#  Absolutely Scintillating:
constraining $\nu$ mass with black hole-forming supernovae

George Parker<EMAIL_ADDRESS>Michael Wurm<EMAIL_ADDRESS>PRISMA+ Cluster of Excellence and Institut für Physik,
Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany

###### Abstract

The terrestrial detection of a neutrino burst from the next galactic core-
collapse supernova (CCSN) will provide profound insight into stellar
astrophysics, as well as fundamental neutrino physics. Using Time-Of-Flight
(ToF) effects, a CCSN signal can be used to constrain the absolute neutrino
mass. In this work, we study the case where a black hole forms during core-
collapse, abruptly truncating the neutrino signal. This sharp cutoff is a
feature that can be leveraged in a ToF study, enabling strict limits to be set
on the neutrino mass which are largely model-independent. If supernova
neutrinos are detected on Earth in liquid scintillator detectors, the
exceptional energy resolution would allow an energy-dependent sampling of the
ToF effects at low neutrino energies. One promising experimental program is
the Jiangmen Underground Neutrino Observatory (JUNO), a next-generation liquid
scintillator detector currently under construction in China. Using three-
dimensional black hole-forming core-collapse supernova simulations, the
sensitivity of a JUNO-like detector to the absolute neutrino mass is
conservatively estimated to be $m_{\nu}<0.39^{+0.06}_{-0.01}$ for a 95% CL
bound. A future-generation liquid scintillator observatory like THEIA-100
could even achieve sub-0.2 sensitivity.

##  I Introduction

Since massless neutrinos were refuted experimentally by the discovery of
neutrino flavour oscillations [1, 2], the neutrino absolute mass scale has
been an open question. Neutrinos are less than $\sim 10^{-6}$ the mass of the
next lightest elementary particle. A non-zero neutrino mass is physics Beyond-
the-Standard Model, and the relative size of this mass suggests a novel
process for mass generation. Therefore, measuring the exact mass of the
neutrino is of great scientific interest and has been a subject of intense
experimental and theoretical work.

Massive neutrinos from core-collapse supernovae (CCSNe) propagate across an
extreme distance before reaching Earth, embedding an energy- and mass-
dependent delay in the signal. This Time-Of-Flight (ToF) effect can be used to
set an upper limit on the absolute neutrino mass. The idea of using neutrinos
from core-collapse supernovae (CCSNe) as a probe of neutrino mass was first
described by Zatsepin [3]. This effect is the delay of a massive neutrino
compared to a massless particle travelling at the speed of light. It is
derived from the delay time between two ultrarelativistic particles (for the
general formulation see [4, 5, 6]). At lowest order, the relation between the
delay $\Delta t$, distance $D$, neutrino mass $m_{\nu}$, and neutrino energy
$E_{\nu}$, is:

$\Delta t=\frac{D}{2c}\left(\frac{m_{\nu}}{E_{\nu}}\right)^{2},$ (1)

which can be rewritten in convenient units, where the D is in units of $10$
kpc, which the average distance to the Galactic centre [7, 8]:

$\Delta
t[\text{s}]=0.515\left(\frac{m_{\nu}[\text{eV}]}{E_{\nu}[\text{MeV}]}\right)^{2}D[\text{10
kpc}].$ (2)

When neutrinos from core-collapse supernova SN1987A were detected on Eart, it
presented the first, and only, opportunity to apply ToF techniques. Kamiokande
II detected 12 events [9], IMB detected 8 [10], and 5 events were detected in
Baksan [11]. Using the data from Kamiokande II and IMB, an upper limit derived
of ${m_{\nu}}<12.0$ (90% CL) [12]. Later, this limit was tightened using a
parameterized model of CCSN neutrino emission and better analysis of
backgrounds (allowing the use of the Baksan events) achieving a limit of
${m_{\nu}}<5.7$ [13]. This was updated to ${m_{\nu}}<5.8$ [14] with an
improved parameterization and likelihood analysis.

Beacom et al [7, 8] first pointed out that if a CCSN evolves into a black
hole, a much tighter neutrino mass limit can be set using the ToF effect.
Black hole formation creates an abrupt cutoff in the neutrino signal, making
ToF effects clearer, and leading to an improved mass sensitivity compared to a
different stellar evolution. Instead of carrying out a fit on the entire
neutrino curve [14], black hole formation allows the consideration of events
only immediately before and after the black hole formation time,
$t_{\textrm{BH}}$. This limits the systematic uncertainty introduced by the
largely experimentally undetermined model for CCSN neutrino emission. Beacom
et al considered a neutrino signal with black hole formation in the context of
the Super-Kamiokande detector [15], deriving a sensitivity of 1.8 for a
canonical CCSN at 10 kpc distance [7, 8].

Lu et al [16] explored the neutrino mass constraint from CCSN neutrinos
detected with a next-generation liquid-scintillator detector. In particular,
they demonstrate that in the Jiangmen Underground Neutrino Observatory (JUNO)
[17] it would be possible to set a mass limit of $m_{\nu}<$ 0.83 $\pm$ 0.24
(95% CL), competitive with current laboratory limits. However, this bound is
obtained without black hole formation. In this work, we show that if the
signal of a black hole-forming core-collapse supernova is observed in a liquid
scintillator detector, the low detection threshold and excellent energy
resolution would allow for a fine-grained analysis of the potential ToF
signature, allowing a model-independent and competitive constraint on the
absolute neutrino mass to be achieved.

