spectra can be
significantly simplified. In such a scenario the spectra of neutrinos for each
channel are proportional to the total number density of dark photons produced
and the corresponding decay channel and are given by:

  * •

Decay into a pair of pions leads to a sharp peak in $\nu_{\mu}$ spectrum

$\frac{dN_{\nu_{\mu}}}{dE}=\text{Br}_{A\rightarrow\pi^{+}\pi^{-}}N_{A^{\prime}}\delta(E-\bar{E_{\nu}}),~{}~{}~{}\bar{E_{\nu}}\approx
29.8\text{ MeV}$ (24)

  * •

Decay into a pair of muons leads to a spectrum of electron and muon flavor
neutrinos, proportional to previously mentioned $f_{I,II}$

$\frac{dN_{\nu_{\mu}}}{dE}=2\text{Br}_{A\rightarrow\mu^{+}\mu^{-}}N_{A^{\prime}}f_{I}$
(25)
$\frac{dN_{\bar{\nu}_{e}}}{dE}=2\text{Br}_{A\rightarrow\mu^{+}\mu^{-}}N_{A^{\prime}}f_{II}$
(26)

Where the $N_{A^{\prime}}$ \- is the number of produced dark photons, factor
$2$ appears from integration over $d\text{cos}\theta$ in the isotropic case.
The examples of the spectra of neutrinos appearing in non-thermalized
scenarios are shown in Fig.4.

Figure 4: Spectra of secondary neutrinos of different flavors from the dark
photon decays for two masses: $m_{A^{\prime}}=300$ MeV (left plot) and
$m_{A^{\prime}}=600$ MeV (right plot) in a scenario, where muons and pions do
not lose their energy e.g. non-thermalized. Neutrinos from the pion channel
have more narrow spectra due to their definite energy in the 2-particle decay
process.

For smaller masses, the dominant channel is decay into a pair of muons while
for the higher mass, decay into pion contributes more to the lower-energy part
of neutrino spectra.

###  V.2 Propagation of neutrinos

Neutrinos, produced in decays will propagate freely outside of the SN media
towards Earth. Due to oscillations, their flavor composition will change.
Those could be regular vacuum oscillations [41] or matter-induced resonant
conversions like the MSW effect [42, 43]. The latter process occurs inside the
SN media at radii $R_{\text{MSW}}\lesssim 10^{5}$ km [44]). Neutrinos, that
travel through the less dense environment, experience only vacuum
oscillations.

The case of vacuum oscillations is independent of the neutrino mass hierarchy.
The flux of resulting neutrinos of electron flavor after oscillation is given
by

$F_{\bar{\nu}_{e},\text{osc}}=P^{\text{vac}}_{ee}F_{\bar{\nu}_{e}}^{0}+P^{\text{vac}}_{e\mu}F_{\bar{\nu}_{\mu}}^{0}$
(27)

where $P^{\text{vac}}_{ee}$ \- survival probability for electron neutrino,
$P^{\text{vac}}_{e\mu}$ \- oscillation probability from a muon flavor,
$F_{\bar{\nu}_{\alpha}}^{0}$ for $\alpha=\\{e,\mu,\\}$ \- spectra of the
secondary neutrinos. Tau flavor is omitted as in DP decays it will not be
produced. The standard expression for vacuum oscillating, time-averaged
probabilities, for the case of vanishing complex phases in the PMNS matrix
(see Ref.[44]), is:

$P_{\nu_{\alpha}\to\nu_{\beta}}=\delta_{\alpha\beta}-2\sum_{i>j}U_{\alpha
i}U_{\alpha j}U_{\beta i}U_{\beta j}$ (28)

here $U_{\alpha\beta}$ \- components of PMNS matrix (see Ref.[41]) with mixing
angles taken as best-fit values from Ref.[45]:

$\displaystyle\sin^{2}\theta_{12}=0.308$
$\displaystyle\sin^{2}\theta_{13}=0.023$
$\displaystyle\sin^{2}\theta_{23}=0.437$

In the case of MSW effect, the resulting $\bar{\nu}_{e}$ spectrum is given by
(c.f. [44]):

$F_{\bar{\nu}_{e}}=\bar{P}^{\text{MSW}}_{ee}F_{\bar{\nu}_{e}}^{0}+[1-\bar{P}^{\text{MSW}}_{ee}]F_{\bar{\nu}_{x}}^{0}$
(29)

where $F_{\bar{\nu}_{e}}^{0}$ – flux of the secondary $\bar{\nu}_{e}$,
$F_{\bar{\nu}_{x}}^{0}$ – flux of muon or tau-flavor secondary neutrinos.
Survival probability $\bar{P}^{\text{MSW}}_{ee}$ depends on the hierarchy of
neutrino masses: $\bar{P}^{\text{MSW}}_{ee}=\cos^{2}\theta_{12}\approx 0.692$
for NH and $\bar{P}^{\text{MSW}}_{ee}=0$ for IH. The example of the flux of
secondary electron anti-neutrino arriving at Earth from the SN at distance
$R_{\text{SN}}=10$ kpc after oscillations taken into account is shown in
Fig.5.

Figure 5: Flux of $\bar{\nu}_{e}$, that can be observed on Earth originating
from muons and pions decay channels, compared to a typical spectrum of SN
anti-neutrinos, approximated by Eqn. (30). Parameters of the dark photons are
$m_{A^{\prime}}=$340\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$,
$U^{2}=10^{-18}$. The blue line corresponds to anti-neutrinos, produced in
muon decays, while the red line corresponds to the anti-neutrinos from the
pion decays. The contribution to the total spectra from the secondary
neutrinos becomes dominant at energy
$E_{\bar{\nu}_{e}}\gtrsim$50\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$. The
contribution from non-thermalized muons (c.f.Section V.1.1) is ignored. The
dashed line corresponds to the total neutrino spectra.

Our resulting (anti)neutrino spectra should be compared to the SN neutrino
spectra. The latter can be described by the following fitting formula, [46]:

$\frac{dN_{\nu}}{dE}=L_{\nu}\frac{(1+\alpha)^{1+\alpha}}{\Gamma(1+\alpha)\braket{E_{\nu}}^{2}}\left(\frac{E}{\braket{E_{\nu}}}\right)^{\alpha}\text{Exp}\left[-(1+\alpha)\frac{E}{\braket{E_{\nu}}}\right]$
(30)

Here $L_{\nu}=0.5\cdot 10^{53}$ erg is the total emission energy in one
neutrino flavor, it is assumed to be the same for all flavors of
neutrinos/anti-neutrinos; $\braket{E_{\nu}}$ are $\braket{E_{\nu}^{2}}$ is the
average energy (energy squared) of the neutrinos; $\alpha$ is the “pinching”
parameter, that indicates how do the spectra differ from the Fermi-Dirac
distribution. It is related to the average energy and average squared energy
as:

$\frac{\braket{E_{\nu}^{2}}}{\braket{E_{\nu}}^{2}}=\frac{2+\alpha}{1+\alpha}$
(31)

The numeric values used to describe the SN neutrino spectrum are given in
Table 1. Finally, $\Gamma(\alpha)$ is the Euler’s Gamma function.

###  V.3 Detection of secondary neutrinos

| $\alpha$ |  $\braket{E_{\nu}}$, $[\text{MeV}]$
---|---|---
$\bar{\nu}_{e}$ | 2.78 | 12.69
$\nu_{e}$ | 2.9 | 10.14
$\nu_{x}/\bar{\nu}_{x}$ | 2.39 | 12.89
Table 1: Fitting parameters for the SN neutrinos spectra, Eqs. (30–31).

