# Prospects for realtime characterization of core-collapse supernova and
neutrino properties

Meriem Bendahman,11footnotetext: These authors contributed equally to this
work. Isabel Goos  Joao Coelho  Matteo Bugli  Alexis Coleiro  Sonia El
Hedri,22footnotetext: Corresponding author Thierry Foglizzo  Davide Franco
Jérôme Guilet  Antoine Kouchner  Raphaël Raynaud  and Yahya Tayalati

###### Abstract

Core-collapse supernovae (CCSNe) offer extremely valuable insights into the
dynamics of galaxies. Neutrino time profiles from CCSNe, in particular, could
reveal unique details about collapsing stars and particle behavior in dense
environments. However, CCSNe in our galaxy and the Large Magellanic Cloud are
rare and only one supernova neutrino observation has been made so far. To
maximize the information obtained from the next Galactic CCSN, it is essential
to combine analyses from multiple neutrino experiments in real time and
transmit any relevant information to electromagnetic facilities within
minutes. Locating the CCSN, in particular, is challenging, requiring
disentangling CCSN localization information from observational features
associated with the properties of the supernova progenitor and the physics of
the neutrinos. Yet, being able to estimate the progenitor distance from the
neutrino signal would be of great help for the optimisation of the
electromagnetic follow-up campaign that will start soon after the propagation
of the neutrino alert. Existing CCSN distance measurement algorithms based on
neutrino observations hence rely on the assumption that neutrino properties
can be described by the Standard Model. This paper presents a swift and robust
approach to extract CCSN and neutrino physics information, leveraging diverse
next-generation neutrino detectors to counteract potential measurement biases
from Beyond the Standard Model effects.

##  1 Introduction

Core-collapse supernovae (CCSNe) play a pivotal role in shaping the dynamics
and composition of galaxies [1, 2, 3]. These cataclysmic events not only
result in spectacular explosions but also give rise to essential cosmic
phenomena such as the production of heavy elements through explosive
nucleosynthesis and the formation of neutron stars and stellar black holes.
These compact objects, in turn, may serve as the seeds for extremely powerful
cosmic phenomena like Active Galactic Nuclei and Gamma-Ray Bursts.
Understanding the underlying mechanism of CCSN explosions and the link between
their outcomes and the properties of their progenitors is thus of paramount
importance.

Despite their profound implications, CCSNe are not fully understood yet [4, 5,
6], as their onset takes place deep inside the star’s core, a region concealed
beneath opaque layers, where matter is subjected to extreme conditions. In
addition to hosting the key processes triggering the CCSN, this core region
provides a unique laboratory for probing high-density, high-magnetic-field,
high-temperature environments and the possibility of exploring physics beyond
the Standard Model (SM). However, the explosion mechanism is also extremely
difficult to model. Although numerical simulations have made substantial
progress in modeling the core of massive stars, accurately replicating the
conditions required for a successful CCSN explosion remains a formidable
computational challenge. Consequently, there is an imperative need for direct
observational insights.

During a CCSN, the star’s core emits an intense burst of neutrinos over a
short but critical timespan. These neutrinos are generated through nuclear
reactions triggered by the collapse and the shockwave’s formation and
propagation. Due to their feeble interactions with matter, neutrinos preserve
most of their initial information during their journey to Earth, offering a
unique window into various phases of the CCSN, with millisecond-level
precision. Moreover, neutrinos are expected to exit the star minutes to hours
before the CCSN becomes visible electromagnetically. The detection of the
$\mathcal{O}(10~{}\mathrm{MeV})$ thermal neutrinos emitted by a CCSN would
hence considerably narrow the search time window for gravitational wave
detectors and provide an early warning for optical telescopes. However, the
detection of CCSN neutrinos is challenging and limited to nearby supernovae in
the Milky Way or the Large Magellanic Cloud. Not only these supernovae are
extremely rare, with about two expected per century [7], but they can also
elude optical telescopes if they are located behind the Galactic Center [8].
If a CCSN occurs, it is therefore imperative to maximize the amount of
information collected in real time by combining observations from all
sensitive neutrino experiments. The Supernova Early Warning System (SNEWS)
serves this purpose [9, 10]. This alert system currently receives data from
ten neutrino experiments, extracts information about the CCSN progenitor, and
sends it to telescopes. It notably deploys algorithms to determine the CCSN’s
angular position (and the associated error box) and its distance from Earth
within minutes after the reception of neutrino data. The CCSN’s angular
position and distance estimates based on neutrino observations, if estimated
quickly enough, will play a major role in optimizing the electromagnetic
follow-up campaigns that will start immediately after the neutrino alert, in
order to locate and study the supernova’s electromagnetic counterpart. In
particular, depending on the size of the error box, the search for the
counterpart will be performed either using large field-of-view / lower
sensitivity instruments or smaller field-of-view facilities which are
generally more sensitive (see e.g. [8]). Similarly, estimating the distance to
the source will constrain whether the supernova exploded in the foreground or
behind the Galactic Center. Such information is fundamental in favouring
infrared rather than optical observations, due to the extinction in the
Galactic plane [8, 11].

A critical challenge for CCSN alert systems, like SNEWS, is disentangling
within minutes three categories of information imprinted in neutrino
observations: CCSN localization (angular position and distance), CCSN
properties (e.g., neutron star Equation of State, progenitor mass, density
profile), and neutrino properties (e.g., mass ordering, interaction nature).
While only the first is time-sensitive, neglecting new physics phenomena such
as neutrino two-body decays, if they are present, or making wrong assumptions
about the ordering of neutrino masses could affect the estimations of the
properties and the location of the CCSN progenitor.

In particular, the detected neutrino signal’s intensity, crucial for CCSN
distance measurements, depends significantly on both CCSN and neutrino
properties. The degeneracy between CCSN distance and CCSN progenitor type has
been extensively studied [12, 13], revealing that the flux of neutrinos during
the first tens of milliseconds of a CCSN exhibits weak dependence on the CCSN
progenitor’s properties. These studies demonstrated that the electron neutrino
emission peak during the supernova neutronization phase, often referred to as
a “standard candle”, offers the potential for CCSN distance measurements with
sub-$10$% precision. However, within the time frames considered in distance
measurement algorithms, the neutrino fluxes highly depend on the neutrino’s
intrinsic properties, primarily their mass ordering (MO) in the Standard Model
[14]. Additionally, several new physics scenarios can significantly distort
CCSN neutrino fluxes while still evading constraints from beam, solar, and
atmospheric neutrino measurements [15, 16, 17, 18, 19]. Since current distance
measurement techniques [12, 20] always assume the SM, the existence of new
physics mechanisms could lead to significant biases, as will be shown in this
work. While methods to constrain neutrino properties have been proposed for
several Beyond the Standard Model (BSM) scenarios, these methods either
consider specific CCSN progenitors [17], assume that the CCSN distance is
known [21], or use neutrino energy measurements [19, 22] which are not
possible with every detector. One generic characteristic signature of these
new physics phenomena, however, is that their effect on neutrino observations
depends on the neutrino type and flavor. The next generation of neutrino and
dark matter experiments will be particularly suited to exploit this feature.

In the next decade, the landscape of neutrino experiments will be radically
transformed: large-scale Water Cherenkov (WC) detectors such as Hyper-
Kamiokande (HK), IceCube, and KM3NeT, sensitive to electron antineutrinos,
will be complemented by the DUNE experiment, sensitive to electron neutrinos,
and by large-scale detectors sensitive to Coherent Elastic Neutrino-Nucleus
scattering (CE$\nu$NS) such as DarkSide-20k and, later, ARGO. Once completed,
this panel of experiments will be sensitive to three diﬀerent ﬂavor
combinations of supernova neutrinos. Combining these different detector types
will provide a unique sensitivity to flavor-dependent phenomena over a wide
range of supernova models and distances.

In this paper, we present a methodology to combine data from diverse neutrino
experiments and concurrently extract information regarding CCSN progenitors,
CCSN locations, and neutrino properties, relying solely on measured neutrino
rates. We demonstrate this methodology using a new physics scenario involving
neutrino two-body decays [19] and illustrate how new physics mechanisms can
bias CCSN distance measurements if only SM neutrino interactions are assumed.
We also propose techniques to identify deviations from the Standard Model and
characterize new physics scenarios, capitalizing on the complementarity
between different types of neutrino experiments. These algorithms offer
realtime analysis capabilities, making them suitable for alert systems and
adaptable to a range of representative new physics models.

This article proceeds as follows. In Section 2, we discuss the neutrino
experiments, CCSN progenitor models, and neutrino flavor conversion processes
considered in our analyses. For the latter, we present both the current SM
flavor-conversion scenario and an overview of possible new physics processes
susceptible to affect the flavor composition of CCSN neutrinos, with an
emphasis on neutrino two-body decays, which will be used as a case study of
the impact of BSM interactions throughout this work. In Section 3, we propose
a likelihood-based approach, inspired by previous studies of the time-
dependence of CCSN neutrino luminosity spectra, to simultaneously constrain
CCSN and neutrino parameters. We apply this approach to the determination of
the neutrino MO and to CCSN distance measurements in Sections 4 and 5,
respectively. Then, Section 6 presents a model-dependent and a model-
independent approach to identify and characterize the effects of physics
beyond the SM on neutrino observations. We conclude in Section 7.

##  2 Methodology

To evaluate the impact of the CCSN progenitor characteristics, the CCSN
localization, and the progenitor properties on observations, we simulate the
neutrino rates expected at large-scale next-generation neutrino experiments
for a wide range of CCSN models. Our simulations incorporate the dominant
neutrino flavor conversion mechanisms expected in the SM as well as an example
of a new physics scenario that may affect the flux and flavor composition of
the neutrinos arriving on Earth. In what follows, we describe our choice of
models and experiments.

###  2.1 CCSN models

This analysis focuses on a comprehensive set of 149 progenitor models,
originally devised and introduced by Segerlund _et al_ [23]. These models are
the product of one-dimensional simulations modelling the evolution of isolated
massive stars with solar metallicity. Their zero-age main sequence (ZAMS)
masses span a broad spectrum, ranging from $9$ to $120$ solar masses. The
neutrino signal was computed by Segerlund _et al_ [20] with 1D core-collapse
simulations using an energy-dependent neutrino transport scheme, state-of-the-
art microphysical electron capture rates and neutrino-matter interactions.
These models were designed to encompass a wide range of pre-supernova
structures typically found in CCSNe with iron cores. In Segerlund _et al_
[20], they were used to identify observables whose values have a weak
dependence on the CCSN model or can be easily parameterized. Consequently,
they constitute an ideal dataset for our analysis.

It is worth noting that the CCSN progenitors represented in these models
exhibit highly mass-dependent occurrence rates in the Universe, with lighter
stars significantly outnumbering their heavier counterparts. To account for
this mass dependence in our work, we attribute prior probabilities to the
different CCSN models. These prior probabilities are taken to be proportional
to the Salpeter Initial Mass Function (IMF), which characterizes the
distribution of stars according to their ZAMS masses [24]:

$\displaystyle p(M)\propto M^{-2.35}.$ (2.1)

###  2.2 Neutrino experiments

For this analysis, we consider a set of large-scale experiments sensitive to
CCSN neutrinos which are expected to be fully built and to start collecting
data within the next decades. In order to lift the degeneracies between CCSN
localization, CCSN properties, and neutrino properties, we consider not only
WC detectors, which have been prominent actors in the search for CCSNe since
the observation of SN1987A, but also experiments sensitive to different
neutrino flavors. Since we want to emulate detector performances as
realistically as possible, we consider only experiments whose detection
efficiencies have been made public, either in the SnowGLoBES software [25] or
in publications by the respective collaborations [26]. The experiments we have
selected for this analysis are projected to be sensitive to three distinct
neutrino flavor combinations: $\bar{\nu}_{e}$, $\nu_{e}$, and the sum of all
neutrino flavors. The detectors and their properties are summarized in table
1.