The present paper is structured as follows: Sec. II.1 is a short review of the
fundamentals of neutrino mass and existing measurements. Sec. II.2 is a brief
explanation of CCSNe evolution with the corresponding structure of the
neutrino emission emphasised in Sec. II.3. Sec. II.4 is a discussion of the
target channel (inverse beta decay), with a short overview of other next-
generation neutrino detectors for similar studies. A description of the
simulations and parameterizations used in this study follows in Sec. III.1.
Sec. III.2 develops the approach used to arrive at an upper $m_{\nu}$ limit
that could be set in case of non-observation. The resulting neutrino mass
sensitivity is presented in Sec. IV.1, with the systematic uncertainties of
this technique evaluated in Sec. IV.2. The effects of the detector mass and
distance to Earth are quantified in Sec. IV.3 and IV.4, respectively. Sec. V
is devoted to systematic effects that have the potential to weaken the derived
mass limit, but in practice do not have a large influence on the derived
bound.

##  II Background

###  II.1 Neutrino Mass Measurements

Neutrino flavour oscillations are dependent on the mass-squared differences,
${\Delta}m_{ij}^{2}={\Delta}m_{i}^{2}-{\Delta}m_{j}^{2}$. However, the sign of
the mass difference ${\Delta}m_{31}$ is not known, leaving two possible
neutrino mass patterns in nature. The first is ’Normal Ordering’ (NO), where
$m_{3}>m_{2}>m_{1}$, and the second is ’Inverted Ordering’ (IO), where
$m_{2}>m_{1}>m_{3}$ [18].

Although oscillation phenomena are not sensitive to the absolute scale of
neutrino masses, we can set lower limits on the sum of neutrino mass states,
$\sum{m_{\nu}}$, using the mass-squared differences. If we set the lightest
mass state to zero, $m_{0}=0$, in Eq. (3) [NO] and Eq. (4) [IO], we calculate
limits of $\sum{m_{\nu}}\gtrsim 0.06$ and $\sum{m_{\nu}}\gtrsim 0.1$ ,
respectively [19].

$\sum{m_{\nu}}=m_{0}+\sqrt{m_{0}^{2}+{\Delta}m_{21}^{2}}+\sqrt{m_{0}^{2}+{\Delta}m_{31}^{2}}$
(3)
$\sum{m_{\nu}}=m_{0}+\sqrt{m_{0}^{2}+{\Delta}m_{31}^{2}}+\sqrt{m_{0}^{2}+{\Delta}m_{31}^{2}+{\Delta}m_{21}^{2}}$
(4)

Cosmological probes including the Cosmic Microwave Background, Type Ia
supernovae, and Baryon Acoustic Oscillation observations are also sensitive to
the sum of the neutrino mass states, giving an extremely strong upper limit,
$\sum{m_{\nu}}<0.11$ (95% CL) [20]. This limit can be improved to
$\sum{m_{\nu}}<0.09$ (95% CL) in the most constraining case [21], putting it
in a weak tension with the lower limit for IO. However, these limits are
strongly dependent on cosmological models and therefore other measurements of
the absolute neutrino mass are needed to corroborate this value [20, 21].

The strongest limit on the neutrino mass from direct measurements is the
results from the KATRIN experiment, which uses the kinematics of tritium
$\beta$-decays to derive an upper mass limit for $\bar{\nu}_{e}$ of 0.8 (90%
CL) [22]. KATRIN is projected to be able to achieve a mass limit approaching
0.2 (90% CL) across its full experimental run [23].

The absolute neutrino mass limit could be further advanced by the Project 8
experiment, an advanced experimental program also targeting tritium
$\beta$-decays with a new spectroscopy method based on cyclotron radiation.
The projected final sensitivity of Project 8 is 0.04 eV [24].

###  II.2 Black Hole-Forming Core-Collapse Supernovae

When a massive star (mass $\gtrsim 8M_{\odot}$, where $M_{\odot}$ is the solar
mass) exhausts its nuclear fuel, its core collapses producing a neutron star
or black hole. Considering historical supernovae, the rate of Galactic
supernova can be calculated to be $R=3.2^{+7.3}_{-2.6}$ (century)-1 [25], but
could also be as low as $R=1.63\pm 0.46$ (century)-1 from a conservative,
combined analysis [26].

The fraction of massive stars that end their life as a black hole compared to
a neutron star is known as $f_{\textrm{BH}}$. This quantity is not
theoretically well-known, with estimates in the region
$f_{\textrm{BH}}\lesssim 0.30–0.35$ [27]. Observational surveys with the Large
Binocular Telescope give a value of $f_{\textrm{BH}}=0.16^{+0.23}_{-0.12}$
(90% CL) with 11 years of data [28].

In this section, we will briefly describe the stages of core-collapse, and
then go into detail on the resulting neutrino flux (for detailed reviews of
the CCSN evolution, see [29, 30, 31]).

#### Pre-Explosion

The chain of nuclear fusion has resulted in an iron core surrounded by layers
of successively lighter elements (’onion-shell’ structure) [29, 30, 31].

#### Core-collapse and -bounce

The outer core collapses supersonically, and the inner collapses subsonically
[31]. As the core density increases, neutrinos become trapped. The inner core
rebounds against the infalling outer core [30], and a shock wave forms at the
interface.