Figure 6: Constraints (95% CL) on dark photon parameters based on the non-
detection of high-energy neutrinos in the event of a supernova explosion.
_Left panel:_ The green-shaded region marks the parameter space where at least
three high-energy neutrino events are expected, assuming the dark photon’s
existence and normal hiererchy of neutrino masses. The difference between the
solid and thin dashed green lines shows the muon thermalization effect, with
the latter assuming no thermalization (see text for details). Thick green
dashed line corresponds for inverted hierarchy. The red shaded area indicates
constraints from SLAC’s E137 experiment [47]. Gray areas show dark photon
limits from SN1987A: darker gray is based on the energy loss arguments [18],
while the lighter gray comes from the absence of electromagnetic signal
fluctuations [17]. The reach of the future SHiP experiment is also shown for
comparison [48]. All SN results assume the Hyper-Kamiokande detector, a 10 kpc
supernova distance, and best-fit neutrino oscillation parameters for the
normal ordering scenario. _Right panel:_ dependence of the sensitivity region
on different assumptions (distance, mass ordering, detector type), as
indicated in the legend, normal mass ordering is assumed here.

Neutrinos from supernova explosions and from dark photon decays will be
detected by terrestrial experiments. The Hyper-Kamiokande, an underground
water Cherenkov detector with around 220 kton of ultra-pure water as its
detection volume, is the most promising in terms of expected event numbers. It
is anticipated to detect approximately $10^{5}$ events from the next supernova
in the galactic center. We will base most of our estimates on this detector.
Two other neutrino detectors of interest are DUNE and JUNO.444Their
characteristics, relevant for our analysis, are summarized in Table 2. The
former is the liquid argon time-projection chamber (LArTPC). It is designed to
have a superior energy resolution compared to water or organic-based
scintillators. The latter is a liquid scintillator detector. Both experiments
are expected to detect around $N_{SN}\simeq(5-10)\times 10^{3}$ events which
include not only electron flavor neutrinos but also neutral current events
from elastic scattering of neutrinos of other flavors. The dominant process
for each experiment is inverse beta decay (IBD).

  * •

For Super-K, Hyper-K and JUNO – the capture of electron anti-neutrino via

$\bar{\nu}_{e}+p\rightarrow e^{+}+n\,.$

  * •

For DUNE – the capture of electron neutrino via

$\nu_{e}+^{40}\text{Ar}\rightarrow e^{-}+^{40}K^{*}\,.$

The estimated number of events per electron (positron) recoil energy, $T_{e}$,
is given by:

$\frac{dN_{e}}{dT_{e}}=\frac{\epsilon\,N_{c}}{4\pi
R_{SN}^{2}}\int\limits_{E_{\rm
break}}^{\infty}dE\,\frac{dN_{\bar{\nu}}(E)}{dE}\frac{d\sigma_{\text{\sc
ibd}}(E,T_{e})}{dT_{e}}\,.$ (32)

Here $\epsilon$ is the detector efficiency; $N_{c}$ is the number of targets
in the detector555Protons in the case of Hyper-K and JUNO and Argon in the
case of DUNE are parameters of the experiment, see Table 2. The differential
cross-section for inverse beta-decay $d\sigma_{\text{\sc ibd}}/dT_{e}$ depends
both on both (anti)-neutrino energy and on the recoil energy $T_{e}$. The
integration over (anti)neutrino energies starts from $E_{\rm break}$ – the
energy above which the contribution of the secondary neutrinos becomes
dominant over the SN neutrinos and the observed spectrum exhibits a break
(c.f. Fig. 5 where the $E_{\rm
break}\approx$52\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$). The $R_{SN}$ is
the distance to the SN. Finally, $\frac{dN_{\nu}(E_{\nu})}{dE_{\nu}}$ is the
spectrum of (anti)neutrinos, as computed in Section V above. The total number
of events will be given by

$N_{\text{total}}=\int\limits_{E_{\rm
break}}^{\infty}dT_{e}\,\frac{dN_{e}}{dT_{e}}\,.$ (33)

Parameters | Hyper-K | Super-K | JUNO | DUNE
---|---|---|---|---
Fiducial volume | $187\text{\,}\mathrm{k}\mathrm{t}\mathrm{o}\mathrm{n}$ | $22.5\text{\,}\mathrm{k}\mathrm{t}\mathrm{o}\mathrm{n}$ | $20\text{\,}\mathrm{k}\mathrm{t}\mathrm{o}\mathrm{n}$ | $40\text{\,}\mathrm{k}\mathrm{t}\mathrm{o}\mathrm{n}$
Efficiency, $\epsilon$ | 0.67 | 0.67 | 0.8 | 0.86
Number of targets, $N_{c}$ | 2.5 $\cdot 10^{34}$ | 1.5 $\cdot 10^{33}$ | 1.2 $\cdot 10^{33}$ | 6 $\cdot 10^{32}$
Refs. | [32] | [49] | [33] | [34]
Table 2: Characteristics of the neutrino detectors, needed to estimate the
number of detected events (see Eq. (32–33)): the detector efficiency
$\epsilon$ and the number of targets, $N_{c}$.

##  VI Results

We express our results as the region in the dark photon parameter space that
can be explored should a supernova explosion be registered by terrestrial
neutrino detectors. Figure 6 (left panel) shows the sensitivity of the Hyper-
Kamiokande experiment. The green shaded area is the region of the parameters
space, where one expects detection of $N_{\text{total}}\geq 3$ high-energy
neutrinos. As the relevant range of neutrino energies
($50\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}100\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$)
is essentially background free,666At these energies, the influence of
supernova neutrinos becomes negligible. Consequently, atmospheric neutrinos
are the sole source of background, with an occurrence rate of a few dozen
events annually [24]. In contrast, we anticipate a sizeable number of events
occurring within a timeframe of approximately 10 sec. such an estimate
corresponds to the 95% confidence level. The upper boundary of the region is
set by the trapping effect. For higher coupling constants, dark photons decay
inside the supernova core, and their decay products thermalize without
influencing the signal. The vertical boundary on the left indicates the mass
threshold $m_{A^{\prime}}\geq 2m_{\mu}$, below which DPs decay to $e^{-}e^{+}$
pairs, which could impact the supernova electromagnetic signal (c.f. [17]).

The neutrino signal of interest primarily arises from the decay of
“thermalized” muons, which have dissipated their energy through
electromagnetic interactions with the plasma, as elaborated in Section IV. The
thermalization makes our findings independent of the temperature and density
profiles outside the SN core. Such profiles have never been measured and bear
large theoretical uncertainties (see e.g. discussion in [50]). The fact that
most of the muons have thermalized, makes our prediction dependent _only_ on
the total number of produced dark photons inside the SN core, but not on the
details of the supernovae’ interior. Only a small “bump” at masses
$m_{A^{\prime}}\sim$250\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$ and
$U^{2}\sim 10^{-22}-10^{-21}$ comes from the remaining “non-thermalized” muons
(notice that this “bump” grows into a noticeable region for a nearby supernova
at distance of 1 kpc, Figure 6, right panel). Additionally, we examine the
scenario where muons do not undergo thermalization (represented by the green
dashed line). This contour is largely model-dependent, serving to highlight
the uncertainty involved in defining the boundary of the green region. All the
results on the left assume the 10 kpc distance to the SN and best-fit neutrino
parameters for the normal mass ordering.

In the right panel of Figure 6, we demonstrate the sensitivity of our results
to these assumptions, by varying the distance, and the type of the experiment
and changing from normal to inverted mass ordering. All limits are 95% CL.