Experiment |  Detected
$\nu$ flavor |  Total mass (kT) |  Efficiency at $\mathbf{20}$ MeV (%) |  Background
rate (Hz)
---|---|---|---|---
Hyper-Kamiokande | $\bar{\nu}_{e}$ | 260 | 100 | 0
IceCube | $\bar{\nu}_{e}$ | $10^{6}$ | 4.8 | $1.5\times 10^{6}$
KM3NeT | $\bar{\nu}_{e}$ | $2.1\times 10^{5}$ | 0.07 | $4.5\times 10^{6}$
DUNE | $\nu_{e}$ | 40 | 100 | 0
DarkSide-20k | all | 0.05 | 95 | 0
ARGO | all | 0.35 | 95 | 0
Table 1: Summary of the properties of the detectors considered in this work.
The detected neutrino flavor is the flavor associated with the with the
dominant interaction for $\mathcal{O}(10~{}\mathrm{MeV})$ neutrinos at each
experiment. The efficiency at $20$ MeV is the average fraction of interactions
detected for a $20$ MeV neutrino. Finally, background rates due to PMT noise
and radioactivity are taken into account for IceCube and KM3NeT and neglected
for all other experiments. The CCSN event selection cuts are harsher at KM3NeT
than at IceCube due to the higher rates of radioactive decays in seawater.

####  2.2.1 Water Cherenkov detectors: KM3NeT, IceCube, Hyper-Kamiokande

Water Cherenkov detectors are primarily sensitive to CCSN electron
antineutrinos. Current and upcoming experiments have large instrumented
volumes and hence would detect particularly high numbers of events if a
Galactic CCSN occurred –e.g. $\mathcal{O}{(50000)}$ events at the Galactic
Center for Hyper-Kamiokande. However, detectors located in natural
environments, such as IceCube and KM3NeT, also suffer from sizable, megahertz-
scale, backgrounds from ambient radioactivity and PMT noise.

##### KM3NeT:

Currently under construction and taking data in the Mediterranean Sea, KM3NeT
[27] is composed of two large arrays of detection modules. When completed, in
2028, the detector will survey a volume of $1.5~{}\mathrm{km}^{3}$ of
seawater. While KM3NeT has been originally designed to detect GeV to PeV
neutrinos, it is also expected to be sensitive to more than $95$% of Galactic
CCSNe [28]. KM3NeT therefore runs a CCSN realtime analysis system and is part
of SNEWS [9]. However, in spite of its large instrumented volume, the
sensitivity of the experiment is limited by important backgrounds from
bioluminescence and ${}^{40}\text{K}$ decays in seawater. For this study, we
consider KM3NeT’s expected final configuration. We use the detection
efficiencies estimated for KM3NeT’s current CCSN analysis; the background rate
after the CCSN event selection if of $4.5\times 10^{6}$ events per second.
Additionally, event-by-event reconstruction of the CCSN neutrino energies is
not possible and only global properties of the time-integrated spectrum can be
constrained [28].

##### IceCube:

Located at the South Pole, IceCube [29] is a cubic-kilometer neutrino detector
consisting of over $5000$ detection modules located deep within the Antarctic
ice sheet. Operational since 2010, it focuses on high-energy neutrino
detection, particularly from astrophysical sources like cosmic rays
interacting with the atmosphere. IceCube’s extensive detection volume of
$1~{}\mathrm{km}^{3}$ also makes it highly sensitive to low-energy neutrinos
from Galactic CCSNe. In particular, the large detection rates expected then
would make IceCube uniquely positioned to determine the neutrino mass ordering
and characterize the signatures of hydrodynamical instabilities inside the
collapsing star [30]. Consequently, IceCube is a major member of SNEWS [9].
Like KM3NeT, however, IceCube suffers from large backgrounds due to
radioactive activity and PMT noise and event-by-event energy reconstruction is
not possible [30]. In what follows, we take IceCube’s CCSN neutrino detection
efficiencies from SnowGLoBES. The background rate after the CCSN event
selection is of $1.5\times 10^{6}$ events per second. [31].

##### Hyper-Kamiokande:

The Hyper-Kamiokande neutrino detector [32], currently under construction at
Mount Ikeno in Japan, will be a large-scale WC detector surveying a volume of
$260$ kT of ultra-pure water. Aimed at the detection of MeV to GeV-scale
neutrinos, this experiment is expected to play a leading role in CCSN neutrino
searches. For a CCSN located at the Galactic Center, for example, HK is
expected to detect $\mathcal{O}(50000)$ events [33]. With a low energy
threshold of around $4$ MeV, HK will trigger on most CCSN neutrinos, with an
efficiency larger than $99\%$ and extremely low backgrounds. In this analysis,
we consider HK’s backgrounds for CCSN neutrino detection to be negligible.

####  2.2.2 $\nu_{e}$ detection experiments: Deep Underground Neutrino
Experiment

The Deep Underground Neutrino Experiment (DUNE) [34], based at the Fermi
National Accelerator Laboratory and at the Sanford Underground Research
Facility, will be a neutrino beam experiment which primarily aims at measuring
the neutrino mass ordering and the CP phase in the lepton sector. However, its
far detector, a liquid argon time projection chamber (LArTPC), comprising four
$10$ kton modules, can also be used to detect atmospheric and cosmic
neutrinos. In the CCSN neutrino energy regime, DUNE would be primarily
sensitive to electron neutrinos and is expected to detect $\mathcal{O}(1000)$
events for a CCSN at the Galactic Center, with negligible backgrounds [35].
DUNE is expected to be completed at the end of the 2020s and its large scale,
low background, and sensitivity to $\nu_{e}$ will make it play a key role in
Galactic supernova searches. In this work, we consider DUNE’s backgrounds for
CCSN neutrino detection to be negligible.

####  2.2.3 Coherent elastic neutrino-nucleus scattering experiments:
DarkSide-20k and ARGO

DarkSide-20k [36], located at the Gran Sasso National Laboratory, is designed
for the detection of dark matter particle candidates. The detector utilizes a
liquid argon time projection chamber (TPC) technology, which detects the
ionization electrons and scintillation light generated by the interaction of
dark matter particles with argon nuclei. Beyond dark matter searches,
DarkSide-20k is sensitive to coherent elastic neutrino-nucleus scattering
(CE$\nu$NS) and would detect $\mathcal{O}(100)$ events for a CCSN at the
Galactic Center, with negligible backgrounds [26]. Its precursor, DarkSide-50
is part of SNEWS [9]. Since CE$\nu$NS treats all flavors equally, DarkSide
could provide unique information on the total flux of active neutrinos emitted
by a CCSN, although, like IceCube and KM3NeT, it cannot perform event-by-event
energy reconstruction. Its main limitation is its small instrumented volume,
leading to particularly low detection rates in the first tens of milliseconds
of the CCSN. We therefore also consider a similar, next-generation,
experimental project, ARGO [26], which will be 7 times larger than
DarkSide-20k. In what follows, we will assume ARGO to differ from DarkSide
only by its size. Moreover, we will neglect the backgrounds associated with
CCSN neutrino detection for both experiments, as these background were
estimated to be more than two orders of magnitude lower than the expected CCSN
signal from a $11~{}M_{\odot}$ progenitor at $10$ kpc [26].

Spanning a diverse range of capabilities and encompassing all the detectable
flavor combinations expected in future-generation experiments, this selection
of detectors, while comprehensive, is not exhaustive. In particular, we
considered Hyper-Kamiokande rather than Super-Kamiokande as the former is more
likely to operate when DUNE and ARGO are also active. Another key experiment
for CCSN investigations in the coming decades is the Jiangmen Underground
Neutrino Observatory (JUNO). JUNO can indeed measure both electron
antineutrinos and the collective sum of all neutrino flavors, effectively
serving as two complementary detectors simultaneously [37, 38]. While
efficiency curves for JUNO’s CCSN detection channels are not currently
publicly available, JUNO, once built, is poised to significantly enhance the
study of the flavor-dependent behaviors of CCSN neutrinos.

###  2.3 Neutrino ﬂavor conversion

The initial tens of milliseconds following the core bounce are characterized
by an intense emission of electron neutrinos, a period known as
“neutronization”. This phenomenon is associated with the initial propagation
of the shock wave through the protoneutron star (PNS) (for a detailed review,
see [4, 5, 6]). At the same time, the accretion of matter onto the PNS leads
to the production of a large number of both electron neutrinos and
antineutrinos by charged-current processes in the PNS hot mantle. This phase,
known as “accretion phase”, can last for a few hundreds milliseconds. Neutrino
emission profiles for both phases are depicted in Figure 1 for different
supernova models. In contrast, muon and tau neutrinos are produced solely by
the cooling of the deepest core regions and their emission flux increases at a
much slower rate. Therefore, during the first hundreds of milliseconds of a
core-collapse supernova, there is a significant disparity in the fluxes of
electron neutrinos, electron antineutrinos, and other neutrino types (see Fig.
1). The observed CCSN neutrino rates on Earth hence strongly depend on the
evolution of neutrino flavor composition, both within the star and along their
journey to Earth. In this section, we discuss the dominant flavor conversion
process in the SM, i.e. the adiabatic Mikheyev-Smirnov-Wolfenstein (MSW) [14]
transitions, and provide an overview of new physics processes susceptible to
modify neutrino rates in the neutronization phase. We focus particularly on
neutrino two-body decays, which will be used as a case study for our analysis.

Figure 1: Neutrino luminosities as a function of time for the $9~{}M_{\odot}$
and $15~{}M_{\odot}$ models from [20]. The solid lines correspond to electron
neutrinos, the dashed lines to electron antineutrinos and the dash-dotted
lines to a species representative of heavy-lepton neutrinos and antineutrinos
$\nu_{x}$. The emission peak for $\nu_{e}$ during the first $30$ ms
corresponds to the neutronization phase. After this period, the fluxes for the
electron neutrinos and antineutrinos become comparable as the star enters the
accretion phase.

####  2.3.1 Adiabatic MSW effect

In the first $\mathcal{O}(100~{}\mathrm{ms})$ of the CCSN, the dominant
neutrino flavor conversion process inside the collapsing star is adiabatic MSW
transitions [14]. These transitions convert electron neutrinos (antineutrinos)
into the heaviest (intermediate) neutrino (antineutrino) mass eigenstate
$\nu_{h}$ ($\bar{\nu}_{m}$), while muon and tau neutrinos become combinations
of the other two eigenstates. The flavor composition of the neutrino fluxes
arriving on Earth can hence be expressed as a function of the electron flavor
survival probability $p$:

$\displaystyle\Phi_{e}^{\mathrm{Earth}}$
$\displaystyle=p\Phi_{e}^{(0)}+(1-p)\Phi_{x}^{(0)}$ (2.2)
$\displaystyle\Phi_{x}^{\mathrm{Earth}}$
$\displaystyle=\frac{1-p}{2}\Phi_{e}^{(0)}+\frac{1+p}{2}\Phi_{x}^{(0)}$ (2.3)

where $\Phi^{(0)}$ is the initially produced neutrino flux and $x$ refers to
the $\mu$ and $\tau$ flavors. A similar formula holds for antineutrinos using
the probability $\bar{p}$. For neutrinos, $p\approx 0.30$ in the inverted mass
ordering (IMO) and $0.02$ in the normal mass ordering (NMO). For
antineutrinos, $\bar{p}\approx 0.02$ in the IMO and $0.70$ in the NMO [14].

####  2.3.2 New physics scenarios: a case study with neutrino two-body decays

At first glance, the possibilities of BSM scenarios altering the
neutronization flux seem limitless. However, BSM physics in the neutrino
sector is already tightly constrained by beam, solar, and atmospheric neutrino
measurements. In order to distort the neutrino flux sufficiently to mislead
CCSN alert systems while still being allowed by current observations, a new
physics process needs to be tied to conditions specific to CCSN measurements:
the high density of the CCSN core, the possible presence of large magnetic
fields with a complex structure, and the particularly long neutrino travel
path from the CCSN to the Earth. To date, three types of processes susceptible
to substantially modify CCSN neutrino fluxes have been proposed: non-standard
neutrino self-interactions [15], the conversion of active neutrinos into
sterile neutrinos [16, 21, 17], and neutrino two-body decays [18, 19]. While
the corresponding new physics models are associated with a wide parameter
space, past studies considered simplified frameworks efficiently modelling
entire classes of BSM scenarios with only two or three parameters. Since the
number of these simplified frameworks is limited by the tight existing
constraints on neutrino properties, it is feasible for an alert system to
consider several representative new physics models to check for possible
departures from the SM. In this work, we assess the impact of such a study
using the example of neutrino two-body decays, which allows studying a highly
diverse landscape of possible CCSN neutrino signal distortions.