#### Shock propagation and stall

The shock propagates through the surrounding material [31]. Electron neutrinos
produced by electron capture on free protons,
$\mathit{{e}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{-}}}+p\rightarrow\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{\\!e}}^{\mspace{-1.0mu}\scriptstyle{}}}+n$,
are suddenly free, causing the ’neutronization’ burst in the
$\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{\\!e}}^{\mspace{-1.0mu}\scriptstyle{}}}$
neutrino signal [29, 30, 31]. The shock continues to lose energy with it
eventually stalling [31, 32, 33].

#### Accretion and neutrino heating

The dense core reaches approximate hydrostatic equilibrium with the
surroundings [32] with matter accreting onto the shock front [29]. Large-
scale, non-radial instabilities such as standing accretion-shock instability
(SASI) can develop [34], where the shock front undergoes a violent sloshing
and spiral oscillation, mixing the material underneath and leading to mass
asymmetry [31, 33].

#### Black hole formation

In the scenario relevant for this work, the explosion is not successful, and
matter continues to accrete onto the ’proto-neutron star’ until it collapses
into a black hole.

###  II.3 Expected Neutrino Signal

Figure 1:  The neutrino luminosity for
$\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{\\!e}}^{\mspace{-1.0mu}\scriptstyle{}}}$
(black), $\bar{\nu}_{e}$ (blue), and $\nu_{x}$ (red). Where $v_{x}$ represents
the other flavours
($\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{\\!\mu}}^{\mspace{-1.0mu}\scriptstyle{}}}$,
$\bar{\nu}_{\mu}$,
$\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{\\!\tau}}^{\mspace{-1.0mu}\scriptstyle{}}}$,
$\bar{\nu}_{\tau}$) which are predicted to act almost identically in the
context of core-collapse. Model (s40) from [33], accessed through the Garching
CCSN Archive [35].

The expected neutrino emission begins at core-bounce, with ’time’ in the CCSN
simulations meaning the post-bounce time, $t_{\textrm{pb}}$ [32]. The sharp
peak in
$\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{\\!e}}^{\mspace{-1.0mu}\scriptstyle{}}}$
luminosity in Fig. 1 is the neutronization burst. Next, we expect a large,
quasistationary flux in the accretion phase, with the potential for SASI to
cause noticeable modulations in the flux corresponding to the SASI frequency.
Finally, black hole formation during the accretion phase abruptly truncates
the neutrino signal at $t_{\textrm{BH}}$ [32, 33]. Black hole formation, the
neutronization burst, and even SASI modulations could be effective targets for
ToF analyses.

If no black hole formed, a successful explosion would be followed by a cooling
phase (see [31]). The cooling phase would constitute the majority of the $\nu$
signal [32] in this case.

###  II.4 Detection for ToF studies

In liquid scintillator and water detectors, electron antineutrinos,
$\bar{\nu}_{e}$, participate in inverse beta decay (IBD) on free protons,

$\bar{\nu}_{e}+p\rightarrow\mathit{{e}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{+}}}+n,$

which is the dominant detection channel. It provides both the largest
statistics ($\mathcal{O}(10^{3})$) for next-generation liquid scintillator and
water Cherenkov detectors [15] for a canonical CCSN at 10 kpc. Crucially for
this study, IBD provides a direct relation between the neutrino and positron
energy of the prompt signal. The delayed capture of the neutron on H (or Gd
[36]) can be used to remove potential backgrounds from elastic neutrino-
electron scattering and other interaction channels. For a review of the
various channels and detectors for CCSN neutrinos, see [32].

Super-K (Super-Kamiokande) and its planned successor experiment Hyper-K
(Hyper-Kamiokande) [37] are Cherenkov detectors with fiducial volumes of 22.5
and 187 of ultrapure water, respectively. Beacom et al derive an upper limit
of $m_{\nu}<1.8\unit{}$ considering the IBD channel in the Super-K detector
[7, 8]. As the mass limit scales with detector mass as $m_{\textrm{lim}}\sim
M_{D}^{-1/2}$, we comment that the result of 1.8 eV in Super-K from Beacom et
al can be trivially scaled for Hyper-K–giving a mass limit of $\sim 0.6$ eV
for a detector $\sim 8$ times larger.

In the present study, we focus on JUNO, the Jiangmen Underground Neutrino
Observatory, a liquid scintillator detector situated southwest of Kaiping city
in Guangdong province in southern China [17]. JUNO has a fiducial mass of 20
and has 17,612 20-inch PMTs. The detector is currently under construction and
is projected to begin data taking in 2024 [38].

While JUNO will provide similar statistics in the IBD channel to Super-K, JUNO
has a better energy resolution $\delta E/E\sim 3\%/\sqrt{E(\unit{\mega})}$ and
an exceptional energy threshold of 0.2 [17]. As the ToF effect, $\Delta
t\propto\frac{1}{E^{2}}$ (Eq. 2), is most pronounced at lower energies, we
expect JUNO to be substantially more sensitive to neutrino mass than Super-K.