The left panel also includes a comparison of our results to both past and
anticipated experiments. The results we have obtained occupy a region of the
dark photon parameter space that has not been previously explored, nor is it
expected to be covered in the near future. The gray shaded areas in the left
panel represent other supernova constraints: those based on arguments
regarding supernova energy loss [18, 20, 17], as well as constraints from the
electromagnetic decays of DPs, which lead to fluctuations in the photon signal
[17]. Additionally, we show the coverage by relevant particle physics
experiments: the SLAC Beam Dump Experiment E137 [47], and the expected
sensitivities from the forthcoming SHiP experiment [48].

##  VII Conclusion and discussion

Supernova interiors are considered potential birthplaces for hypothetical
feebly interacting particles (axions, dark photons, sterile neutrinos, etc),
which can have masses up to several hundred MeVs, see e.g. [6]. In the past,
SN constraints have been mostly driven by the energy-loss arguments that aim
to restrict the energy emission rate from the SN via the FIP channel to limit
the potential shortening of active neutrino emission (see e.g. [7]). However,
the neutrino signal from a supernova contains more information (a form of the
spectrum, duration of the burst, etc) that can also be used to provide
constraints on FIP parameters.

In this study, we investigated the generation of dark photons in supernovae
and followed their subsequent escape from the supernova core and decay into
secondary particles (muons and pions). The latter subsequently decay into
neutrinos. This sequence of decays, with neutrinos as tertiary particles, is a
novel area of exploration in the literature. We show that these neutrinos
manifest as a high-energy “tail” in the neutrino distribution, making them
distinctly identifiable. Furthermore, such neutrino events occur in a part of
the spectrum that is largely free from any background neutrino events, thereby
facilitating their detection.

Consequently, a supernova explosion within our Galaxy could enable the probing
of dark photon parameters using detectors such as Hyper-Kamiokande, DUNE, and
to a lesser extent, Super-Kamiokande and JUNO. For supernovae occurring near
the Galactic Center, the derived constraints would mainly pertain to neutrinos
from the decays of thermalized muons, providing constraints that are largely
independent of the supernova’s temperature and density profiles and thus are
robust. The contribution of neutrinos from non-thermalized muons (or from pion
decay products) is minor at distances equivalent to the Galactic Center but
becomes significant for supernova explosions much closer than the Galactic
Center, offering the potential to explore an even broader portion of the
parameter space. We demonstrate that such an analysis would allow us to
explore the region of dark photon’s masses up to $m_{A^{\prime}}=500$ MeV and
values of mixing parameter $U^{2}\sim 10^{-21}-10^{-16}$ which makes our
result complementary for the future SHiP experiment and the excessive gamma-
emission constraints, covering previously unexplored region. The developed
novel approach will allow for the study of other short-living FIP species,
that can be produced in the supernova environment and subsequently decay into
neutrinos. In the case of a galactic center supernova, it can provide higher
sensitivity to the parameters of the FIP and less model dependence than the
scenario of energy-loss/energy-transfer constraint.

##  Appendix A Distribution function of the decay products

Sections III and V of this paper use the formula for the transformation
between the phase-space distribution function of the parent particles in
laboratory frame and the corresponding distribution function of one of the
decay products. Such a relation is given by the following integral (see e.g.
[31, 24]):

$\frac{dN_{x}}{dE}=\int d(\cos\theta)\int
dE_{p}\frac{1}{\gamma(1+\beta\cos\theta)}\frac{dN_{p}}{dE_{p}}f_{\star}\left(E_{\star},\cos\theta\right)$
(34)

Here $dN_{p}/dE_{p}$ is the distribution function of the parent particle in
the laboratory frame, $\gamma$ is the Lorentz gamma factor for transforming
between the rest and laboratory frames, with

$\gamma=\frac{1}{\sqrt{1-\beta^{2}}},$ (35)

and $E_{\star}$ is the energy of the decay product in the rest frame of the
parent particle, corresponding to the energy $E$ in the laboratory frame, such
that

$E=\gamma(1+\beta\cos\theta)E_{\star}$ (36)

. In the case of the two-body decays, the distribution function of the decay
product $x$ does not depend on the angle $\theta$ and is given by the
$\delta$-function:

$f_{\star}\left(E_{\star},\cos\theta\right)=\frac{1}{2}\delta(E_{\star}-\bar{E})$
(37)

where $\bar{E}$ is the energy of the decay product $x$ in the rest frame of
the parent particle. Using $E_{\star}$ as a variable instead of $\cos\theta$

$d\cos\theta=-\frac{1}{\beta\gamma}\frac{E}{E_{\star}^{2}}dE_{\star}$ (38)

we can remove the delta function and one integration resulting in Eqn 11.
Since the delta function does not always have a solution, there appears a
minimum value of the energy of the parent particle that can produce a decay
product with given energy $E$. It corresponds to the case $\cos\theta=1$ and
appears as a solution of

$E=\frac{E_{p}+\sqrt{E_{p}^{2}-m_{p}^{2}}}{m_{p}}\bar{E}$ (39)

with respect to $E_{p}$, where $m_{p}$ \- is the mass of the parent particle.
In the case of 3-body decay, we can not simplify the integration and have to
take both integrals.

##  Appendix B Integrands for NP and PP processes

Following the [20], introduce variables:

$x=\frac{E_{A}}{T};~{}~{}~{}y=\frac{m_{\pi}^{2}}{m_{N}T};~{}~{}~{}q=\frac{m_{A^{\prime}}}{T};~{}~{}~{}z=cos(\theta_{ij})$
(40)

where $E_{A}$ \- the energy of the dark photon and $\theta_{ij}$ \- scattering
angle between nuclei in the center of the mass frame. In those terms, the
energy emission integral can be simplified (using the approximation
$E_{A},T\ll m_{N}$). PN-process integrand can be given as:

$\mathcal{I}_{pn}=\sum_{j}C_{j}\mathcal{I}_{j}$ (41)

where the components $\mathcal{I_{j}}$ and $C_{j}$ are:

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\frac{(u+v-2z\sqrt{uv})^{3}}{(u+v-2z\sqrt{uv}+y)^{2}}$ (42)
$\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\frac{(u+v-2z\sqrt{uv})(u+v+2z\sqrt{uv})^{2}}{(u+v+2z\sqrt{uv}+y)^{2}}$
(43) $\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle\frac{(u+v-2z\sqrt{uv})(-u^{2}-v^{2}+(6-z^{2})uv)}{(u+v+y)^{2}-4z^{2}uv}$
(44) $\displaystyle I_{4}$ $\displaystyle=$ $\displaystyle
x\frac{-u^{2}-v^{2}+(6-4z^{2})uv)}{(u+v+y)^{2}-4z^{2}uv}\frac{u-v}{u+v+y+2z\sqrt{uv}-\frac{T}{m_{N}}q^{2}}$
(45) $\displaystyle I_{5}$ $\displaystyle=$ $\displaystyle
x\frac{(u+v+2z\sqrt{uv})^{2}}{(u+v+2z\sqrt{uv}+y)^{2}}\frac{u-v}{u+v+y+2z\sqrt{uv}-\frac{T}{m_{N}}q^{2}}$
(46) $\displaystyle I_{6}$ $\displaystyle=$ $\displaystyle
x^{2}\frac{(u+v+2z\sqrt{uv})^{3}}{(u+v+2z\sqrt{uv}+y)^{2}}\frac{1}{(u+v+y+2z\sqrt{uv}-\frac{T}{m_{N}}q^{2})^{2}}$
(47) $\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle 1$ (48)
$\displaystyle C_{2}$ $\displaystyle=$ $\displaystyle
4\left[1+\frac{6x^{2}-4x(u-v)+2q^{2}}{(u+v)^{2}-4z^{2}uv}\right]$ (49)
$\displaystyle C_{3}$ $\displaystyle=$
$\displaystyle-2\left[1+\frac{2x(u-v)}{-u^{2}-v^{2}+(6-4z^{2})uv}\right]$ (50)
$\displaystyle C_{4}$ $\displaystyle=$ $\displaystyle-4$ (51) $\displaystyle
C_{5}$ $\displaystyle=$ $\displaystyle-16$ (52) $\displaystyle C_{6}$
$\displaystyle=$ $\displaystyle 16$ (53)