In the SM, heavy neutrinos species can decay into lighter species only through
a loop-level process, with a lifetime larger than the age of the Universe.
However, this lifetime can be considerably shortened in simple extensions of
the SM where a heavy neutrino mass eigenstate can undergo a two-body decay
into a lighter neutrino and a new scalar field. These BSM scenarios have been
the focus of thorough investigation and have been constrained by a wide range
of measurements, as extensively described in [19] and [39]; the latter, in
particular, derived new stringent constraints on certain classes of neutrino
decay models. Overall, models where at least two neutrino species decay with
similar rates have therefore been already strongly constrained up to distance
scales of the order of $10$ kpc. However, models featuring significant
discrepancies in the lifetimes of neutrino species, particularly with one
species having a notably shorter lifetime, remain an intriguing realm of
exploration.

In this study, we focus on a scenario discussed in [19], where the heaviest
neutrino or antineutrino mass eigenstate $\nu_{h}$ decays into the lightest
mass eigenstate $\nu_{\ell}$ and an invisible scalar $\phi$:

$\displaystyle\nu_{h,L}\rightarrow\nu_{\ell,L/R}+\phi$ (2.4)

where $L$ and $R$ represent the neutrino helicity. We consider the $\phi_{0}$
model of [19], where neutrinos are Dirac particles and lepton number is
conserved. In this model, active CCSN (anti)neutrinos decay into either active
or sterile (anti)neutrinos, depending on whether the interaction conserves or
flips helicity. We parameterize this model using two quantities. First, we
define the _normalized supernova distance_ as $\bar{r}=d/r_{0}$, where $d$ is
the supernova distance to Earth and $r_{0}$ is the characteristic decay length
for $10$ MeV neutrinos, defined as

$\displaystyle r_{0}=c\tau\left(\frac{10~{}\mathrm{MeV}}{m}\right),$ (2.5)

with $\tau$ and $m$ being the lifetime and the mass of the heavy neutrino
eigenstate, respectively. In the SM, the $\bar{r}$ parameter is equal to $0$.
Second, we introduce the _branching ratio of visible decays_ , $\zeta$, which
represents the fraction of helicity-conserving neutrino decays [19]. For
$\zeta=1$, all heavy CCSN neutrinos decay into light active neutrinos, while
for $\zeta=0$ all decay products are undetectable.

When travelling through the collapsing star, as discussed in Section 2.3.1,
adiabatic MSW conversions turn electron neutrinos into the heaviest mass
eigenstate $\nu_{h}$. On their way to Earth, $\nu_{h}$ ($\bar{\nu}_{h}$) will
partially decay into the lightest active neutrino $\nu_{\ell}$
($\bar{\nu}_{\ell}$), or into sterile neutrinos. The impact of these decays on
the neutrino rates during the first tens of milliseconds of the CCSN have been
extensively described in [19]. The variations of the expected neutrino signal
time profiles as a function of $\bar{r}$ and $\zeta$ are shown in Figure 2 for
different experiments. In what follows, we provide a concise recap of the most
significant effects of neutrino decays in the $\phi_{0}$ model:

  * •

_Appearance/disappearance of the $\nu_{e}$ neutronization peak_: in the NMO,
the $\nu_{e}$ emission peak associated with the neutronization phase is not
expected to be visible in $\nu_{e}$ detection experiments such as DUNE.
However, if heavy neutrinos decay into light visible neutrinos, which have a
$70$% probability of being seen as $\nu_{e}$, this peak can reappear.
Conversely, in the IMO, neutrino decays will cause the neutronization peak
expected at DUNE to disappear since the electron neutrino component of the
lightest species is of only $2\%$. These effects can be seen in the left
panels of Figure 2.

  * •

_Increase/decrease of the $\bar{\nu}_{e}$ rate_: in the NMO, the
$\bar{\nu}_{e}$ rate expected at WC detectors will receive contributions from
the visible decays of heavy antineutrinos, driven by the higher likelihood of
the lightest neutrino species being identified as $\bar{\nu}_{e}$. Conversely,
the electron antineutrino rate would decrease by up to $30\%$ in the IMO when
$\bar{r}$ increases. These effects are illustrated in the middle panel of
Figure 2.

  * •

_Decrease of the total neutrino rates_ : in the Dirac $\phi_{0}$ scenario
considered here, active neutrinos could decay into sterile neutrinos which
would escape detection. Moreover, even for visible decays, the produced
neutrinos will be softer and hence less likely to interact. For experiments
sensitive to the sum of all neutrino flavors, such as DarkSide-20k or ARGO,
the neutrino rate is expected to fall short of the SM predictions. Decays into
sterile neutrinos could also lead to a decrease of the $\nu_{e}$ and
$\bar{\nu}_{e}$ rates expected at DUNE and at WC detectors, regardless of the
mass ordering. These effects are shown in Figure 2 for the different detector
types. Note that for DUNE and WC detectors, $\zeta$ variations either have a
weak impact on detection rates or can lead to degeneracies with the SM. These
degeneracies, however, can be lifted by considering experiments sensitive to
all neutrino flavors such as DarkSide-20k and ARGO.

For sufficiently short neutrino lifetimes, as shown in Figure 2, the
aforementioned effects could potentially alter the expected neutrino rates by
several tens of percents, thus substantially impacting CCSN distance
measurements. In the NMO and for $\zeta$ close to one, rate increases at DUNE
and in WC detectors might lead to an underestimation of the CCSN distance.
Nevertheless, this bias could potentially be mitigated by the observation of a
neutrino deficit at DarkSide-20k or ARGO. Conversely, within the IMO or for
small values of $\zeta$, the decrease of the neutrino rates across all
experiments could lead to overestimating the CCSN distance, although the size
of this decrease depends on the type of experiment considered. Note that the
flux suppression induced by neutrino decays can mimic the one observed in
other new physics scenarios such as the ones involving active-sterile neutrino
conversions. In either case, a meticulous comparison of neutrino rates among
different detector types is imperative to identify deviations from the SM and
to make necessary corrections to distance measurements.

Figure 2: Expected neutrino rates for a $11~{}\text{M}_{\odot}$ progenitor
[23] at $10$ kpc at DUNE (left), HK (middle), and ARGO (right), for the normal
(top) and inverted (bottom) mass orderings, and for different values of
$\bar{r}$ and $\zeta$. The SM case ($\bar{r}=0$) is shown in blue, and decay
scenarios with $\bar{r}=1$ and $5$ are shown in orange and green,
respectively. Solid (dashed) lines represent models with $\zeta=1$
($\zeta=0$).

###  2.4 Analysis pipeline

We generate neutrino ﬂuxes and compute the expected neutrino detection rates
using SNEWPY v1.3b1 [40]. Interaction cross-sections, as well as the detection
efficiencies and energy reconstruction effects, are modelled using the
SnowGLoBEs software [25]. For DarkSide-20k and ARGO, which have not been
included in SnowGLoBEs yet, we use the detection efficiencies from [26]. For
IceCube and KM3NeT, where radioactivity and PMT noise cannot be neglected, we
add Poissonian backgrounds to the signal, with constant rates of $1.5$ MHz
[31] and $3$ MHz [28], respectively. Finally, we modified the SNEWPY public
version to include the neutrino decay model discussed in Section 2.3. Semi-
analytical expressions for the neutrino fluxes expected on Earth for these
models are given in Appendix A.

For each CCSN model and neutrino detector considered, expected neutrino rates
are evaluated both in the NMO and the IMO, for the following $(\bar{r},\zeta)$
values:

$\displaystyle\bar{r}$ $\displaystyle=\begin{cases}0.1k\quad\text{for
}k\in\llbracket 0~{};~{}9\rrbracket\\\ k\quad\text{for }k\in\llbracket
1~{};~{}10\rrbracket\end{cases}$ (2.6) $\displaystyle\zeta$
$\displaystyle=0.1k\quad\text{for }k\in\llbracket 0~{};~{}10\rrbracket$

For $\bar{r}=10$ all heavy neutrinos and antineutrinos will have decayed
before reaching Earth, and therefore we do not study models with larger
$\bar{r}$ values. To evaluate the neutrino rates for any value of $\bar{r}$
and $\zeta$, we then approximate the $(\bar{r},\zeta)$ dependence of the CCSN
rate by a polynomial function, as described in Appendix B.

##  3 Constrain all CCSN properties at once

In this section, we present a likelihood-based analysis framework which could
be used by a CCSN alert system to simultaneously estimate the CCSN distance
and the properties of the neutrinos, by fitting the neutrino event counts in
suitably chosen time windows for multiple experiments.

###  3.1 Choosing time windows

We base the choice of time windows for this study on two categories of
observables, which have been shown in the literature to either have a weak
dependence on the CCSN model, or to be easily parameterizable as a function of
the supernova progenitor properties.

##### Neutrino rates in the neutronization phase:

the neutrino rates measured during the first tens of milliseconds postbounce
have been shown to only weakly depend on the stellar progenitor properties
[12, 13]. This model dependence is expected to be minimal when considering the
first $10$ ms after the onset of the CCSN neutrino signal, a time region which
contains the bulk of the $\nu_{e}$ neutronization peak. However, for most
experiments, the number of events expected in this narrow window is
particularly small for most Galactic supernovae and, hence, to reduce
statistical uncertainties, longer time periods should be considered. Using the
fluxes measured up to $18$ ms postbounce, Kachelriess _et al_ [12] showed that
the CCSN distance could be measured with a precision of $6\%$ in a
hypothetical megaton-scale WC detector. For smaller detectors, enlarging this
window up to $50$ ms has been proposed by Segerlund _et al_ [20] in the
context of analyses at SNEWS. The expected signal in this larger window,
however, would significantly depend on the CCSN type, as illustrated in Figure
3 (left) for the case of Hyper-Kamiokande.

##### Neutrino rate ratio fΔ:

in [20], a new observable, $f_{\Delta}$, was defined to mitigate the residual
model dependence of the $N(0-50~{}\mathrm{ms})$ rates. This observable is
defined as the ratio of neutrino rates between the accretion and the
neutronization phases:

$f_{\Delta}=\frac{N(100-150~{}\mathrm{ms})}{N(0-50~{}\mathrm{ms})}$ (3.1)

and its dependence on $N(0-50~{}\mathrm{ms})$ is close to linear, as shown in
Figure 3 (right). Exploiting this quasi-linear dependence, Segerlund _et al_
[20] proposed an algorithm to infer the CCSN distance from a
$(f_{\Delta},N(0-50~{}\mathrm{ms}))$ measurement at a single detector.
Accounting for both variations between CCSN models and the CCSN distance
distribution in the galaxy, the associated distance measurement uncertainties
are of around $5\%$ at HK and IceCube and around $8\%$ at DUNE.

Figure 3: Left: CCSN neutrino event counts in the Hyper-Kamiokande detector
for a CCSN at $10$ kpc as a function of progenitor masses, for the first $10$
ms (green), $20$ ms (blue), and $50$ ms (purple) postbounce. Right:
$f_{\Delta}$ as a function of the number of CCSN neutrino events within the
first $50$ ms after bounce. For both panels, the light and dark shades
represent the NMO and the IMO, respectively. Note that, for the first $20$ ms
postbounce, the NMO and IMO predictions do not overlap, even when no
assumption is made about the CCSN model. For the first $50$ ms, however, the
IMO and NMO scenarios can be distinguished only if the range of possible CCSN
models is constrained.