We also consider THEIA-100, a proposed 100 detector containing water-based
liquid scintillator (WbLS) [39]. By doping ultrapure water with liquid
scintillator, it is possible to achieve a balance between Cherenkov light
which allows impressive directional resolution, and scintillation light which
gives excellent energy resolution. In [40], it has been shown that WbLS with
5% organic loading would be sufficient to achieve an MeV-level threshold.
Given the large target mass of THEIA-100, we expect an extremely tight limit
to be set on the neutrino mass, exceeding the projected final sensitivity of
KATRIN.

##  III Methods

###  III.1 Simulated Supernova Models

The sharp cutoff of the neutrino signal by black hole formation is largely
model-independent and therefore provides an excellent feature to search for a
mass-induced ToF delay. We use three-dimensional black hole-forming CCSN
simulations to generate realistic IBD spectra and statistics, but the general
analysis approach does not depend on the models used.

The two non-rotating black hole-forming models s40 [41] and u75 [42] originate
from progenitors of mass 40$M_{\odot}$ and 75$M_{\odot}$, respectively.
According to the common notation of CCSN simulations, the prefix ’s’ denotes
solar metallicity ($Z_{\odot}\sim 0.0134$) and the prefix ’u’ denotes ultra-
low metallicity ($Z\sim 10^{-4}Z_{\odot}$) (additionally the prefix ’z’
denotes zero metallicity) [27].

The models s40 and u75 are realised in three-dimensional hydrodynamical
simulations with three neutrino flavours
($\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{\\!e}}^{\mspace{-1.0mu}\scriptstyle{}}}$,
$\bar{\nu}_{e}$, and $\nu_{x}$) [33]. The simulations were carried out with
PROMETHEUS-VERTEX code [43] which approximates general relativity effects with
a modified Tolman-Oppenheimer-Volkoff potential [44]. Further details of the
simulation can be found in [33].

Access to these models is possible with the Garching CCSN Archive [35]. The
neutrino data is given in the parameters of neutrino luminosity, $L$, mean
energy $\langle E\rangle$, and second moment of energy, $\langle
E^{2}\rangle$, for each time-step and flavour. The pinching factor, $\alpha$,
which is a measure of the spectral pinching (with $\alpha=2$ being a Maxwell-
Boltzmann spectrum), can be calculated using [45]:

$\frac{\langle E^{2}\rangle}{\langle E\rangle^{2}}=\frac{2+\alpha}{1+\alpha}.$
(5)

The (normalized) pinched-thermal functional form [46, 18] can be used to
construct the neutrino flux information, $\phi(E)$, for each flavour:

$\displaystyle\Phi^{0}(E)$ $\displaystyle=\left(\frac{L}{\langle
E\rangle}\right)\frac{(\alpha+1)^{(\alpha+1)}}{\Gamma(\alpha+1)}\left(\frac{E}{\langle
E\rangle}\right)^{\alpha}\exp\left(-\frac{(\alpha+1)E}{\langle
E\rangle}\right).$ (6)

The Mikheyev-Smirnov-Wolfenstein (MSW) effect occurs at the outer envelope,
due to the supernova density profile and resonant flavour conversions [47, 48,
49, 50, 51]. Assuming adiabatic transitions, this can be written in the form
[52] for NO:

$\displaystyle\Phi_{\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{\\!e}}^{\mspace{-1.0mu}\scriptstyle{}}}}=$
$\displaystyle\Phi^{0}_{\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{\\!e}}^{\mspace{-1.0mu}\scriptstyle{}}}},$
(7a) $\displaystyle\Phi_{\bar{\nu}_{e}}=$
$\displaystyle\Phi^{0}_{\bar{\nu}_{e}}{\cos^{2}\theta_{12}+\Phi^{0}_{\nu_{x}}\sin^{2}\theta_{12}},$
(7b) $\displaystyle\Phi_{\nu_{x}}=$
$\displaystyle\Phi^{0}_{\nu_{x}}\frac{2+\cos^{2}\theta_{12}}{4}+\Phi^{0}_{\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{\\!e}}^{\mspace{-1.0mu}\scriptstyle{}}}}\frac{1}{4}+\Phi^{0}_{\bar{\nu}_{e}}\frac{\sin^{2}\theta_{12}}{4}$
(7c)

and for IO we have:

$\displaystyle\Phi_{\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{\\!e}}^{\mspace{-1.0mu}\scriptstyle{}}}}=$
$\displaystyle\Phi^{0}_{\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{\\!e}}^{\mspace{-1.0mu}\scriptstyle{}}}}{\sin^{2}\theta_{12}+\Phi^{0}_{\nu_{x}}\cos^{2}\theta_{12}},$
(8a) $\displaystyle\Phi_{\bar{\nu}_{e}}=$ $\displaystyle\Phi^{0}_{\nu_{x}},$
(8b) $\displaystyle\Phi_{\nu_{x}}=$
$\displaystyle\Phi^{0}_{\nu_{x}}\frac{2+\sin^{2}\theta_{12}}{4}+\Phi^{0}_{\bar{\nu}_{e}}\frac{1}{4}+\Phi^{0}_{\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{\\!e}}^{\mspace{-1.0mu}\scriptstyle{}}}}\frac{\cos^{2}\theta_{12}}{4},$
(8c)

where $\theta_{12}$ is the solar mixing angle, which is taken to be
$33.45^{+0.77}_{-0.75}$∘ [53].