In similar terms, the PP-integrand after the approximation and reduction will
be:

$\mathcal{I}_{pp}=\frac{(u+v-2z\sqrt{uv})^{3}}{2(u+v-2z\sqrt{uv}+y)^{2}}$ (54)

##  Appendix C Thermalization in $e\mu$ and $ep$ scatterings

The squared element for the electron muon scattering [51] is given by

$\frac{1}{4}\sum_{\text{spins}}|\mathcal{M}|^{2}=\frac{2e^{4}}{t^{2}}\left[u^{2}+s^{2}+4t(m_{e}^{2}+m_{\mu}^{2})-2(m_{e}^{2}+m_{\mu}^{2})^{2}\right]$
(55)

where s,t, and u are Mandelstamm variables. The differential cross-section in
the lab frame:

$\frac{d\sigma}{d\Omega}=\frac{1}{64\pi^{2}}\frac{1}{|\vec{p_{\mu}}|m_{e}}\frac{1}{4}\sum_{\text{spins}}|\mathcal{M}|^{2}\frac{|\vec{p^{\prime}_{\mu}}|}{E_{\mu}+m_{e}-\frac{|\vec{p_{\mu}}|E_{\mu}^{\prime}}{|\vec{p^{\prime}_{\mu}}|}\cos\theta}$
(56)

where $\vec{p_{\mu}},\vec{p^{\prime}_{\mu}}$ \- 3-momenta of muon before and
after the scattering, $E_{\mu},E_{\mu}^{\prime}$ \- energies of muon. Value of
$|\vec{p^{\prime}_{\mu}}|$ is found from energy-momentum conservation:

$E_{\mu}+m_{e}=E_{\mu}^{\prime}+\sqrt{|\vec{p_{\mu}}|^{2}-2|\vec{p_{\mu}}||\vec{p^{\prime}_{\mu}}|\cos\theta+|\vec{p^{\prime}_{\mu}}|^{2}+me^{2}}$
(57)

the total cross-section requires regularization as it diverges at $\theta\to
0$. For simplicity of the estimates we use a relation between the impact
parameter and the scattering angle in a classical case

$b=\frac{\alpha}{2E_{\mu}}\sqrt{\frac{1+\cos\theta}{1-\cos\theta}}$ (58)

and use the maximum value of the impact parameter as $b_{max}=n_{e}^{-1/3}$
where $n_{e}$ \- is the number density of electrons. Since we are interested
in the energy-loss rate, the actual cutoff will not change the result
significantly. The value of the averaged energy loss per reaction is given in
Fig.7

Figure 7: Left panel: Relative energy loss $\frac{\Delta E_{\mu}}{E_{\mu}}$ of
muon in electron-muon scatterings averaged with differential cross-section.
Typical muons from dark photons decays will lose $\approx 2-3\%$ of their
energy in each scattering, which is higher, than in the case of Thomson
scattering where their energy loss at relevant SN radius with $T\sim 0.1$ MeV
is an order of magnitude less.

In the case of point-like protons, the mass of the electron in the above
expression has to be replaced with the mass of a proton. The energy-loss rate
on protons is significantly lower. the comparison between energy-loss on
protons and electrons is presented in 8, where we plot a ratio:

$R_{\text{therm}}=\frac{\sigma_{p\mu}\Delta
E_{\mu}^{p\mu}}{\sigma_{e\mu}\Delta E_{\mu}^{e\mu}}$

where indices $e\mu,p\mu$ correspond to the process with a given cross-section
$\sigma$ and energy loss $\Delta E$, averaged over angles. Since the number
densities of protons and electrons are equal or very close in a major part of
the SN due to electroneutrality, this ratio will also show the ratio between
energy-loss rates in those two processes.

Figure 8: The ratio of muon energy loss rates in reactions with protons to the
same process, but with electrons. The difference is at the level of several
orders of magnitude, therefore we do not need to improve the calculations,
considering the proton as a non-point-like particle as it will not change the
outcome significantly and we can conclude that muon-electron interaction is
the dominant process for the muon thermalization.

##  Appendix D Supernova simulation profiles

To estimate the thermalization effects we used SN simulation data for a
specific scenario of 8.8 $M_{\odot}$ simulation. The non-thermalized part of
the final result remains model-dependent. Examples of the temperature,
density, and electron fraction profiles are given in Fig.9

Figure 9: Temperature, density, and electron fraction $Y_{e}=n_{e}/n_{B}$
profiles in the reference model used to estimate the thermalization effects -
[38, 39].

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# ADCNet: End-to-end perception with raw radar ADC data

Bo Yang Ishan Khatri Michael Happold Chulong Chen
Motional
{bo.yang, ishan.khatri, michael.happold<EMAIL_ADDRESS>
###### Abstract

There is a renewed interest in radar sensors in the autonomous driving
industry. As a relatively mature technology, radars have seen steady
improvement over the last few years, making them an appealing alternative or
complement to the commonly used LiDARs. An emerging trend is to leverage rich,
low-level radar data for perception. In this work we push this trend to the
extreme – we propose a method to perform end-to-end learning on the raw radar
analog-to-digital (ADC) data. Specifically, we design a learnable signal
processing module inside the neural network, and a pre-training method guided
by traditional signal processing algorithms. Experiment results corroborate
the overall efficacy of the end-to-end learning method, while an ablation
study validates the effectiveness of our individual innovations.

##  1 Introduction

Figure 1: ADCNet prediction samples. The camera images are provided for
reference only. The Range-Azimuth map is generated by the RADIal SDK. The
small red squares in the RA image denote ground truth, while the blue squares
denote network predictions

Perception systems for autonomous driving have been undergoing fast progress
over the last decade. This is well-motivated, as the perception system
provides input for subsequent systems such as motion prediction and planning.
Accurate perception of the surroundings is thus paramount for the operation of
an autonomous vehicle (AV).

Compared to LiDAR, radar has been under-investigated for AV perception tasks.
Indeed, much of perception system’s progress is built upon LiDAR sensors.
LiDAR sensors provide accurate measurements in common driving scenarios,
particularly those with sunny and dry weather. However, the shortcomings of
LiDAR cannot be overlooked. Among other things, LiDAR sensors are less
physically robust and much more costly than radars, and LiDARs are usually not
robust against adverse weather, such as rain and fog [14]. The hefty cost of
LiDARs undermines the cost-effectiveness of AVs, while non-robustness poses
safety risks.

While radars have long been applied in automotive applications such as blind
spot monitoring and adaptive cruise control, their application in assisting
semantic understanding of the scene have been limited. This is due to the fact
that traditional automotive radars mostly work at 24GHz, with a limited usable
frequency bandwidth of 200MHz [27], and even this usable band is expiring [1].
This small frequency band limits the attainable range resolution, and the low
carrier frequency limits attainable angle resolution. Together these factors
render traditional radars unsuitable for AVs, as they usually require high
resolution sensing of the environment.