Based on the observables and studies described above, we selected four time
windows as follows. First, exploiting the results from [12], we split the $50$
ms postbounce time interval into three windows: $[0,10]$, $[10,20]$, and
$[20,50]$ ms. As a fourth window, following the methodology detailed in [20],
we choose the $[100,150]$ ms time interval. This choice of time windows allows
considering the whole $50$ ms neutronization period while resolving the
neutronization peak, for experiments sensitive to $\nu_{e}$, and mitigating
the CCSN model dependence.

###  3.2 Likelihood analysis

For a given set of neutrino detectors taking data at the time of the CCSN, we
define the following likelihood function:

$\displaystyle\log\mathcal{L}(\\{\mathcal{O}_{\mathrm{obs}}\\}|d,M,\bar{r},\zeta,\mathrm{MO})=\sum_{i}\log
P[N_{i}(10\text{ms})]$ $\displaystyle+\log P[N_{i}(10-20\text{ms})]$ (3.2)
$\displaystyle+\log P[N_{i}(20-50\text{ms})]$ $\displaystyle+\log
P[N_{i}(100-150\text{ms})]$ (3.3)

where the index $i$ refers to each neutrino detector, $d$ and $M$ are the
supernova distance and progenitor mass, respectively, and $(\bar{r},\zeta)$
are the neutrino decay parameters. $P$ is the Poisson probability distribution
to observe a given number of events:

$\displaystyle P(N_{i})=\frac{\lambda_{i}^{N_{i}}e^{-\lambda_{i}}}{N_{i}!},$
(3.4)

with $\lambda_{i}(d,M,\bar{r},\zeta,\mathrm{MO})$ being the expected value of
$N_{i}$ for the CCSN and neutrino properties considered.
$\\{\mathcal{O}_{\mathrm{obs}}\\}$ represents the set of measurements
performed at the different detectors. When constraining a given parameter
$\theta$, e.g. the CCSN distance, we treat the others as nuisance parameters
$\Xi$, and define a profile likelihood:

$\displaystyle\mathcal{L}_{\mathrm{prof}}(\\{\mathcal{O}_{i}\\}|\theta)$
$\displaystyle=\mathrm{max}_{\Xi}\mathcal{L}(\\{\mathcal{O}_{i}\\}|\theta,\Xi)$
(3.5)

which can be used for either parameter fitting or hypothesis testing.
Depending on the parameter of interest, the nuisance parameters can be any
subset of $(d,M,\bar{r},\zeta)$. Since we are considering a discrete, finite,
set of CCSN models, the likelihood optimization over $M$ can be performed by
testing each model individually. To optimize the likelihood over
$(\bar{r},\zeta)$ we use a regular grid with $\Delta\bar{r}=0.05$ and
$\Delta{\zeta}=0.1$. We will confirm in Section 6 that this grid spacing is
far below the resolution of the best-performing detectors considered in this
study, and therefore this grid search is exhaustive. Finally, to optimize over
the CCSN distance, we exploit the dependence of the number of signal events in
the inverse CCSN distance squared, computing the optimal distance either
analytically or numerically as described in Appendix C.

In the rest of this paper, we will describe how to exploit this profile
likelihood method to constrain the CCSN distance, the CCSN models, and the
neutrino properties. To investigate the performances of the different
detectors, we will evaluate the likelihood $\mathcal{L}$ either for a large
number of pseudo-experiments or for a “typical” measurement for which the
Poisson probabilities $P$ are maximal. In the latter case, we take each
observed number of events $N_{i}$ to the equal to the integer part of the
expectation value $\lambda_{i}$ of its underlying Poisson distribution. For
non-integer expectation values, $\lfloor\lambda_{i}\rfloor$ indeed corresponds
to the mode of the associated Poisson distribution.

##  4 Breaking degeneracies: neutrino mass ordering

In this section, we evaluate the rejection p-value of the IMO (NMO) case
assuming the real mass ordering is the NMO (IMO). For this study, we will
first consider the SM, then investigate the impact of new physics processes on
the detectors’ performances using an example BSM scenario. This time, in order
to account for uncertainties on the CCSN distance and progenitor properties,
we perform hypothesis testing using the likelihood function from equation 3.3.
For a given set of observations $\\{\mathcal{O}_{\mathrm{obs}}\\}$, we
optimize the likelihood over the nuisance parameters for each ordering
hypothesis and use the ratio of the resulting likelihoods as a test statistic.
This test statistic will be defined as:

$\displaystyle
t(\\{\mathcal{O}_{\mathrm{obs}}\\})=\frac{\mathrm{max}_{d,M,\bar{r},\zeta}\left[\mathcal{L}(\\{\mathcal{O}_{\mathrm{obs}}\\}|d,M,\bar{r},\zeta,\mathrm{NMO})\right]}{\mathrm{max}_{d,M,\bar{r},\zeta}\left[\mathcal{L}(\\{\mathcal{O}_{\mathrm{obs}}\\}|d,M,\bar{r},\zeta,\mathrm{IMO})\right]}$
(4.1)

where the likelihoods are optimized over the CCSN distance and the CCSN
models, setting $\bar{r}=0$ and $\zeta=1$ if only SM neutrino interactions are
assumed. If the possibility of neutrino decays is accounted for, the
likelihoods will be also optimized over the $\bar{r}$ and $\zeta$ parameters.

To evaluate the significance associated with a given observation, we use a
hybrid Bayesian-Frequentist method which has been introduced by Highland and
Cousins [41] and used in the context of LHC studies [42, 43]. Specifically, to
evaluate the probability distribution of $t$ for a given mass ordering
hypothesis, we sample possible observations from a prior probability
distribution defined as:

$\displaystyle\pi(\mathcal{O}_{\mathrm{obs}},\mathrm{MO})=\int\mathcal{L}(\\{\mathcal{O}_{\mathrm{obs}}\\}|d^{\prime},M^{\prime},\bar{r}^{\prime},\zeta^{\prime},\mathrm{MO})\,\pi_{d}(d^{\prime})\,\pi_{M}(M^{\prime})\,\pi_{\bar{r}}(\bar{r}^{\prime})\,\pi_{\zeta}(\zeta^{\prime})\;\mathrm{d}d^{\prime}\,\mathrm{d}M^{\prime}\,\mathrm{d}\bar{r}^{\prime}\,\mathrm{d}\zeta^{\prime}$
(4.2)

where the $\pi$ functions represent the prior probabilities of the different
parameters. The use of priors allows estimating the distribution of the test
statistics $t$ before an observation is made. Since this step can be
particularly computationally intensive, the use of the hybrid Bayesian-
Frequentist method could allow an alert system to get a quick first estimate
of the neutrino mass ordering less than a few seconds after a CCSN
observation.

For the CCSN distance, we take a uniform prior in $[0.5,60]$ kpc –thus
accounting for CCSNe both in the Milky Way and in the Large Magellanic Cloud.
Since the CCSN distance distribution is expected to peak near the Galactic
Center [11] and no supernova is expected between the Milky Way and the LMC,
the results presented here are expected to be conservative, with the
experiments’ reach being underestimated. For the CCSN model, we take a prior
proportional to the Salpeter initial mass function $\pi(M)\propto M^{-2.35}$,
as discussed in Section 2.1. If only SM processes are assumed, we set
$\bar{r}$ and $\zeta$ to $0$ and $1$, respectively. Conversely, when
accounting for neutrino decays we choose flat priors on $\bar{r}$ and $\zeta$
with $\bar{r}\in[0,10]$ and $\zeta\in[0,1]$ —these priors could later be
updated by alert systems in collaboration with theorists.

To assess the performance of various detectors or combinations of experiments,
we determine the CCSN distance at which the median p-value under the IMO
hypothesis exceeds $3\sigma$ when the null hypothesis is the NMO, and vice
versa. Here, in order to obtain conservative estimates of this $3\sigma$
distance horizon, we consider a $9~{}M_{\odot}$ progenitor. Indeed, this
progenitor type is associated with both the highest prior probability and the
lowest neutrino fluxes in our study.

###  4.1 Standard Model case

Figure 4 shows the $3\sigma$ distance horizon for rejecting the IMO hypothesis
for individual experiments and for pairs of detectors. We see that pairing
Hyper-Kamiokande and IceCube, two large-scale WC detectors, increases this
horizon from $20$ kpc (at HK alone) to $23$ kpc. However, the largest increase
is obtained when pairing HK with DUNE, with a $27$ kpc horizon covering the
whole galaxy. Similarly, the IMO $3\sigma$ distance horizon increases from
$11$ to $18$ kpc and $14$ to $19$ kpc when pairing DUNE with ARGO and IceCube,
respectively. Finally, note that combining KM3NeT or DarkSide-20k with another
detector leads to no increase of the horizon. Indeed, due to the small size of
DarkSide-20k, only a few events are expected in each bin for CCSNe beyond $10$
kpc at this experiment [26], while the high KM3NeT noise rates hinder signal
characterization beyond a few kpc [28].

The $3\sigma$ distance horizons for NMO rejection are shown in Figure 5. Here,
similarly to the IMO rejection case, combining DUNE and HK leads to the
highest significances, with a distance horizon extending up to the LMC.
Combining DUNE with IceCube also leads to a mild improvement of the distance
horizon, from $31$ to $35$ kpc. Note that, while these results demonstrate the
advantage of combining flavor-complementary neutrino experiments for mass
ordering studies, the distance horizons obtained for the NMO rejection
scenario will likely shrink when considering more realistic detector models,
with non-negligible background rates. Such a study, however, is beyond the
scope of this paper since searches for distant supernovae require dedicated
analyses (see for example [44] at Super-Kamiokande) which have not been
designed yet for the experiments we consider.

Figure 4: $3\sigma$ distance horizon (in kpc) for rejecting the IMO when the
NMO is the true mass ordering. The diagonal represents individual detectors
while off-diagonal elements show detector pairs. The hatched regions show
experiments or pairs of experiments with no sensitivity to the mass ordering,
such as DUNE in the presence of neutrino decays and experiments targeting
CE$\nu$NS such as DarkSide-20k and ARGO. Left: SM scenario with a SM
likelihood. Right: BSM scenario with $\bar{r}=5,\zeta=1$, optimizing over
$\bar{r}$ and $\zeta$ in the likelihood.

Figure 5: $3\sigma$ distance horizon (in kpc) for rejecting the NMO when the
IMO is the true mass ordering. The diagonal represents individual detectors
while off-diagonal elements show detector pairs. The hatched regions show
experiments or pairs of experiments with no sensitivity to the mass ordering,
such as DUNE in the presence of neutrino decays and experiments targeting
CE$\nu$NS such as DarkSide-20k and ARGO. Left: SM scenario with a SM
likelihood. Right: BSM scenario with $\bar{r}=5,\zeta=1$, optimizing over
$\bar{r}$ and $\zeta$ in the likelihood.

###  4.2 Impact of neutrino decays on mass ordering determination

To assess the impact of the presence of neutrino decays on the determination
of the mass ordering, we consider a CCSN with a $9~{}M_{\odot}$ progenitor and
a neutrino decay scenario with $\bar{r}=5$ and $\zeta=1$. Here again, to
evaluate the rejection significance of the NMO (IMO) hypothesis, we consider
the median p-value of possible IMO (NMO) measurements. We first consider a
situation where, in spite of the presence of neutrino decays, we optimize the
likelihoods from equation 4.1 assuming the SM. Due to this (wrong) SM
assumption, the distance horizons at the best-performing experiments shrink
significantly. For individual WC detectors, the distance horizons for IMO
rejection shrink down to $\sim 10$ kpc for HK and IceCube and $2$ kpc for
KM3NeT. Combining HK and IceCube leads to a horizon of $17$ kpc, while, in the
SM, the reach of this detector pair extended to $23$ kpc. A similar behavior
is observed for NMO rejection. WC detectors retain partial discriminating
power, with horizons of $36$ kpc and $16$ kpc at HK and IceCube, respectively.
HK therefore retains its ability to probe the mass ordering for any CCSN in
the Milky Way. On the other hand, DUNE’s capability to reject either mass
ordering is completely lost regardless of the mass ordering hypothesis. DUNE’s
poor performance in this context arises from significant distortions in the
expected electron neutrino rates due to neutrino decays. These distortions can
lead to observations in the NMO that resemble a Standard Model IMO scenario,
and vice versa.