The event rate is calculated for the canonical distance to a galactic
supernova, $D=10$ kpc. For JUNO, we use the number of free protons in the
detector, $N_{p}=1.5\times 10^{33}$, and the IBD cross-section,
$\sigma_{\textrm{IBD}}$ [54].

$\dfrac{\mathrm{d}^{2}N}{\mathrm{d}E\mathrm{d}t}=N_{p}\frac{1}{4\pi
D^{2}}\Phi_{\bar{\nu}_{e}}\sigma_{\textrm{IBD}},$ (9)

Figure 2:  Detected IBD ($\bar{\nu}_{e}$) spectrum in JUNO as a function of
energy and time, with colours representing normalized number of neutrino
events. A projection of the energy-integrated event profile is included. Black
hole formation with a massless neutrino ensures no ToF delay, producing the
sharp cutoff of events at $t_{\mathrm{BH}}\sim$ 570 ms. See text for more
information.

where input variables and smearing effects have been dropped for simplicity.
The nominal JUNO energy resolution and threshold have been used when
calculating energy smearing [17]. Note that in this case energy is the
detected positron energy
$E_{\mathit{{e}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{+}}}}=E_{\nu}-1.293$
.

For s40 we expect $\sim$ 7000 IBD events in JUNO, with $\sim$ 4000 events for
u75. Although s40 has a lower flux rate at $t_{\mathrm{BH}}$ compared to u75,
the integrated event number is higher because the black hole forms at a later
time ($\sim$ 570 ms post-bounce for s40, and $\sim$ 250 ms post-bounce for
u75).

As part of this study, we used the supernova neutrino package SNOwGLoBES [55],
which includes multiple detector configurations and detection channels and can
be used to approximate the fluxes, mixing effects, and expected event rates.

Fig. 2 is a 2D plot of the detected IBD ($\bar{\nu}_{e}$) event spectrum in
JUNO to highlight the energy-time structure of the signal. A projection is
given at the top of the plot to show the classic energy-integrated event
profile. For all of the following plots we use model s40 from [33] accessed
from the Garching CCSN Archive [35], with the canonical distance from Earth
(10 kpc) and normal neutrino mass ordering (NO), unless stated otherwise.

###  III.2 Statistical Procedure

In this paper, we use the numerical method in [16] as a guide. In this work,
Lu et al also constrain the absolute neutrino mass using CCSN neutrinos
detected in JUNO, but without black hole formation. Their method involves a
parameterization of neutrino emission [56] with eight parameters: one for the
absolute start time, one for the early or ’rising’ time, three for the
accretion phase, and three for the cooling phase. The method in [16] fits all
eight parameters to artificial data with varying neutrino mass delays, and
performs a likelihood analysis to derive an upper limit of the absolute
neutrino mass. This provides a model-dependent bound of 0.83 $\pm$ 0.24 (95%
CL) for NO.

However, black hole formation allows a great simplification of this analysis,
and ensures the mass limit is largely independent of the supernova model.
Since the black hole forms during the accretion phase, the neutrino flux and
spectrum can be assumed as quasistationary for a time period of several 10 ms
before $t_{\mathrm{BH}}$. In our analysis, we use a period of 75 ms before
black hole formation to extract the spectral shape and rate of neutrino signal
prior to its cutoff. This fitted region is presented in Fig. 3. While our fit
procedure assumes the fit to be stationary, flux and energy are expected to
vary considerably to effects such as SASI oscillations. Our fit window is
chosen sufficiently long to average out these effects.

Figure 3:  Fitted region of detected IBD ($\bar{\nu}_{e}$) spectrum in JUNO as
a function of energy and time, with colours representing normalized number of
neutrino events. A projection of the energy-integrated event profile is
included. Black hole formation and ToF delay corresponds to a 1.0 eV neutrino
mass. See text for more information.

For this period, we create artificial data sets with a uniform spectra using
the pinched-thermal functional form in Eq. (6). We then induce a ToF delay in
the artificial data sets consistent with absolute neutrino masses from
$0.0-1.5$ in steps of $0.025$ . A binned likelihood function with Poisson
statistics is used to compare the data from the simulation and the artificial
data set to derive a median sensitivity. For each neutrino mass step, we
calculate the $\Delta\chi^{2}$ with 4 free parameters: the time of black hole
formation $t_{\textrm{BH}}$ (to mitigate the effect of $t_{\textrm{BH}}$ being
wrongly characterised), the pinching parameter, $\alpha$, the mean energy,
$\langle E\rangle$, and the normalisation, $\mathcal{N}$.

##  IV Results

###  IV.1 Constraint

To derive limits for the $m_{\nu}$ sensitivity, we apply the technique
outlined in Sec. III.2 to the model signal spectra discussed in Sec. III.1.
For this, we investigate two progenitor models (s40 and u75), both possible
neutrino mass orderings in nature (NO and IO), as well as different
assumptions for visibility of SASI in the signal. We present both a mass limit
from where the neutrino emission has been spherically averaged (Fig. 2 and 3),
and from the specific observer direction corresponding to the highest number
of events (denoted ’Direction 1’ for s40 and the ’Strong Modulations’
direction for u75). The specific observer direction and spherically averaged
cases are compared in Fig. 4 for model s40. A 2D energy-time plot of neutrino
events for the observer direction is presented in Fig. 5 to be contrasted with
the spherically averaged case in Fig. 2.