In the past few years, imaging radars for autonomous driving started to
emerge. Compared to traditional automotive radars, imaging radars usually
operate at around 77GHz, with roughly 4GHz available bandwidth – a much larger
band than that of traditional automotive radars. In addition, these imaging
radars are often equipped with multiple transmit and receiving antenna.
Utilizing Multiple-Input-Multiple-Output (MIMO) technology [28], these new
radars achieve much improved range and angle resolution.

Capitalizing on this improved sensing capability of imaging radar, several
works [19, 12, 36, 17, 26, 16, 24] have attempted building perception models
on these radars. One common theme of these works is that perception is done
with low-level radar data, instead of the radar detection data as in
traditional automotive radars. Low-level radar data refers to any radar sensor
data that have not gone through the peak detection & thresholding step of the
signal processing chain (see Figure 2). There are several low-level radar data
representations: range-doppler (RD) radar cube, range-azimuth-doppler (RAD)
cube, etc., that have gone through various stages of signal processing, but
not the final peak detection step that leads to very sparse radar detections.

In this work, we explore the possibility of end-to-end perception from raw
radar data. That is, the perception model is built upon the ADC data, without
any pre-defined signal processing steps. Although signal processing steps,
such as applying discrete Fourier transform (DFT) and windowing functions, are
well-motivated, they could also introduce signal model mismatch that hurts
final perception performance. On the other hand, radar ADC data is quite
intricate, where waveform design (e.g. chirp firing sequence, time-division,
code-division, and Doppler-division multiplexing, etc.) choices could make
unraveling the data challenging. In other words, it remains an open question
whether a system learned directly on ADC data alone can produce competitive or
even state-of-the-art results.

To the best of our knowledge, this work provides the first affirmative answer.
In order to circumvent the intricacy of radar ADC data, we develop several
techniques, specifically in model architecture and training method. At a high
level, our contributions are

  * •

We showcase that better perception performance can be obtained using radar ADC
data;

  * •

We design a model architecture that effectively incorporates trainable signal
processing modules into the neural network such that end-to-end learning
starts from a good starting point;

  * •

We design a signal processing (SP) guided pre-training method that assists in
learning a powerful representation with unlabelled data. To the best of our
knowledge, this is the first pre-training method design for low-level radar
data.

To clarify, while we devise an end-to-end learning-based approach on raw ADC
data, signal processing techniques are exploited on various aspects of the
model architecture and training method. We view this as a happy marriage of a
traditional, theory-based paradigm with a data-driven, learning-based
paradigm. We show that superior performance can be achieved when wisdom from
both worlds is judiciously employed.

##  2 Background on radar signal processing

This work is mainly concerned with a type of automotive radar termed Frequency
Modulated Continuous Wave (FMCW) radar, see e.g. [3]. At each measuring cycle,
these radars send out a series of rapid “chirps” – short waves with increasing
frequency. At the receiving antenna, the reflections are captured, and sampled
by an ADC device. This digitized signal is then passed to other software
modules in the radar sensor for signal processing. A typical radar signal
processing chain is shown in Figure 2. The readers are referred to [30] for a
more in-depth coverage of radar basics.

Figure 2: A simplified radar signal processing chain Figure 3: Illustration of
ADC and RD arrays. Left: ADC, Right: RD

The ADC data is usually organized as arrays as shown in Figure 3. Each frontal
slab has dimension (number of samples per chirp, number of chirps), and holds
all the samples for one receiving antenna. Signal processing operations
(mainly DFTs) transform these two dimensions into range and Doppler
dimensions, forming the RD array, as shown in the right part of Figure 3.

##  3 Related works

Radars for AV perception. There has been a considerable amount of work
utilizing radar detections for AV perception [38, 33, 20]. In these papers,
the radar detections are usually fused with another sensor modality, such as
camera and LiDAR. The main reason is that radar detections are very sparse,
and reliable semantic understanding of the driving scene is hard to achieve
with radar alone. In [38], the authors propose a fusion scheme that combines
radar detections and LiDAR point clouds, and obtain improved performance for
object detection and velocity estimation. In [33], the authors combine radar
detections and LiDAR point clouds and achieve improved performance for
trajectory prediction, compared to using LiDAR alone.

There is a growing body of work attacking the AV perception problem by means
of low-level radar data [19, 12, 36, 17, 26, 16, 24]. CramNet [12] proposes a
method for fusing radar Range-Azimuth (RA) images with camera images using the
attention mechanism. In [19], a graph convolution network is developed to work
with radar RA data. In [26], a radar-LiDAR fusion method is developed to
improve perception robustness against adverse weather. In [17], the authors
propose a method to cope with the missing modality problem for a radar-LiDAR
fusion system.

It is also worth highlighting the low-level radar datasets that have been
facilitating the research in this direction. These include RADIATE [34],
RADIal [29], ORR [2], CRUW [35], while new ones (e.g. K-Radar[23]) continue to
emerge. Different datasets often use different radar sensors, so their radar
data representations tend to be different. For example, RADIATE provides radar
radio-frequency (RF) images, ORR provides radar intensity maps, CRUW provies
RA maps, while RADIal provides both ADC and Range-Doppler (RD) data. Since we
are interested in exploring ADC data, we adopt the RADIal dataset to conduct
our experiments, as among the currently ready-to-use radar datasets, RADIal is
the only one that provides radar ADC data.

Pre-training Pre-training has been proven quite successful in several
important domains, such as computer vision [22, 10, 5], natural language
understanding [8, 4], speech and audio processing [6, 15, 41], etc. These
methods usually entail designing a surrogate learning task, such as predicting
masked part of the input signal, so that the neural network learns intrinsic
structure of the data. While deceptively simplistic, many works in this genre
have shown that powerful representations, that support impressive performance
on downstream tasks (such as image classification, question answering with
natural language, automatic speech recognition), can be learned from a large
collection of data.

End-to-end learning. End-to-end learning from raw sensor data has been gaining
traction in several fields. In these works, traditional feature
representations, which are usually designed with SP techniques, or handcrafted
with human intuition, are replaced with carefully designed neural networks. In
audio and speech processing, deep learning from the raw audio waveform has
shown impressive performance for several important tasks, such as music
analysis [9], phone classification and speaker recognition [22], speech
recognition [32] etc. In computer vision, learning from raw pixel values has
been the mainstream paradigm, after [13] showed impressive performance on the
difficult ImageNet dataset [7].

Figure 4: Two data samples from the RADIal dataset. Top row: camera images,
Middle row: radar Range-Azimuth map, Bottom row: radar Range-Doppler map.
Figure 5: The ADCNet model. A learnable signal processing module, as shown in
the dashed box, is introduced within the network to enable end-to-end
learning. The backbone and head networks are adopted from [29] for comparison,
while other types of backbone and head networks can be easily incorporated.

##  4 The RADIal dataset

To assist discussion, we provide a brief introduction to the RADIal [29]
dataset, and readers are referred to the original publication for more
details.

The RADIal dataset provides synchronized radar, LiDAR and camera images for
about 2 hours of driving. Labels for vehicle detection and freespace
segmentation are also provided. In this work, we are interested in the radar
data. For this, the dataset provides ADC data and RD data. The RADIal SDK
111https://github.com/valeoai/RADIal also enables generating RAD maps from the
ADC data. Figure 4 shows two samples from the dataset.

The radar sensor used in the RADIal dataset is an imaging radar with 12
transmit antennas and 16 receiving antennas. Doppler division multipliexing
(see e.g. [37]) is employed in the radar waveform design – this can be seen
from Figure 4, where a single target produces multiple equal-spaced bright
spots in the RD map. The radar is mounted at the front of the vehicle, and has
an approximate field-of-view (FOV) of 180 degrees, and a range of around 100
meters.