We now evaluate the capability of neutrino experiments to distinguish the NMO
and IMO scenarios in the presence of neutrino decays when the likelihoods in
Equation 4.1 are optimized over $\bar{r}$ and $\zeta$ in addition to the other
parameters. The right panels of Figures 4 and 5 show the associated distance
horizons for rejecting the SM IMO and NMO hypotheses, respectively. Here
again, we consider a measurement corresponding to a neutrino decay scenario
with $\bar{r}=5,\zeta=1$ but we now allow $\bar{r}$ and $\zeta$ to vary when
optimizing likelihoods. For NMO rejection, no sizable increase of the distance
horizons are obtained for pairs of experiments compared to single detectors.
The slight shrinking of the horizon observed for HK+DUNE compared to HK alone
is due to our choice of priors333In DUNE, when only event counts are
considered, the IMO and NMO cannot be distinguished if $\bar{r}$ and $\zeta$
are unknown. Due to this degeneracy and to our choice of priors, DUNE’s test-
statistics distribution for NMO measurements is slightly biased towards “IMO-
like” (larger than $1$) values. HK observations are not sufficient to correct
this bias., which does not affect the interpretation of the results since no
CCSN is expected in the $25-50$ kpc region. Conversely, for IMO rejection the
highest distance horizons are obtained by pairing DUNE with HK ($22$ kpc) and
HK with ARGO ($21$ kpc). These distance horizons include more than $99\%$ of
the CCSN candidates in the Milky Way [11].

The results obtained in this section demonstrate the capability of most future
experiments to discriminate neutrino mass ordering hypotheses for more than
$99\%$ of the CCSN candidates in the Milky Way, even when new degrees of
freedom allowing departures from SM physics are introduced. Moreover, several
of the experiments considered in this analysis, such as DUNE, KM3NeT, and HK,
also aim at measuring the neutrino mass ordering using neutrino beams or
atmospheric neutrinos [34, 27, 32]. In the rest of this paper, when estimating
other parameters, we assume the mass ordering to be known unless stated
otherwise. Additionally, due to the limited performance of KM3NeT and
DarkSide-20k, in the following sections we will focus on the DUNE, HK,
IceCube, and ARGO experiments.

##  5 Supernova distance measurement

If a nearby CCSN occurs, the neutrino burst will be detected minutes to hours
before the electromagnetic emission [45]. It is therefore essential for
neutrino detectors to accurately locate the CCSN in real time and transmit
this information to telescopes. Estimating the CCSN distance is all the more
necessary when the supernova takes place in the Galactic plane, where dust can
hinder optical measurements [8]. However, distance measurement algorithms
proposed in the literature assume that neutrino properties are given by the SM
[12, 20]. In this section, we evaluate the bias in CCSN distance estimates
introduced by ignoring neutrino decays for the $\phi_{0}$ scenario introduced
in de Gouvêa _et al_ [19] and discussed in Section 2.3.

Up to now, the methods proposed in the literature [12, 20], and considered by
SNEWS [9], to evaluate the CCSN distance make use of the measured CCSN
neutrino rates during the first $50$ ms following the supernova observation at
a given detector. One approach also incorporates information from the early
accretion phase –the first $100$ to $150$ ms after the CCSN trigger– to reduce
the dependence in the progenitor properties [20]. Here, to estimate the CCSN
distance and the associated uncertainties, we maximize the likelihood shown in
equation 3.3 over the CCSN and neutrino model parameters –assuming the mass
ordering is known– and study its variations as a function of the CCSN
distance.

###  5.1 Methodology: distance fit and uncertainty evaluation

We consider two possible assumptions for the likelihood optimization: the SM
assumption, where $\bar{r}$ is set to $0$ and $\zeta$ to $1$, and the BSM
assumption where $\bar{r}$ and $\zeta$ are allowed to vary. These scenarios
are described by the following likelihood profiles:

$\displaystyle\mathcal{L}_{\mathrm{BSM}}(d,\mathrm{MO})=\mathrm{max}_{\bar{r},\zeta,M}\mathcal{L}(d,M,\bar{r},\zeta,\mathrm{MO})$
(5.1)
$\displaystyle\mathcal{L}_{\mathrm{SM}}(d,\mathrm{MO})=\mathrm{max}_{M}\mathcal{L}(d,M,\bar{r}=0,\zeta=1,\mathrm{MO})$
(5.2)

For either assumption, we consider a typical measurement, as defined in
Section 3.2 for a given set of CCSN and neutrino parameters and evaluate the
corresponding likelihood profile for different CCSN distances. We then
evaluate the confidence interval on the CCSN distance using Wilks’ theorem.

###  5.2 Results

We first evaluate the impact of the CCSN model choice on distance measurements
by estimating the CCSN distance for four progenitors with $9M_{\odot}$,
$11M_{\odot}$, $20M_{\odot}$, and $40M_{\odot}$. The first two models feature
typical, light, progenitors while the last two involve heavier progenitors
with neutrino fluxes up to $\sim 100\%$ larger. Figure 6 shows the $90\%$ and
$68\%$ confidence intervals normalized by the true distance for these
progenitors at $10$ kpc, for measurements made at DUNE and HK. The size and
location of the confidence interval vary significantly between models. In
particular, the confidence intervals are skewed towards higher distances for
the lighter models, associated with lower rates, and towards lower distances
for the heavier models. However, the $68\%$ confidence intervals always
contain the true distance value, thus demonstrating the robustness of our fit.
This robustness has also been verified for other detector pairs and for
individual experiments. In what follows, we will hence focus on a single
model, the $11~{}M_{\odot}$ progenitor, which is associated with both a high
probability and a small bias.

We first consider SM measurements under the SM-only assumption. Figure 7 shows
the median value and the $90\%$ confidence interval for the measured distance
as a function of the true CCSN distance for the four best-performing
experiments from Section 4, under the NMO. At $10$ kpc, the width of the
$90\%$ confidence interval ranges from about $1.5$ kpc at HK and IceCube to
$3.5$ kpc at ARGO. At $25$ kpc, the furthest edge of the galaxy, this width
ranges from $6$ kpc at HK to more than $15$ kpc at ARGO. For the latter,
however, distance estimates are biased towards larger values; hence, setting
an upper bound of $25$ kpc on the CCSN distance would significantly shrink the
confidence range. For a CCSN at $10$ kpc, we then study the impact of
combining flavor-complementary detectors on the distance measurement
precision. Figure 8 shows the confidence bands for a $9M_{\odot}$ progenitor
in the SM, under both the NMO and the IMO, for both individual experiments and
detector pairs. Similarly to [20], we find that the IMO is associated with the
smallest uncertainties. When considering detector pairs, the largest precision
increase on the CCSN distance is observed in the IMO for the HK+IceCube
combination. However, combining flavor-complementary experiments does not lead
to a significant precision increase.

To study the impact of neutrino decays on CCSN distance measurements we
consider a typical measurement for a $11~{}M_{\odot}$ progenitor and for an
example neutrino decay scenario with $\bar{r}=5$ and $\zeta=1$, under which a
large fraction of heavy neutrinos and antineutrinos will have decayed before
reaching Earth. This model is associated with a sizable enhancement
(reduction) of the $\nu_{e}$ and $\bar{\nu}_{e}$ rates in the NMO (IMO), and
with a reduction of the total rate expected at DarkSide-20k or ARGO. As
discussed in Section 2.3, the case of neutrino decays in the IMO illustrates
particularly well the range of possible outcomes for new physics models, as a
wide range of new physics scenarios, such as active-sterile neutrino
conversions, are also associated with a flavor-dependent flux suppression [16,
21, 17]. In what follows, we hence focus on neutrino decay scenarios occurring
in the IMO. The NMO case, which, for the time windows considered, leads to
significantly milder flux distortions compared to the SM, is discussed in
Appendix D. In this scenario, when considering neutrino decays, the size of
the $90$% C.I. on the CCSN distance can be reduced by up to $60$% by combining
HK or IceCube with either DUNE or ARGO. The superior performance of the
HK+ARGO and IceCube+ARGO pairs in the NMO can be explained by the opposite
evolution of the total CCSN rates with $\bar{r}$ in WC and CE$\nu$NS
detectors, which allow breaking the degeneracy between $\bar{r}$ and the CCSN
distance. These results therefore highlight the synergy potential of the three
types of detectors considered in this paper.

Figure 9 shows the measured CCSN distance measured and the $90\%$ confidence
interval as a function of the true distance for the IMO $(\bar{r}=5,\zeta=1)$
scenario described above, for the DUNE and HK detectors as well as for their
combination. In this Figure, the distance estimated using the SM-only
likelihood is compared to the distance obtained by allowing $\bar{r}$ and
$\zeta$ to vary. For DUNE and HK, a clear bias in the distance measurement is
observed under the SM assumption. This bias can be at least partially
corrected by including $\bar{r}$ and $\zeta$ in the CCSN distance fit.
However, for individual detectors, the addition of these extra degrees of
freedom leads to factors of $2$ to $3$ increases of the uncertainties compared
to the SM measurements shown in Figure 7. This uncertainty, however, can be
significantly reduced by combining DUNE and HK. To assess the universality of
this behavior, we study how the size of the $90\%$ C.I. varies with $\bar{r}$
for DUNE, HK, and DUNE+HK. Here, we keep $\zeta=1$ as, in the IMO, the
influence of this parameter on neutrino rates is expected to be negligible for
both DUNE and HK. The dependence of the CCSN distance measurement precision in
$\bar{r}$ is shown in Figure 10. For HK, we observe a continuous increase in
the size of the confidence interval as $\bar{r}$ grows. This increase can be
attributed to the fact that the neutrino decays under consideration primarily
affect the normalization of the time-dependent rate observed at HK, without
significantly altering its shape, as can be seen in Figure 2. In the case of
DUNE, where decay-induced flux distortions are more pronounced than at HK but
the total event count is lower, the uncertainty size initially grows more
rapidly compared to HK before decreasing for large $\bar{r}$ values.
Interestingly, when we combine data from both DUNE and HK, we maintain
distance measurement uncertainties below $25\%$. This combination results in a
substantial improvement over using HK alone, with the size of the confidence
interval shrinking by $40$% to $100$%. This notable behavior underscores how
the synergy between these two experiments enables our analysis to capture the
most significant characteristics of neutrino decay models.

To investigate variations of the CCSN distance precision with the choice of
experiment, we compute the $90\%$ C.I. on the CCSN distance for the same
experiments as the ones considered in Figure 8 for the SM. In Figure 11, we
compare these confidence intervals for a SM measurement, both under the SM and
BSM assumptions. We also revisit the scenario discussed earlier, that is,
neutrino decays with $\bar{r}=5$ and $\zeta=1$, under the BSM assumption. For
each individual detector, we observe a consistent trend: the uncertainties
grow as $\bar{r}$ increases. This trend aligns with our previous observations
in the DUNE and HK cases. As previously discussed, combining detectors that
are sensitive to different neutrino flavors proves effective in mitigating
this uncertainty growth. Specifically, when $\bar{r}=5$, we see that combining
HK or IceCube with DUNE reduces uncertainties by a factor of two. Finally,
note that the introduction of uniform priors on $\bar{r}$ and $\zeta$ for the
BSM likelihood causes the confidence intervals to be tilted towards small
distances, particularly towards smaller distances, for HK and IceCube.
Encouragingly, this bias towards small distances can be mitigated by pairing
either detector with DUNE or ARGO.

While the confidence limits discussed above have been obtained assuming the
neutrino mass ordering is known, we verified that relaxing this assumption
does not change our conclusions. Specifically, when treating the mass ordering
as another nuisance parameter, the precision of the distance measurements does
not change when considering Galactic CCSNe and SM neutrino interactions. When
neutrino decays are introduced, not knowing the mass ordering leads to a
significant loss of precision for DUNE but the confidence intervals for the
other experiments remain stable. Indeed, as discussed in Section 4, HK and
IceCube can discriminate between the IMO and NMO scenarios well beyond the
Galactic Center, while DUNE can distinguish the two mass orderings only in the
SM.