Figure 4:  Electron antineutrino ($\bar{\nu}_{e}$) flux for ’Direction 1’ and
the spherically averaged case for 1) no mixing, 2) normal ordering (NO) and 3)
inverted ordering (IO). Model s40 [33] accessed from [35].

The signals from the specific observer directions have the imprint of violent
SASI activity. By also obtaining a mass limit from these signals we highlight
the robustness of this method. As our constant fit averages over several
oscillations we expect that SASI will not influence our neutrino mass
sensitivity.

We try different SASI realizations but always obtain very similar mass limits
because our constant fit averages over several oscillations. Hence, SASI is
not expected to affect our neutrino mass sensitivity.

Fig. 4 also shows the effect of the adiabatic MSW effect for NO and IO. This
effect lowers the $\bar{\nu}_{e}$ flux, with the IO having the larger decrease
when compared to NO. The plot of luminosities (Fig. 1) shows in the accretion
phase
$\Phi^{0}_{\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{\\!e}}^{\mspace{-1.0mu}\scriptstyle{}}}}\approx\Phi^{0}_{\bar{\nu}_{e}}>\Phi^{0}_{x}$.
Equations (7b) and (8b) show that $\bar{\nu}_{e}$ is a full swap with
$\nu_{x}$ for IO, and a superposition of
$\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{\\!e}}^{\mspace{-1.0mu}\scriptstyle{}}}$
and $\nu_{x}$ for NO, explaining the discrepancy in Fig. 4. This reduction in
the total IBD events for IO results in a less restrictive mass limit compared
to NO. Conversely, for
$\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{\\!e}}^{\mspace{-1.0mu}\scriptstyle{}}}$
in DUNE for example, the relevant equations would be (7a) and (8a), and IO
would ensure a tighter constraint than NO.

Table 1:  Calculated upper limit for $m_{\nu}$ at 95% CL for the two spherically averaged models and possible mass orderings. All values in . Mass Ordering | 40$M_{\odot}$ model | 75$M_{\odot}$ model
---|---|---
NO | 0.40 | $0.32$
IO | 0.45 | $0.35$

Exact upper limits in Table I are derived using the method described in the
previous section (in Sec. IIIB); producing a $\Delta\chi^{2}$ curve and then
requiring $\Delta\chi^{2}>3.84$ for 95% CL, using the $\Delta\chi^{2}$
distribution with one degree of freedom. It should be noted we do not follow
the more complex procedure from [16] where the deviation from the true
$\Delta\chi^{2}$ distribution is investigated which means that we are placing
a slightly less stringent limit as would be the case in the more accurate
treatment.

Figure 5:  Plot of Direction 1 to be compared with the spherically averaged
plot (Fig. 2). Detected IBD ($\bar{\nu}_{e}$) spectrum in JUNO as a function
of energy and time, with a projection of events in time. Normalized neutrino
events are represented by colours. Black hole formation with a massless
neutrino ensures no ToF delay.

Model u75 provides a tighter constraint of 0.32 eV compared to 0.40 eV for s40
(for NO), as it has a higher event rate in the detector immediately before
black hole formation. Note that progenitor mass and neutrino luminosity are
not related in a simple way. The neutrino output is dependent on many
interrelated factors including metallicity, stellar structure, mass accretion
rate, and other complexities in the stellar evolution. These factors and their
effect on the post-bounce dynamics can be encapsulated by a parameter called
’compactness’ [27] which has been shown to correlate with neutrino emission in
1D [57].

Table 2: Relative differences in $m_{\nu}$ upper limit for the specific direction compared to the spherically averaged case at 95% CL.  Mass Ordering | 40$M_{\odot}$ model | 75$M_{\odot}$ model
---|---|---
NO | -2.9% | $0.1\%$
IO | -1.5% | $0.3\%$

Table II shows the effect of modulations on the derived neutrino mass limit.
The model s40 has a larger relative difference in mass bound between modulated
and average signal of -2.9% compared to 0.1% for u75 (for NO). We expect this
result as the model s40 has a significantly stronger SASI oscillation in the
signal. For s40, the SASI oscillation lowers the limit, and for u75,
modulations in the signal increase the mass bound, but it is clear the
magnitude of this effect on the derived limit is small.

###  IV.2 Statistical Uncertainties from Bootstrapping

Figure 6:  Histogram of the mass sensitivity at 95% CL with 1000 pseudo-
experiments simulated using the s40 model with NO as the representative
’Asimov’ dataset [58, 59].

The exact limit on $m_{\nu}$ will be dependent on the statistical fluctuations
in the signal. Here, we quantify the level to which the upper limit on
$m_{\nu}$ is affected by these fluctuations by performing a statistical
bootstrap. The spherically averaged s40 model with NO is chosen as a
representative case. The event rate for the model is used as a ’true’
distribution from which 1000 simulated pseudo-experiments (toy MC) are
produced. A histogram is generated from the calculated neutrino mass limit for
each pseudo-experiment (Fig. 6). The median value of this dataset is 0.39 eV,
which is approximately equal to the bound of 0.40 eV derived in the previous
section. Uncertainty can then be estimated by considering 1$\sigma$ statistics
around the median, leading to a result of $0.39^{+0.06}_{-0.01}$ eV for a 95%
CL bound (see Fig. 6). This distribution is clearly asymmetric with a maximum
of 0.96 . Therefore, the uncertainty is also asymmetric, showing that
statistical fluctuations could have a large effect on the derived limit.