The RADIal dataset contains a relatively balanced distribution of driving
scenes in city, country side, and highway. For the object detection task and
the freespace segmentation task, the labels are generated by models running on
images and LiDAR, followed by human verification. Only vehicles are labeled
for object detection.

Note that, out of the total 25,000 synced data frames, only about 8300 frames
have objects and segmentation label. In Section 5.3 we will propose a method
to utilize the unlabeled dataset for pre-training and learn a meaningful
representation.

Figure 6: The signal processing steps for generating RD map in the RADIal
dataset

##  5 The ADCNet

In this section, we present our novel techniques that enable learning from
radar ADC data.

Figure 7: A learnable DFT layer. The dashed box is shared. The two modules
inside the dashed box represent parameter matrices of the neural network, and
are initialized as (perturbed) real and imaginary part of the DFT matrix.

###  5.1 A trainable DSP module

In [29], the authors presented a model termed FFT-RadNet. This model learns to
perform object detection and freespace segmentation, using radar RD data. For
the RADIal dataset, the RD data is generated from ADC data by applying DFT and
windowing functions, as commonly done in radar signal processing. The specific
signal processing steps for the RADIal dataset are summarized in Figure 6.

We hypothesize that end-to-end learning from radar ADC data needs to start
from signal processing. That is, we need to include layers that mimics signal
processing steps in the early part of the neural network, while these SP-like
layers will be trained end-to-end together with other layers. In Section 6.5,
we detail experiments to examine this hypothesis.

Following this intuition, we design an SP module inside the proposed ADCNet,
as shown in Figure 5. The complex DFT operations are implemented with real
weight matrices as shown in Figure 7. At initialization, this module mimics
the signal processing operations of RADIal as shown in Figure 6. If we
initialize the module to be the exact signal processing chain used in FFT-
RadNet, then the ADCNet is the same as FFT-RadNet at initialization, while the
SP module is left trainable during the training process.

However, we suspect that initializing the SP module exactly as the signal
processing chain of FFT-RadNet can hurt performance, since the training
process may fail to drive the weights away from the local minima to which they
are initialized. To alleviate the issue, we developed a perturbed-DFT
initialization strategy:

$\displaystyle\mathbf{M}=\mathbf{M}_{\text{DFT}}+\mathcal{N}(0,\gamma)$ (1)

In Eq. 1, $\mathbf{M}_{\text{DFT}}$ denotes the DFT matrix, and
$\mathcal{N}(0,\gamma)$ denotes a random Gaussion matrix (with the same shape
as the DFT matrix), with i.i.d. elements each has mean 0, and variance
$\gamma$. Intuitively, this $\gamma$ parameter should be small, such that the
final matrix $\mathbf{M}$ used for initializing the SP module is perturbed,
but still resemble the DFT matrix. In our experiments, the $\gamma$ is set to
0.1. The effect of this perturbation is examined in Section 6.5

###  5.2 Supervised model training

To facilitate comparison with FFT-RadNet, we adopt the same multi-task
learning setup. Specifically, the ADCNet is trained to perform both object
detection and freespace segmentation at the same time.

For model training, the object detection part of the loss is

$\displaystyle\mathcal{L}_{\text{det}}=\text{focal}(\mathbf{y}_{\text{cls}},\hat{\mathbf{y}}_{\text{cls}})+\alpha\text{smooth-L1}(\mathbf{y}_{\text{reg}},\hat{\mathbf{y}}_{\text{reg}}),$
(2)

where $\mathbf{y}_{\text{cls}}$ denotes the ground-truth classification labels
on the feature map, with $1$ indicates presence of object while $0$ otherwise.
The shape of $\mathbf{y}_{\text{cls}}$ is $N_{\text{range-bins}}\times
N_{\text{azimuth-bins}}$.The $\mathbf{y}_{\text{reg}}$ denotes the regression
targets. Following [29], the network predicts a range and an azimuth value.
The regression targets are thus the remainders of range and azimuth modulo
range and azimuth bin sizes. The shape of $\mathbf{y}_{\text{reg}}$ is
$N_{\text{range-bins}}\times N_{\text{azimuth-bins}}\times 2$, where the last
dimension corresponds to range and azimuth. The $\text{focal}()$ function is
from [18], and the $\text{smooth-L1}()$ function is also known as Huber loss
[11]. The $\alpha$ is a hyper-parameter balancing the two parts.

The freespace segmentation part of the loss is

$\displaystyle\mathcal{L}_{\text{seg}}=\sum_{r,a}\text{BCE}(\mathbf{y}_{\text{seg}}(r,a),\hat{\mathbf{y}}_{\text{seg}}(r,a)),$
(3)

where $\mathbf{y}_{\text{seg}}(r,a)$ denotes the ground truth at location
$(r,a)$, with $1$ denotes object presence and $0$ otherwise. The
$\text{BCE}()$ term denotes the binary cross entropy loss.

Combining the two parts, the loss function for training is thus

$\displaystyle\mathcal{L}=\mathcal{L}_{\text{det}}+\beta\mathcal{L}_{\text{seg}},$
(4)

where $\beta$ is a hyper-parameter controlling the weights on the freespace
segmentation task.

###  5.3 A signal processing-guided pre-training scheme

As discussed in Section 4, only a small part of all the frames from RADIal are
labeled. This is likely to hold true for other (future) radar datasets as
well, since collection of data samples is relatively easy while obtaining
high-quality labels is difficult and costly. This is especially true with raw
radar data as shown in [23] since radar data tends to be less human
interpretable and requires sensor fusion techniques for accurate 3D labels. We
envision that leveraging a large collection of unlabeled data will help with
end-to-end learning. This is because learning from ADC data directly means
disregarding important prior information (e.g. the fact that fast-time
frequency correspond to target range, and slow-time frequency correspond to
relative radial velocity), in order to leave larger parameter space for data-
driven learning.

While exposing the model to a larger dataset is a good motivation, the
proposed pre-training method is further designed to exploit advanced
techniques in SP. In particular, we would like to pre-train the network such
that it learns to estimate angle-of-arrival (AoA) of targets. Together with
estimated range and Doppler information, we would like to get the Range-
Azimuth-Doppler (RAD) cube from the radar ADC data. This is because, as
spatial information, range and azimuth help localize the object in the scene,
while Doppler information is helpful distinguishing moving and static objects.
Such a data representation should be quite useful for downstream perception
tasks.

Learning from the time-tested SP algorithms is well motivated. A considerable
amount of effort has been devoted to study the angle estimation problem (core
step in generating the RAD map) in the SP community, see e.g. [40, 31, 25].
These algorithms are designed to achieve some desirable properties, such as
noise robustness, outlier rejection, and interference mitigation capability.
Intuitively, many of these algorithms can be a good fit to generate RAD, and
serve as the training signal to train the trunk of the neural network.

As such, the proposed pre-training method consists of two main steps:

  1. 1.

Using an SP algorithm, generating RAD from all the ADC data;

  2. 2.

Using the collected (ADC, RAD) pairs as input and (pseudo) labels to train the
trunk network in a supervised learning fashion.

Unlike unsupervised pre-training techniques, which usually ask the neural
network to predict missing parts of the signal (e.g. predictive coding [22] in
the audio domain), or to differentiate “real” and “fake” samples as in the
contrastive learning regime, the proposed method is designed to learn from an
algorithm. That is, we task a neural network to learn the mapping between ADC
and RAD, where the RAD is generated by a classical SP algorithm.