In the context of neutrino decays, we demonstrated that the presence of new
physics in the neutrino sector can significantly bias CCSN distance estimates,
even for models allowed by current neutrino observations. In particular, new
physics scenarios leading to a neutrino flux suppression, such as neutrino
decays in the IMO but also active-sterile neutrino conversions [16, 21, 17],
can lead to overestimating the CCSN distance by up to $70\%$ for a CCSN in the
Galactic bulge. Such an estimate would place the CCSN in a region obscured by
dust, accessible only by telescopes with a deep but narrow field of view like
the Vera Rubin Observatory [8]. This result underscores the necessity for all
types of telescopes to investigate CCSN alerts, even when they initially
appear beyond their observational reach. Additionally, when we introduce new
physics parameters for combinations like DUNE+HK and DUNE+IceCube, we observe
limited growth in uncertainties. This suggests that with the next generation
of neutrino experiments, SNEWS could contemplate incorporating a range of BSM
scenarios into its CCSN parameter fits. Importantly, our algorithm’s
efficiency, with runtimes of under $10$ seconds for detector pairs in a
Jupyter notebook, allows for parallel testing of multiple models without
compromising the alert system’s performance. Finally, we note that this
testing procedure would greatly benefit from preliminary analyses capable of
identifying and characterizing deviations from the SM before conducting the
distance fits. We present examples of such analyses in the next section.

Figure 6: $90\%$ and $68\%$ confidence interval on the measured CCSN distance,
normalized by the true distance, for a supernova at 10 kpc. The measurement is
made by combining DUNE and Hyper-Kamiokande. Here, CCSN models with progenitor
masses of $9$, $11$, $20$, and $40$ solar masses are considered. The black
lines show the median distances. The confidence interval always contains the
true CCSN distance. Neutrino properties are described by the SM. Figure 7:
Median value and $90\%$ confidence interval for the measured CCSN distance at
DUNE (top left), IceCube (top right), Hyper-Kamiokande (bottom left), and ARGO
(bottom right), using the SM likelihood from equation 5.2, assuming the mass
ordering is known to be the NMO and considering a $11~{}M_{\odot}$ progenitor.
The dashed line represents the true CCSN distance. Figure 8: $90\%$ confidence
interval on the measured CCSN distance, normalized by the true distance, for a
supernova with a $11M_{\odot}$ progenitor at $10$ kpc. Here, the performances
of individual experiments and of pairs of detectors are compared. Neutrino
properties are described by the SM. Figure 9: Median values and $90\%$
confidence intervals for the measured CCSN distance, as a function of the true
distance for a $11M_{\odot}$ progenitor and for a neutrino decay model with
$\bar{r}=5,\zeta=1$. The IMO is assumed. The grey band shows the results of
the fit under the SM hypothesis ($\mathcal{L}_{\mathrm{SM}}$ in equation 5.2)
and the orange band shows a fit where $\bar{r},\zeta$ are optimized along with
the other parameters ($\mathcal{L}_{\mathrm{SM}}$ in equation 5.2). The
experiments considered are DUNE (left), HK (middle), and DUNE+HK (right).
Figure 10: Size of the $90\%$ confidence interval on the measured CCSN
distance as a function of $\bar{r}$ for $\zeta=1$, normalized by the true
distance, for a supernova at 10 kpc with a $11~{}M_{\odot}$ progenitor. The
experiments shown here are DUNE (dash-dotted orange), Hyper-Kamiokande (dashed
green), and the combination of the two detectors (solid blue). Figure 11:
$90\%$ confidence intervals on the measured CCSN distance, normalized by the
true distance, for a supernova at 10 kpc, for a $11M_{\odot}$ progenitor and
different detector combinations. The hollow rectangles with a dashed border
show the confidence intervals for the SM under the SM hypothesis. The light
and dark orange rectangles show the confidence intervals for the SM and for a
neutrino decay model with $\bar{r}=5,\zeta=1$, respectively, under the BSM
assumption: both measurements are fitted by optimizing the CCSN progenitor
mass and location, and the $\bar{r},\zeta$ parameters. The IMO is assumed in
both situations. Different detectors and pairs of detectors are considered.

##  6 Neutrino decay characterization

We now describe how to constrain neutrino decay parameters using the
likelihood function introduced in equation 3.3. Since neutrino decay
signatures can be similar to SM scenarios with a wrong mass ordering or CCSN
distance [19], we propose algorithms to either identify deviations from the SM
without making assumptions about the possible BSM processes, or directly
constrain new physics parameters when considering a specific model. Here
again, we assume that the neutrino mass ordering is known.

###  6.1 Identifying beyond-the-SM phenomena

Since accounting for new physics scenarios when locating CCSNe can be a
computationally intensive procedure, we propose to first search for the
presence of neutrino two-body decays using hypothesis testing. For a given set
of measurements $\\{\mathcal{O}_{\mathrm{obs}}\\}$, we take the optimized SM
likelihood $\mathcal{L}_{\mathrm{SM}}$ as a test statistic:

$\displaystyle\mathcal{L}_{\mathrm{SM}}=\mathrm{max}_{d,M}[\mathcal{L}(\\{\mathcal{O}_{\mathrm{obs}}\\}|d,M,\bar{r}=0,\zeta=1)]$
(6.1)

Similarly to the procedure described in Section 4, we generate pseudo-
experiments under the SM hypothesis to evaluate the probability distribution
of $\mathcal{L}_{\mathrm{SM}}$. Here again, we randomly select CCSN models
from [23] with probability $w(M)$. Since we expect this search to be sensitive
only to CCSNe in the Milky Way, we only consider CCSN distances in the
$0.5-25$ kpc range. For each set of decay parameters $(\bar{r},\zeta)$, we
then compute the p-value associated with the typical measurement defined in
Section 3.2.

Figures 12 and 13 show the CCSN distance corresponding to a $3\sigma$
exclusion of the SM as a function of $\bar{r}$ and $\zeta$, in the NMO and the
IMO respectively, for different combinations of measurements at DUNE, HK, and
ARGO. Here, we showcase the performance of our approach and the impact of
combining experiments by selecting the detector (or set of detectors) with the
highest reach over most of the parameter space for each combination of $1$,
$2$, and $3$ experiments. For both mass orderings, DUNE is the individual
detector associated with the largest $3\sigma$ distance horizons over a large
fraction of the $(\bar{r},\zeta)$ space. In the NMO, in particular, models
with large $\bar{r}$ and $\zeta$ can be distinguished from the SM due to the
partial reappearance of the neutronization peak, as discussed in Section 2.3.
For lower values of $\zeta$, however, this neutronization peak is suppressed
again, resulting in a near-degeneracy with the SM, particularly around
$\zeta=0.25$. This degeneracy, also present at HK, can be effectively resolved
by combining DUNE with ARGO. ARGO’s sensitivity to the sum of all neutrino
flavors allows it to discriminate between active neutrino oscillations and
decays into sterile neutrinos. Finally, a combined measurement using DUNE,
ARGO, and HK, leads to an extended horizon at small $\zeta$ but slightly
reduces the horizon at large $\zeta$. For $\bar{r}\gtrsim 3$, distance
horizons exceeding $5$ kpc can be achieved regardless of $\zeta$. Conversely,
in the IMO, there is no degeneracy region with the SM at large $\bar{r}$ for
DUNE as the primary signature of neutrino decays is the suppression of the
neutronization peak, which affects different time windows unequally. While
combining DUNE with Hyper-Kamiokande increases the distance horizon for all
$\zeta$ values, the inclusion of ARGO results in a decreased horizon,
especially for large $\zeta$. Nonetheless, the final reach with the 3-detector
combination remains larger than that of DUNE alone for all $\zeta$ values.

The $(\bar{r},\zeta)$ dependence of the $3\sigma$ horizons shown in Figures 13
and 12, and the dependency of these horizons in the types of detectors
considered, reveal several noteworthy aspects. First, combining multiple
detectors can sometimes lead to a reduction in the search’s reach. Indeed, in
our analysis, we exclusively examine deviations from the SM rather than
fitting neutrino decay parameters. Assuming that we lack prior knowledge of
the CCSN progenitor and its location, adding a new detector could potentially
make observations more similar to an SM scenario. However, note that this
reduction in performance remains limited for the models we have considered.
Second, while combining experiments does not always result in a performance
boost, it allows to identify new physics scenarios which, for individual
experiments, could be completely degenerate with the SM. By combining DUNE, a
WC detector, and ARGO, we ensure that we can identify all flavor-dependent
distortions in the neutrino flux if the event counts are high enough. Third,
the method we propose here stands out for its speed and simplicity, taking
less than a second to execute. It provides a single easily interpretable
output: the p-value of the measurement within the SM framework. This metric
can be assessed not only by organizations like SNEWS but also by individual
experiments. Lastly, although our approach is straightforward, rapid, and does
not rely on assumptions about the possible BSM processes, it is most suitable
for nearby CCSNe. While the choice of a $3\sigma$ criterion can be adjusted
based on the requirements of alert systems and telescopes, exploring regions
like the Galactic Center and beyond may necessitate the consideration of
specific BSM scenarios.

Figure 12: Distance at which a “typical“ observation would correspond to a
$3\sigma$ deviation from the SM, using the test statistic defined in equation
6.1, as a function of $\bar{r},\zeta$ for a $11M_{\odot}$ progenitor under the
NMO. The distance horizons are shown for DUNE alone (left), DUNE+ARGO
(middle), and DUNE+ARGO+HK (right). Figure 13: Distance at which a “typical“
observation would correspond to a $3\sigma$ deviation from the SM, using the
test statistic defined in equation 6.1, as a function of $\bar{r},\zeta$ for a
$11M_{\odot}$ progenitor under the IMO. The distance horizons are shown for
DUNE alone (left), DUNE+ARGO (middle), and DUNE+ARGO+HK (right).

###  6.2 Fitting neutrino decay parameters

In addition to searching for deviations from the SM without assuming a
specific new physics scenario, we investigate the impact of combining flavor-
complementary detectors when fitting the parameters of a given BSM scenario,
again focusing on neutrino two-body decays. As discussed above, this model-
dependent approach could allow identifying deviations from for the SM for
CCSNe beyond a few kiloparsecs. To evaluate the $(\bar{r},\zeta)$ parameters,
we maximize the likelihood from equation 3.3 over $d$ and $M$, assuming that
the neutrino mass ordering is known.

To assess the capability of our approach to identify BSM scenarios, we
consider a model with $\bar{r}=2.5$ and $\zeta=0.5$. With the generic approach
described in Section 6.1, this model could be distinguished from the SM at
$3\sigma$ only for CCSNe closer than $4-5$ kpc. Here, similarly to the SM
scenario discussed above, we show the best-fit point and the
$-2\log\mathcal{L}_{\mathrm{max}}+2.3$, $4.6$, $6.2$, and $11.8$ contours
(which would have corresponded to the $1\sigma$, $90\%$ confidence level,
$2\sigma$ and $3\sigma$ values if Wilks’ theorem were applicable) for the NMO
and the IMO in Figures 14 and 15, respectively, for a $11~{}M_{\odot}$
progenitor. Here again, combining flavor-complementary experiments
significantly increases the precision on $\bar{r}$ and $\zeta$ and breaks the
degeneracies observed for DUNE and HK in the NMO. For both mass orderings,
including ARGO sizeably increases the sensitivity in $\zeta$. Moreover, for
the DUNE+HK+ARGO combination, the likelihood for $\bar{r}=0$ lies outside the
$-2\log\mathcal{L}_{\mathrm{max}}+11.8$ contour, revealing a considerable
performance increase compared to the generic approach from Section 6.1. Hence,
for a wide range of neutrino decay models, evidence for BSM phenomena could be
detected for CCSNe as far as the Galactic bulge. Finally, note that one
possible caveat of this approach is its runtime: in Python, generating the
contours shown in Figures 14 and 15 for a combination of $3$ detectors takes
$\sim 100$ s. However, for these Figures, a particularly fine
$(\bar{r},\zeta)$ grid was used, with steps of $0.05$ for both parameters.
Simply dividing the size of this grid by $4$ would bring the runtime of this
algorithm under one minute with minimal loss of information.