###  IV.3 Detector Mass

Figure 7:  The mass sensitivity of the black hole-forming CCSN at 95% CL is
shown for a range of detector masses. This range is chosen to include KamLAND,
JUNO and THEIA(-100). The theoretical expectation scaled from the JUNO limit
with Eq. (10) is a good fit for the other detectors.

In the rest of this work, we consider neutrinos being detected in the liquid
scintillator detector JUNO, which currently under construction in southern
China and is the most promising LS observatory in the imminent future.
However, there are many other liquid scintillator detectors that would be
suitable for a study of this type, from the past, the present and from
proposals, such as Borexino [60], SNO+ [61], and LENA [62].

Along with JUNO, we select two representative detector setups to cover a range
of relevant liquid scintillator target masses. We select a 1 detector setup
which we label ”KamLAND” and corresponds to the capabilities of KamLAND-Zen
[63, 64] and SNO+. We also consider a 100 setup which we call ”THEIA”
corresponding to THEIA-100 [39]. In our setup, where we do not consider
realistic detector efficiencies and response, the effect of different energy
resolutions was found to have a negligible impact the $\nu$ mass limit
derived. Therefore, we use the JUNO value of $3\%/\sqrt{E(\unit{\mega})}$
instead of $6.5\%/\sqrt{E(\unit{\mega})}$ and $\sim
7\%/\sqrt{E(\unit{\mega})}$ for KamLAND [65] and THEIA, respectively. For CCSN
without black hole formation, the mass limit scales with detector mass as
$m_{\textrm{lim}}\sim M_{D}^{-1/4}$. With the sharp cutoff induced by the
formation of a black hole, Beacom et al [7, 8] show that the derived mass
limit instead goes with

$m_{lim}\sim\left(L_{BH}M_{D}\right)^{-1/2}.$ (10)

In Fig. 7, we show the KamLAND mass limit to be $1.99$ eV for a 95% CL bound
compared to a limit of $0.40$ eV for JUNO. To directly compare this to the
limit from Beacom et al, we have to account for both detector mass, $M_{D}$,
and neutrino luminosity per flavour at the cutoff, $L_{BH}$ (Eq. 10). In [7,
8], the $L_{BH}$ is $10^{52}$ erg/s, which is $\sim 1$ order of magnitude
smaller than the neutrino luminosity in Fig. 1. Therefore, the limit derived
by Beacom et al can be rescaled to $0.57$ eV showing that the analysis
technique enabled by liquid scintillator detectors is a $\sim 30\%$
improvement. For THEIA, we derive a neutrino mass constraint of $0.16$ eV,
which is below the projected final sensitivity for the KATRIN experiment of
$0.2$ eV.

###  IV.4 Distance from Earth

In the earlier sections, we assume the canonical distance of a CCSN from Earth
of 10 kpc, which is slightly farther than the average distance to the galactic
centre [66]. However, it is possible that the next detected CCSN could occur
anywhere in the Galaxy and its satellites. SN1987A was determined to be in the
Large Magellanic Cloud (LMC), $\sim 50$ kpc from Earth [56]. In the simulation
paper for the s40 and u75 models [33], it is suggested that the neutrino
signal from these models would even be detectable at a few sigma significance
from near-neighbour galaxies [33]. However, it is not expected that there
would be mass sensitivity at this distance due to a lack of statistics.

In this section, we quantify the effect of distance on the neutrino mass
constraint. The distance, $D$, has two effects on the neutrino emisssion: 1)
the ToF delay scales with $\propto D$, and 2) the flux rate scales with
$\propto\frac{1}{D^{2}}$.

For CCSN without black hole-formation, the mass limit is independent of the
distance from Earth, $D$. With the sharp cutoff, Beacom et al [7, 8] show that
the derived mass limit instead goes with

$m_{lim}\sim D^{1/2}.$ (11)

Figure 8:  The mass sensitivity of the black hole-forming CCSN at 95% CL is
shown from 0.2-50 kpc. This range is chosen because it is the distance from
Betelgeuse to the LMC, where it is thought SN1987A originated. The purple
crosses denote the classic scenario with four free parameters, (formation
time, $t_{\textrm{BH}}$; pinching parameter, $\alpha$; mean energy, $\langle
E\rangle$; and the normalisation, $\mathcal{N}$). The maroon crosses are
generated by fixing the black hole formation time, and leaving the other three
parameters free. The theoretical expectations scaled from the 10 kpc limit in
each scenario are the blue and orange lines.

The distance from Earth in Fig. 8 ranges from 0.2 to 50 kpc. The distance to
SN1987A is 50 kpc, where mass sensitivity at 95% CL is $\sim 1$ , larger than
the current direct experimental limit of 0.8 . The distribution of Galactic
supernovae is within a narrow band, with 10% of SNe within 5 kpc and 90%
within 15 kpc [67]. A limit rivalling KATRIN’s projected sensitivity of of 0.2
(90% CL) [23] could be obtained for our CCSN model (s40) less than $\sim 5$
kpc from Earth.