Specifically, using the RADIal SDK, we generate RAD map for all the available
ADC data. As such, we get a collection of (ADC, RAD) pairs. We use these pairs
to pre-train the SP module and backbone network, with the following loss

$\displaystyle\mathcal{L}=\text{smooth-L1}(\mathbf{Y}_{\text{RAD}}-\hat{\mathbf{Y}}_{\text{RAD}}),$
(5)

where $\mathbf{Y}_{\text{RAD}}$ denotes the generated RAD tensor and
$\hat{\mathbf{Y}}_{\text{RAD}}$ denotes the prediction of the neural network.
From the RADIal SDK, the generated RAD tensor is of shape $512\times 751\times
256$. We downsample the tensor to have shape $128\times 248\times 256$ for
easier training. After pre-training the trunk of the ADCNet, we proceed to
fine-tune the network with the multi-task setting as describe in Section 5.2.

Table 1: Comparing ADCNet with baselines. AP: Average Precision, AR: Average
Recall, RE: Range Error on detected objects, AE: Azimuth Error on detected
objects. ADCNet-PTFT: ADCNet with pre-training and fine-tuning. The full
testset is broken down into an Easy and a Hard subset for detailed comparison.
*: results cited from [29].

|  | All | Easy | Hard
---|---|---|---|---
| Radar Input | F1 | AP | AR | RE (m) | AE (°) | F1 | AP | AR | RE (m) | AE (°) | F1 | AP | AR | RE (m) | AE (°)
FFT-RadNet * | RD | 0.89 | 0.97 | 0.82 | 0.11 | 0.17 | 0.95 | 0.98 | 0.92 | 0.10 | 0.13 | 0.76 | 0.93 | 0.65 | 0.13 | 0.26
Pixor - PC * | Point cloud | 0.48 | 0.96 | 0.32 | 0.17 | 0.25 | 0.45 | 0.99 | 0.29 | 0.15 | 0.19 | 0.55 | 0.93 | 0.39 | 0.19 | 0.33
Pixor - RA * | RA | 0.89 | 0.97 | 0.82 | 0.10 | 0.20 | 0.92 | 0.97 | 0.88 | 0.09 | 0.16 | 0.81 | 0.96 | 0.70 | 0.12 | 0.27
FFT-RadNet-optimized | RD | 0.90 | 0.92 | 0.89 | 0.15 | 0.11 | 0.96 | 0.95 | 0.97 | 0.14 | 0.10 | 0.80 | 0.87 | 0.74 | 0.17 | 0.13
ADCNet | ADC | 0.91 | 0.96 | 0.87 | 0.12 | 0.10 | 0.97 | 0.98 | 0.97 | 0.11 | 0.10 | 0.78 | 0.91 | 0.68 | 0.16 | 0.12
ADCNet-PTFT | ADC | 0.92 | 0.95 | 0.89 | 0.13 | 0.11 | 0.97 | 0.96 | 0.98 | 0.12 | 0.11 | 0.81 | 0.91 | 0.73 | 0.16 | 0.12

Table 2: Freespace segmentation comparison. The performance is measured by mIOU, the higher the value the better. *: results cited from [29]. | All | Easy | Hard
---|---|---|---
FFT-RadNet * | 74.00% | 74.60% | 72.30%
PolarNet [21] * | 60.6% | 61.9% | 57.4%
FFT-RadNet-optimized | 76.74% | 77.83% | 73.96%
ADCNet | 74.45% | 75.85% | 70.89%
ADCNet-PTFT |  78.59% |  79.63% |  75.90%

##  6 Experiment results

In this section, we detail experiment results to verify the effectiveness of
the proposed techniques. We present an overall comparison in Section 6.3,
while design and present ablation studies in the following subsections.

###  6.1 Baseline methods

For the proposed method, the most relevant baseline to compare against is FFT-
RadNet. This is because the FFT-RadNet method is specifically designed for the
RADIal dataset: a MIMO pre-encoder module (a dilated convolution layer) is
designed to fully exploit the Doppler division multiplexing (DDM) scheme used
on this radar, while other methods such as those in [19, 12, 36, 17, 26, 16,
24] are designed for other types of radar data, such as radar intensity map,
radar radio frequency image, etc., so they are unlikely to be competitive on
the RADIal dataset without major changes. Futhermore, to highlight the effects
of our novel techniques, the proposed ADCNet is following FFT-RadNet, with a
novel learnable SP module incorporated into the neural network. We emphasize
that the proposed techniques can be easily combined with other type of
backbone and head networks, while the ADCNet present in this work is
intentionally kept simple and similar to FFT-RadNet, for the purpose of
comparison.

The Pixor [39] method is also included for comparison as the detection head
used in Section 5.2 is inspired from it. For this method, two different radar
data representations are used: point cloud and RA map. For the point cloud-
based method, the point cloud are voxelized into a 3D volume before feeding
into the network; for the RA-based method, the RA input is formed by
collapsing (summing over) the Doppler dimension of the RAD tensor.

In order to ensure a fair comparison, we first optimize the hyper-parameters
for FFT-RadNet. Surprisingly, we get much improved performance for FFT-RadNet
with a different batch-size and learning-rate than the original paper (see
Table 1). We thus use our own tuned FFT-RadNet as the baseline.

###  6.2 Experiment setup

We use the same data partitioning as that of [29]. The whole labelled dataset
is split into train, val, and test set by sequence (driving session). That is,
data samples from a sequence can only appear in one of the splits. This is to
avoid allocating closely-resembling data samples (e.g. two consecutive data
samples in the same driving session) to the same split, which can lead to
inflated performance metrics.

For evaluation, we also follow the setup as in [29]. Object detection
performance is measured by average-precision, average-recall and average-f1
scores, where average is done across a range of detection thresholds. The
freespace segmentation results are evaluated with mIOU metric: the average IOU
(intersection over union) score across test samples.

For the pre-training experiments in Section 6.4, it is important to note that
we make sure any samples from the labeled test set sequences are not included
in the pre-training stage. This avoids giving unfair advantages to the ADCNet-
PTFT, so that it is not allowed to see test samples even in the pre-training
stage.

###  6.3 Comparing ADCNet to baselines

In this section we compare the ADCNet to baselines. We consider two variant of
ADCNet – with and without pre-training. To differentiate these, we denote the
method with pre-training as ADCNet-PTFT, where PTFT stands for pre-training &
fine-tuning.

The object detection performance results are shown in Table 1, and the
freespace segmentation results are presented in Table 2. Overall, we can see
that the ADCNet-PTFT method out-performs FFT-RadNet for both tasks. In
particular, the ADCNet-PTFT method achieves about 2 percentage higher
freespace segmentation mIOU, while at the same time outperforming the well
tuned FFT-RadNet-optimized method on the detection task. The benefit of pre-
training is apparent when comparing ADCNet and ADCNet-PTFT. For example,
compared with ADCNet, ADCNet-PTFT achieves about 4 points improvement on the
freespace segmentation task on the labeled test set.

We present several model prediction samples in Figure 1. It can be seen from
the samples that the network is able to correctly recognize the cars while
rejecting other bright spots in the RA images. In the second sample, the
network misses one car, possibly due to the road sign on the right, which
causes a bright spot in the RA map, at a similar range of the missed car.

###  6.4 RAD prediction accuracy in pre-training

In Table 1 and Table 2, the effect of pre-training is evident by comparing
ADCNet and ADCNet-PTFT. In this section, we are interested in examining the
prediction accuracy of RAD tensor in the pre-training stage. This is a valid
question, as both the ADC tensor and the RAD are large: ADC is of shape
$512\times 256\times 8$ while RAD is $128\times 248\times 256$ for the RADIal
dataset. It is unclear whether learning a mapping between these two high-
dimensional vectors is feasible.

While the results from Table 1 and Table 2 hint that pre-training is
successful, here we observed that the lowest validation loss (calculated on
2014 samples) during training is about 1.6. This is a very small error, as the
ground truth (power spectrum) values of RAD is distributed around 60.