Figure 14: Best-fit point and isolikelihood contours corresponding to
$\mathcal{L}_{\mathrm{max}}+2.3,4.6,6.2$, and $11.8$ (from light to dark) in
the $(\bar{r},\zeta)$ space for a $11M_{\odot}$ progenitor and a neutrino
decay scenario with $\bar{r}=2.5,\zeta=0.5$, assuming the NMO. The contours
are shown for DUNE alone (left), DUNE+ARGO (middle), and DUNE+ARGO+HK (right).
The white dot and white cross show the best-fit and the true values of
$\bar{r}$ and $\zeta$, respectively. Figure 15: Best-fit point and
isolikelihood contours corresponding to
$\mathcal{L}_{\mathrm{max}}+2.3,4.6,6.2$, and $11.8$ (from light to dark) in
the $(\bar{r},\zeta)$ space for a $9M_{\odot}$ progenitor and a neutrino decay
scenario with $\bar{r}=2.5,\zeta=0.5$, assuming the IMO. The contours are
shown for DUNE alone (left), DUNE+HK (middle), and DUNE+HK+ARGO (right).The
white dot and white cross show the best-fit and the true values of $\bar{r}$
and $\zeta$, respectively.

##  7 Discussion

With a minimal set of observables, we have demonstrated that harnessing the
capabilities of the next generation of neutrino experiments can enable CCSN
alert systems to derive meaningful constraints on CCSN properties without the
need for event-by-event energy reconstruction. Taking neutrino decays as an
example, we have illustrated how the presence of new physics in the neutrino
sector can introduce significant biases into CCSN distance estimates. Such
biases could have consequences for electromagnetic follow-ups. In particular,
if the source is estimated to be in the foreground of the Galactic Center,
optical telescopes could be the first to try to identify it, but might fail to
do so because of the significant absorption on the line of sight (see e.g.
[8]). However, we have also shown that when new physics affects neutrino
fluxes in a flavor-dependent manner, it is possible to correct distance
estimates while keeping uncertainties under control. This correction can be
achieved by combining data from WC detectors like HK and IceCube with data
from DUNE. Furthermore, we have highlighted that deviations from the SM can be
efficiently characterized by combining information from WC detectors, DUNE,
and ARGO. While a generic search for anomalies would only detect these
deviations for CCSNe within a few kiloparsecs, tailored fits targeting new
physics models could potentially extend our reach to CCSNe in the Galactic
bulge. The findings presented here also point to several promising avenues for
future exploration, aimed at optimizing realtime CCSN identification.

##### CCSN model dependence of early neutrino rates

Because of the need to compute a large number of progenitors, the supernova
modelling used for our study is restricted to a one-dimensional treatment and
is therefore approximate in several aspects. Multi-dimensional hydrodynamic
instabilities such as PNS convection, neutrino-driven convection and the
Standing Accretion Shock Instability (SASI) [6] are missing from this
modelling. Note, however, that the instabilities do not grow immediately after
bounce: PNS convection starts $\sim 50$ ms after bounce [46], prompt post-
shock convection starts $\sim 50-100$ ms after bounce [6] and SASI reaches
significant amplitudes at still later times of
$\mathcal{O}(100~{}\mathrm{ms})$ [6, 47]. As a consequence, the early-time
dynamics and neutrino rates during the neutronization phase should not be
affected by the restriction to one dimension. On the other hand, the later
accretion could be impacted by the multidimensional dynamics, and its impact
on our results would deserve to be addressed in future studies.
Another important assumption of the supernova modelling is the neglect of
rotation and magnetic fields, which are not expected to play a dominant role
in the explosion dynamics for most stellar progenitors [4]. However, it has
been shown that neutrino-driven post-shock convection [48] and moderate
rotation rates [49] can amplify weak magnetic field seeds up to the point
where they can assist the neutrino-heating mechanism and quantitatively affect
the supernova explosion. Since these results have to rely on computationally
expensive three-dimensional CCSN models, more studies are still needed in
order to accurately assess the impact of rotation and magnetic fields on the
neutrino emission from stellar explosions.

##### Parameterization of new physics effects

As discussed in Section 2.3, new physics scenarios with the potential to
impact early CCSN neutrino rates can be currently classified into three broad
categories. Each of these categories has been studied using simplified
frameworks involving only two or three new parameters [19, 15, 16, 17, 18,
39]. Consequently, incorporating the effects of new physics into realtime CCSN
analyses necessitates the establishment of a strategy for exploring these
diverse frameworks. One approach could involve an alert system simultaneously
fitting several representative BSM scenarios. However, considering the limited
information employed by our algorithm, as well as the current approaches for
CCSN distance measurements [20], it is likely that multiple new physics models
will yield similar predictions. Therefore, the development of a generic
parameterization capable of encompassing multiple classes of BSM scenarios
would greatly enhance the analytical capabilities of alert systems. It is also
worth noting that our analysis exclusively considered flat prior probabilities
for new physics parameters. While transitioning to other realistic prior
distributions is unlikely to qualitatively alter our results, determining the
optimal prior choices at alert systems, in collaboration with theorists, will
be essential for implementing the final analysis.

##### Neutrino energies

While the approach discussed here used only the timing information of neutrino
events, it could be extended to include neutrino energy information. The
inclusion of this energy information, however, would need to be done on an
experiment-by-experiment basis since detectors such as IceCube, KM3NeT,
DarkSide-20k, and ARGO, could only constrain global properties of CCSN
neutrino spectra.

##### Large-scale CE$\nu$NS detectors

In this study, we have demonstrated the potential of experiments sensitive to
the sum of all neutrino flavors (through CE$\nu$NS) for CCSN studies, focusing
on DarkSide-20k and ARGO. While DarkSide-20k may lack the capacity to
constrain CCSN and neutrino properties effectively, ARGO, when combined with
other experiments, could significantly enhance the sensitivity to mass
ordering and the investigation of active neutrino decays into sterile
neutrinos. It is important to note, however, that ARGO is currently in its
early planning stages, and its commissioning date remains uncertain. An
alternative option worth considering is the JUNO experiment, which possesses
sensitivity to both $\bar{\nu}_{e}$ and CE$\nu$NS interactions. Specifically,
for a CCSN located at the Galactic Center, we anticipate approximately $2000$
CE$\nu$NS interactions at JUNO [37, 38]. Depending on its detection
efficiency, JUNO has the potential to achieve a sensitivity level comparable
to, or even surpassing, that of ARGO. To facilitate realistic performance
forecasts for alert systems, it is essential for the JUNO collaboration to
release detailed, energy-dependent detection efficiency curves for CE$\nu$NS.
This data would enable accurate assessments of the experiment’s capabilities.

##### A unified strategy at alert systems

In this study, we have presented various applications of the analysis method
detailed in Section 3. For an alert system like SNEWS, consolidating these
diverse applications into a cohesive procedure is of utmost importance. Our
proposed approach begins by searching for and characterizing deviations from
the SM using the methodologies outlined in Section 6. Furthermore, if
measurements allow for the existence of BSM scenarios capable of significantly
distorting CCSN neutrino fluxes, we can implement dedicated CCSN distance
fits. These fits can follow a similar framework to the one described in
Section 7 for neutrino decays. Additionally, the neutrino MO can be determined
separately using the approach discussed in Section 4, if this MO has not been
already measured by experiments targeting neutrino oscillations. This step,
which takes less than a second, would allow verifying the validity of the
constraints on BSM neutrino interactions derived earlier and would yield a key
result for a wide variety of neutrino searches. The layered approach proposed
here is facilitated by the efficiency of the algorithms involved, with each
step requiring less than a minute of computation using the current Python
implementation.

By conducting a straightforward likelihood analysis, our research underscores
the capability of upcoming experiments to bolster the effectiveness of CCSN
alert systems. Our study notably evaluates the impact of new physics on CCSN
distance estimations and highlights the importance of combining data from
various detectors in order to simultaneously constrain supernova properties
and the physics of the neutrino sector. Consequently, our research sets the
stage for a more accurate realtime CCSN identification process.

## Acknowledgments

This work is supported by LabEx UnivEarthS (ANR-10-LABX-0023 and
ANR-18-IDEX-0001) and Paris Region (DIM ORIGINES). We thank Gwenhaël de
Wasseige, Maria Cristina Volpe and Joachim Kopp for interesting and helpful
discussions. This project has received funding from the European Union’s
Horizon Europe research and innovation programme under the Marie Skłodowska-
Curie grant agreement No 101064953 (GR-PLUTO). Joao Coelho, Alexis Coleiro,
Sonia El Hedri, Davide Franco, Isabel Goos and Antoine Kouchner are supported
by Centre National de la Recherche Scientifique (CNRS). The Moroccan Ministry
of Higher Education, Scientific Research and Innovation is acknowledged.

##  Appendix A Impact of neutrino two-body decays on detected rates

We follow the approach described in [19] and consider only decays of the
heaviest (active) neutrino species $\nu_{h}$ into the lightest (active or
sterile) $\nu_{\ell}$. We consider the following processes:

$\displaystyle\nu_{h,L}$ $\displaystyle\rightarrow\nu_{\ell,L/R}+\phi$ (A.1)
$\displaystyle\nu_{h,L}$ $\displaystyle\rightarrow\bar{\nu}_{\ell,L/R}+\phi$
(A.2)

where $\phi$ is a scalar field and the helicities of the final-state neutrinos
and antineutrinos depend on the model considered. In the Dirac case, if the
neutrinos (antineutrinos) are left-handed (right-handed), they will be visible
in neutrino experiments. Otherwise, they will be sterile and will escape
detection. In the case of Majorana neutrinos all final-state neutrinos will be
visible.

In the analysis shown in this paper, we consider the Dirac $\phi_{0}$ model
introduced in [19], where the neutrinos are Dirac and lepton number is
conserved. In this case, a heavy active (anti)neutrino can decay to either
another active (anti)neutrino via a _helicity-conserving_ interaction or to a
sterile (anti)neutrino via a _helicity-flipping_ interaction. While the
products of the latter will not be detectable at current and upcoming
experiments, helicity-conserving decays lead to an increase in the flux of
light active neutrinos. The fluxes of the heaviest and lightest active
neutrinos as a function of the distance travelled can be expressed as [19, 18,
39]:

$\displaystyle\Phi_{h}(\bar{r},E_{\nu},t)=$ $\displaystyle
e^{-\bar{r}\frac{E_{0}}{E_{\nu}}}\Phi_{h}^{(0)}(E_{\nu},t)$ (A.3)
$\displaystyle\Phi_{\ell}(\bar{r},E_{\nu},t)=$
$\displaystyle\zeta\int_{E_{\nu}}^{\infty}\left(1-e^{-\bar{r}\frac{E_{0}}{E^{\prime}}}\right)\psi(E^{\prime},E_{\nu})\Phi_{h}^{(0)}(E^{\prime},t)\,\mathrm{d}E^{\prime}$
$\displaystyle+\Phi_{\ell}^{(0)}(E_{\nu},t)$

where $\zeta$ is the branching ratio associated with the helicity-conserving
decays of $\nu_{\ell}$ and $\Phi^{(0)}$ represents the neutrino flux right
outside the star. Neglecting the masses of $\nu_{\ell}$ and $\phi$, the
kinematic factor $\psi(E_{h},E_{\ell})$ associated with helicity-conserving
decays is:

$\displaystyle\psi(E_{h},E_{\ell})=\frac{2E_{\ell}}{E_{h}^{2}}.$ (A.4)

Injecting the expression for $\psi$ into equation A and performing the
variable change $y=\dfrac{E^{\prime}}{E_{\nu}}$, the lightest neutrino flux
can be rewritten as