Furthermore, we visualize several RAD prediction samples in Figure 8. To
visualize this 3D RAD tensor, we summed over the third dimension, and get a 2D
image of dimension range and azimuth. To get a quantitative error measure, we
compute the Relative Absolute Error (RAE) for each RAD entry as

$\displaystyle\text{RAE}(i,j,k)=\frac{|\mathbf{Y}_{\text{RAD}}(i,j,k)-\hat{\mathbf{Y}}_{\text{RAD}}(i,j,k)|}{|\mathbf{Y}_{\text{RAD}}(i,j,k)|}.$
(6)

As such, we get a relative error measure for each entry, and we report the
maximum and mean of these errors in Figure 8.

Figure 8: RAD prediction examples. The mean and max of Relative Absolute Error
(RAE) for each sample are reported. As can be seen these examples, after pre-
training, the network is able to predict RAD tensors to a very high accuracy.
Table 3: Comparing Exact-DFT and Perturbed-DFT initialization method for the
object detection task

|  | All | Easy | Hard
---|---|---|---|---
| Initialization | F1 | AP | AR | RE (m) | AE (°) | F1 | AP | AR | RE (m) | AE (°) | F1 | AP | AR | RE (m) | AE (°)
ADCNet | Exact-DFT | 0.90 | 0.96 | 0.84 | 0.13 | 0.10 | 0.95 | 0.98 | 0.93 | 0.12 | 0.09 | 0.77 | 0.91 | 0.67 | 0.15 | 0.12
ADCNet | Perturbed-DFT | 0.91 | 0.96 | 0.87 | 0.12 | 0.10 | 0.97 | 0.98 | 0.97 | 0.11 | 0.10 | 0.78 | 0.91 | 0.68 | 0.16 | 0.12
ADCNet-PTFT | Exact-DFT | 0.89 | 0.93 | 0.86 | 0.13 | 0.10 | 0.96 | 0.95 | 0.97 | 0.12 | 0.10 | 0.75 | 0.86 | 0.67 | 0.15 | 0.12
ADCNet-PTFT | Perturbed-DFT | 0.92 | 0.95 | 0.89 | 0.13 | 0.11 | 0.97 | 0.96 | 0.98 | 0.12 | 0.11 | 0.81 | 0.91 | 0.73 | 0.16 | 0.12

It can be seen from Figure 8 that the network can predict the RAD tensor to a
very high accuracy: the ground truth and predicted images are visually close,
and the RAE measures confirm this observation.

###  6.5 Different methods for SP module initialization

Table 4: Comparing different initialization methods for ADCNet. The numbers represent the performance on the full test set. | F1 | AP | AR | RE(m) | AE (°) | mIOU
---|---|---|---|---|---|---
Exact-DFT | 0.87 | 0.91 | 0.84 | 0.12 | 0.10 | 75.60%
Random-Doppler | 0.82 | 0.91 | 0.74 | 0.16 | 0.10 | 74.43%
Perturbed-DFT | 0.91 | 0.96 | 0.87 | 0.12 | 0.10 | 74.50%
Random | Failed to converge

In this section, we present an ablation study regarding the initialization
method for the learnable SP module. In particular, we compare these methods:

  * •

Perturb-DFT initialization: perturbed DFT matrices as shown in Equation 1 are
used to initialize both the range and Doppler DFT module in Figure 5

  * •

Exact-DFT initialization: the exact DFT matrices are used to initialize both
range and Doppler DFT modules

  * •

Random-Doppler initialization: the exact DFT matrix is used to initialize the
range DFT, while a randomly generated matrix is used for Doppler

  * •

Random: randomly generated matrices are used for both range and Doppler
modules

These methods employ a various amount of SP knowledge: the Exact-DFT method
relies fully on SP, and the Random method almost completely disregard SP,
while Perturbed-DFT and Random-Doppler lie in between.

The results are shown in Table 4. As can be seen, the Perturbed-DFT yields the
best results. As we suspected, using the Exact-DFT initialization method can
lead to sub-optimal performance. We hypothesize that this is because the
network gets stuck in a local minima when initialized exactly. The Random-
Doppler method yields considerably worse performance, while the Random method
failed to converge to any meaningful model.

Since the Exact-DFT and Perburbed-DFT initialization methods get better
performance than other choices, we provide a more in-depth comparison between
the two. Moreover, we implement these initialization methods in the pre-
training stage, then fine-tune to get the final detection and segmentation
results. For this experiment, the multi-task learning setup as described in
Section 5.2 is performed, with and without pre-training. All the hyper-
parameters are kept the same, and only the initialization method is varied.

The results are reported in Table 3 and Table 5. As can be seen from these
tables, with and without pre-training, the Perturbed-DFT initialization method
always yields better performance than Exact-DFT. The improvement by using
Perturbed-DFT is especially visible for ADCNet, confirming our intuition that
a larger dataset is needed for end-to-end learning, as limited training data
may not be able to nudge the network too far from the SP-based initialization.

Table 5: Comparing Exact-DFT and Perturbed-DFT initialization method for the freespace segmentation task | Initialization | All | Easy | Hard
---|---|---|---|---
ADCNet | Exact-DFT | 70.39% | 70.81% | 69.31%
ADCNet | Perburbed-DFT | 74.45% | 75.85% | 70.89%
ADCNet-PTFT | Exact-DFT | 77.28% | 78.37% | 74.50%
ADCNet-PTFT | Perturbed-DFT | 78.59% | 79.63% | 75.90%

As discussed in Section 5.1, the $\gamma$ parameter in 1 should be chosen with
care: a large $\gamma$ can destroy the DFT structure, while a too small
$\gamma$ can leave the SP module in a non-optimal local minimum. Here we
present an experiment, where different $\gamma$ is applied in the Perturb-DFT
initialization scheme, and an ADCNet is trained (without pre-training) on the
labeled RADIal dataset using the supervised multi-task learning setup in
Section 5.2. Table 6 shows the results: a large $\gamma$ (e.g. $\gamma=2$)
brings significant degradation to both object detection and freespace
segmentation, while a small $\gamma$ like 0.1 achieves better performance than
if no perturbation ($\gamma=0$) is applied.

Table 6: The effect of the $\gamma$ parameter in Perturb-DFT initialization $\gamma$ | F1 | AP | AR | RE(m) | AE (°) | mIOU
---|---|---|---|---|---|---
0 | 0.90 | 0.96 | 0.84 | 0.13 | 0.1 | 70.39%
0.1 | 0.91 | 0.96 | 0.87 | 0.12 | 0.1 | 74.45%
0.5 | 0.84 | 0.88 | 0.81 | 0.16 | 0.13 | 70.03%
2 | 0.65 | 0.75 | 0.57 | 0.18 | 0.12 | 69.47%

##  7 Conclusion

We presented an end-to-end learning based model on raw radar ADC data. To
overcome the difficulty associated with learning from the ADC data, we
designed a learnable signal processing module with an improved initialization
scheme as well as an effective pre-training strategy. The combination of these
techniques results in improved object detection and freespace segmentation
results on the RADIal dataset within a unified multi-task model.

The designed learnable SP sub-network is modular, such that it can be used
with other backbone or head networks to tackle other AV perception tasks. In
the same vein, the pre-training technique is generic and can be adapted for
other radar-based perception tasks as well as other RAD estimation techniques.
We showcased the merit of the proposed techniques, and we expect these
techniques to be more powerful in more challenging environments (e.g. dense
urban roads), or when using more unlabeled data for pre-training. We leave
these explorations for future works.

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