$\displaystyle\Phi_{\ell}(\bar{r},E_{\nu},t)=$ $\displaystyle
2\zeta\int_{1}^{\infty}\frac{1-e^{-\bar{r}\frac{E_{0}}{yE_{\nu}}}}{y^{2}}\Phi_{h}^{(0)}(yE_{\nu},t)\,\mathrm{d}y$
(A.5) $\displaystyle+\Phi_{\ell}^{(0)}(E_{\nu},t).$

When traveling through the star, neutrinos undergo adiabatic MSW flavor
transitions which turn electron neutrinos (antineutrinos) into $\nu_{h}$
($\nu_{\ell}$) states and turn muon and tau neutrinos and antineutrinos into
the other mass eigenstates. Since the muon and tau emissions during the first
$\mathcal{O}(100~{}\mathrm{ms})$ of the CCSN can safely be assumed to be the
same, once outside the star we can model the energy spectra of the different
neutrino mass eigenstates using pinched Fermi-Dirac distributions [50]:

$\displaystyle\Phi_{i}(E,t)=\frac{(\alpha+1)^{\alpha+1}}{\langle
E\rangle^{2}\Gamma(\alpha+1)}L(t)\left(\frac{E}{\langle
E\rangle}\right)^{\alpha}e^{-(\alpha+1)\frac{E}{\langle E\rangle}}$ (A.6)

where $L(t)$ is the luminosity, $\langle E\rangle$ is the mean energy, and
$\alpha$ is a dimensionless parameter. The lightest neutrino flux can
therefore be expressed as:

$\displaystyle\Phi_{\ell}(\bar{r},E_{\nu},t)=$ $\displaystyle
2\zeta\frac{(\alpha+1)^{\alpha+1}}{\langle
E\rangle^{2}\Gamma(\alpha+1)}L(t)\left(\frac{E_{\nu}}{\langle
E\rangle}\right)^{\alpha}$ (A.7)
$\displaystyle\times\int_{1}^{\infty}\left(1-e^{-\bar{r}\frac{E_{0}}{yE_{\nu}}}\right)y^{\alpha-2}e^{-(\alpha+1)\frac{yE_{\nu}}{\langle
E\rangle}}\,\mathrm{d}y$ $\displaystyle+\Phi_{\ell}^{(0)}(E_{\nu},t).$

##  Appendix B Analytical approximation for likelihoods

To compute the likelihood shown in equation 3.3 we evaluate the detection
rates $N_{i}$ for each detector, for all the models described in Section 2.1,
and for the $(\bar{r},\zeta)$ values given by Equation 2.6. To interpolate
between the values from the grid, we fit the number of neutrino events in each
time bin by a fifth-degree polynomial in $\bar{r}$ whose coefficients depend
linearly on $\zeta$:

$\displaystyle N(\bar{r},\zeta)$
$\displaystyle=\sum_{i=0}^{5}C_{i}(\zeta)\bar{r}^{i}$ (B.1) $\displaystyle
C_{i}(\zeta)$ $\displaystyle=A_{i}\zeta+B_{i}.$ (B.2)

Each CCSN model, detector, mass ordering, and time window will be associated
to a set of polynomial coefficients $(A_{i},B_{i})$. An example of this
polynomial fit is shown in Figure 16 for the number of events expected within
the first $10$ ms after the detection of a CCSN with a $9~{}M_{\odot}$
progenitor at the DUNE detector, in the NMO. In the case of DUNE,
discrepancies between the analytical model and the exact rate values are below
$1\%$. The largest discrepancies have been observed for event rate predictions
at DarkSide-20k and ARGO within the first $10$ ms of the CCSN and remain below
$5\%$. We therefore neglect the associated errors in our subsequent analyses.

Figure 16: Interpolation of the number of events expected at DUNE in the NMO
for a $9~{}M_{\odot}$ progenitor model, within the first $10$ ms after the
beginning of the CCSN. Left: expected number of events as a function of
$\bar{r}$ with a 5-degree polynomial fit. Right: $C_{i}$ coefficients of the
5-degree polynomial fit of $\bar{r}$ as a function of $\zeta$.

##  Appendix C Likelihood optimization over the supernova distance

To optimize the likelihood over the CCSN distance, we use the fact that the
expected number of signal events at a given experiment is proportional to the
inverse of the CCSN distance squared. We can therefore rewrite the expectation
values $\lambda_{i}$ in equation 3.4 as:

$\displaystyle\lambda_{i}=S_{i}\left(\frac{d_{0}}{d}\right)^{2}+B_{i}$ (C.1)

where $B_{i}$ is the expected number of background events and $S_{i}$ is the
expected number of signal events at $d_{0}=10~{}$kpc. Optimizing the
likelihood from equation 3.4 then amounts to solving the following equation:

$\frac{\mathrm{d}\log\mathcal{L}(\\{\mathcal{O}_{\mathrm{obs}}\\}|d,M,\bar{r},\zeta,\mathrm{MO})}{\mathrm{d}X}=\sum_{i}S_{i}\left(\frac{N_{i}}{S_{i}X+B_{i}}-1\right)=0$
(C.2)

where $X=\left(\dfrac{d_{0}}{d}\right)^{2}$ and the $i$ index runs over all
time bins and detectors considered. If all the backgrounds are negligible,
this equation can be solved analytically, giving:

$\displaystyle X=\frac{\sum_{i}N_{i}}{\sum_{i}S_{i}}.$ (C.3)

If not all backgrounds can be neglected, we solve equation C.2 numerically.
Since the left-hand side of the equation decreases monotonically for $X\geq
0$, we use Newton’s method with the following initial guess:

$\displaystyle
X_{\mathrm{init}}=\mathrm{max}\left\\{\frac{\sum_{i}N_{i}-B_{i}}{\sum_{i}S_{i}},X_{\mathrm{min}}+1\right\\}$
(C.4)

where $X_{\mathrm{min}}=\left(\frac{d_{0}}{d_{\mathrm{max}}}\right)$ and
$d_{\mathrm{max}}$ is the maximal CCSN distance considered for a given
analysis (we define $X_{\mathrm{min}}$ only when we use priors on CCSN
distances, in Sections 4 and 6.1). Additionally, the left-hand side of
equation C.2 can diverge for $X\rightarrow 0$. In these cases, if Newton’s
method converges towards $X\ll\mathrm{min}_{i}\dfrac{B_{i}}{S_{i}}$, for
$B_{i}>0$, we perform a first-order expansion of equation C.2 in $X$:

$\frac{\mathrm{d}\log\mathcal{L}(\\{\mathcal{O}_{\mathrm{obs}}\\}|d,M,\bar{r},\zeta,\mathrm{MO})}{\mathrm{d}X}\approx-X\sum_{i}\frac{N_{i}S_{i}^{2}}{B_{i}^{2}}+\left(\sum_{i}S_{i}\frac{N_{i}}{B_{i}}-\sum_{i}S_{i}-\sum_{j}S_{j}\right)+\frac{\sum_{j}N_{j}}{X}=0$
(C.5)

where the indices $i$ and $j$ run over observables at experiments with non-
zero and negligible backgrounds, respectively. We then turn this equation into
a degree 2 polynomial and solve it analytically. Finally, independently of the
distance optimization method, we set

$\displaystyle
X_{\mathrm{optimal}}=\mathrm{max}\left\\{X_{\mathrm{sol}},X_{\mathrm{min}}\right\\}$
(C.6)

where $X_{\mathrm{sol}}$ is the numerical solution found by our algorithm.

##  Appendix D CCSN distance measurements: neutrino decays in the NMO

Section 7 presented CCSN distance measurements in the presence of neutrino
decays in the IMO. In this appendix, we present results obtained by applying
the same approach for decay scenarios in the NMO.

Figure 17 shows the median measured CCSN distance and the $90\%$ confidence
interval as a function of the true distance for the $(\bar{r}=5,\zeta=1)$
scenario considered in Section 7, in the NMO, for the DUNE and HK detectors as
well as for their combination. In this Figure, the distance estimated using
the SM-only likelihood is compared to the distance obtained by allowing
$\bar{r}$ and $\zeta$ to vary. For all detectors, a bias towards distances
smaller than the true value is observed for the SM-only likelihood. However,
this bias is milder than for the IMO, with a measured distance of $9$ kpc
found for a CCSN located at $10$ kpc. As in Section 5, we then study how the
size of the $90\%$ C.I. varies with $\bar{r}$ for DUNE, HK, and DUNE+HK.
Again, here, we keep $\zeta$ fixed to one as, for lower values of $\zeta$, the
CCSN neutrino flux will be suppressed leading to effects similar to the ones
observed in the IMO. The dependence of the CCSN distance measurement precision
in $\bar{r}$ is shown in Figure 18. For both HK and DUNE, we now observe a
continuous decrease in the size of the uncertainty as $\bar{r}$ grows. As with
the IMO, combining data from both DUNE and HK, stabilizes distance measurement
uncertainties, maintaining them below $\sim 20\%$ and reducing their size by
up to $60\%$.

To investigate variations of the CCSN distance precision with the choice of
experiment, we compute the $90\%$ C.I. on the CCSN distance for the individual
experiments and detector pairs considered in Figure 11. In Figure 19, we
compare these confidence intervals for a SM measurement, both under the SM and
BSM assumptions. We also revisit the scenario discussed earlier, $\bar{r}=5$
and $\zeta=1$, under the BSM assumption. For DUNE, HK, and IceCube, we observe
again that the uncertainties decrease as $\bar{r}$ increases. Conversely, for
ARGO, as in the IMO case, the uncertainties grow as $\bar{r}$ increases. This
difference in behaviors can be explained by the fact that, in the NMO,
neutrino fluxes increase with $\bar{r}$ at WC detectors and decrease at ARGO,
as can be seen in Figure 2. Here, combining detectors that are sensitive to
different neutrino flavors proves effective in mitigating the growth of
uncertainties at low $\bar{r}$, with reductions of up to $60\%$ with all
combinations for $\bar{r}=0$. For all detector pairs considered, the resulting
uncertainty bands are now comparable to the ones expected in the SM with the
SM-only likelihood, for both high and low $\bar{r}$. A notable finding for
this ordering is the importance of ARGO. While the role of this detector was
limited in the IMO, its effect on mitigating measurement uncertainties in the
NMO is now similar to the one of DUNE. This improved performance can be
explained by the opposite evolution of the neutrino fluxes with $\bar{r}$ at
ARGO compared to the other detectors. Indeed, since the neutrino flux
normalization is strongly correlated with the CCSN distance, this opposite
evolution can significantly narrow down the range of possible CCSN locations.

Figure 17: Median values and $90\%$ confidence intervals for the measured CCSN
distance, as a function of the true distance for a $11M_{\odot}$ progenitor
and for a neutrino decay model with $\bar{r}=5,\zeta=1$. The NMO is assumed.
The grey band shows the results of the fit under the SM hypothesis
($\mathcal{L}_{\mathrm{SM}}$ in equation 5.2) and the orange band shows a fit
where $\bar{r},\zeta$ are optimized along with the other parameters
($\mathcal{L}_{\mathrm{SM}}$ in equation 5.2). The experiments considered are
DUNE (left), HK (middle), and DUNE+HK (right).  Figure 18: Size of the $90\%$
confidence interval on the measured CCSN distance as a function of $\bar{r}$
for $\zeta=1$, normalized by the true distance, for a supernova at 10 kpc with
a $11~{}M_{\odot}$ progenitor. The mass ordering is assumed to be the NMO. The
experiments shown here are DUNE (dash-dotted orange), Hyper-Kamiokande (dashed
green), and the combination of the two detectors (solid blue). Figure 19:
$90\%$ confidence intervals on the measured CCSN distance, normalized by the
true distance, for a supernova at 10 kpc, for a $11M_{\odot}$ progenitor and
different detector combinations. The hollow rectangles with a dashed border
show the confidence intervals for the SM under the SM hypothesis. The light
and dark orange rectangles show the confidence intervals for the SM and for a
neutrino decay model with $\bar{r}=5,\zeta=1$, respectively, under the BSM
assumption: both measurements are fitted by optimizing the CCSN progenitor
mass and location, and the $\bar{r},\zeta$ parameters. The NMO is assumed in
both situations. Different detectors and pairs of detectors are considered.

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