Although there are several CCSN candidates within 0.1-1 kpc, these events are
rare. There are at least six supernova candidate stars $<0.3$ kpc from Earth,
including Betelgeuse in the Orion constellation at a distance of $\sim 0.2$
kpc [68]. In Fig. 8, we show that for our model (s40) located $0.2$ kpc from
Earth, we derive a mass limit of 0.06 eV. However, for very near-Earth
supernova, there is a concern about the possibility of data loss as this
extremely large rate could overload the data-acquisition systems [32]. Note
that the data-acquisition system of JUNO will be designed to accommodate this
possibility.

The orange curve in Fig. 8 is the relationship between the distance from Earth
and neutrino mass limit expected from Eq. (11). The neutrino mass limits
derived for different distances (the maroon crosses) follow this trend if the
black hole-formation time is fixed. However, if the black hole-formation time
allowed to be a free parameter, as in the rest of this work, we find
significant deviation from the theoretical expectation (blue curve) and the
mass limits (purple crosses). This is because the relationship between the
cutoff time, normalisation, and distance from Earth is not captured by Eq.
(11). This also exemplifies the importance of properly identifying the black
hole formation time, which relies on the laudable timing resolution of LS
detectors.

##  V Discussion

During the analysis, we have neglected certain ’higher-order’ effects that
could potentially alter the mass sensitivity. In this section, we attempt to
quantify these effects, including Earth-matter effects, non-radial neutrino
emission, and neutrino echoes.

#### Earth-matter effects

As neutrinos travel through the Earth on their journey to a detector, they
undergo the MSW effect. In this work, we took the most optimistic case, where
the CCSN is in the same hemisphere as JUNO, and Earth-matter effects are not
relevant. However, if Earth-matter effects are relevant they would cause a
slight oscillation in the $\bar{\nu}_{e}$ energy spectrum for NO and would
have no effect for IO [18, 69]. Of all current or near-future detectors, it is
proposed that JUNO, as a liquid scintillator detector with excellent energy
resolution, is best poised to disentangle this effect [70]. We can safely
neglect this effect, as although it may cause a small oscillation in the
neutrino spectrum, we have shown this technique is robust against even violent
SASI oscillations.

#### Rotation and non-radial neutrino emission

In this work, we have only considered non-rotating models. It is conjectured
that rotation would alter the flavour conversion and broaden the peaks of SASI
activity [71], although it is not expected that rotation would have a large
effect on the mass sensitivity alone, it is important to consider rotation in
combination with non-radial neutrino emission. In a simple picture, where
neutrinos are emitted only radially from a non-rotating CCSN that undergoes
black hole formation, the neutrinos either would not escape the expanding
event horizon or continue on their radial trajectories. However, in a more
realistic model neutrinos will be emitted in all directions at
$t_{\textrm{BH}}$. As a consequence, some neutrinos will be trapped spiralling
out in unstable circular orbits before escaping the gravitational field. The
additional path length in case of this non-radial emission could soften the
neutrino cutoff and therefore, introduce a tail of events similar to the mass-
induced ToF effect. For non-rotating systems, the number of trailing events
after the cutoff can be estimated to 0.7 events in JUNO, which is negligible
compared to the ToF effect [72]. However, for strongly rotating models, it is
thought that this effect is much more important, and altering the size and the
shape of the tail after black hole formation [72]. Although the strongly
rotating case is thought to be rare, this effect makes it important to
characterize the energy-dependence of the cutoff, leveraging the unique low-
energy signature of ToF delay.

#### Neutrino Echoes

Neutrino echoes are neutrinos emitted just before black hole formation which
scatter on infalling material on their journey to Earth, thus being delayed
[73]. If this effect is present on the scale predicted in [73] then in JUNO we
expect $\sim 5$ events after $t_{\textrm{BH}}$ from neutrino echoes. However,
as coherent scattering has a strong energy dependence, it is thought that
these delayed neutrinos will mostly appear at energies higher than the mass-
delayed neutrinos, making this effect easier to identify. To assess the effect
of a neutrino echoes on the derived mass limit, we use the neutrino echo data
from Gullin et al [73], which uses the same solar-metallicity 40$M_{\odot}$
progenitor (s40) [41] evolved with GR1D code [74, 75] instead of PROMETHEUS-
VERTEX code. First, we introduce a neutrino mass delay into a signal with a
neutrino echo, and then we compare this to a signal without a neutrino echo.
Specifically, by carrying out a log-likelihood fit similar to the one
described in Sec. IIIB, although a neutrino echo can ’fake’ a larger neutrino
mass delay, it has a noticeably larger absolute $\chi^{2}$, showing that this
feature is distinguishable. Further studies are needed to derive a mass limit
in the presence of a neutrino echo. We speculate that developing a more
sophisticated fit which additional parameters to approximate the neutrino echo
could break any degeneracy between these effects.

Based on this survey of possible effects superimposed on the ToF signature, we
conclude that they are subdominant effects that can either be neglected or
recognized and potentially included in the $m_{\nu}$ analysis. This
corroborates that the characteristic energy-dependent delay beyond the sharp
neutrino cutoff is a well-recognizable signature that can be exploited to set
a tight neutrino mass limit with only a minimum of assumptions on the original
neutrino signal required.

##  VI Conclusion

In this work, we have derived a mass sensitivity of $0.39^{+0.06}_{-0.01}$ for
a 95% CL bound using neutrinos from a black hole-forming core-collapse
supernova detected in a JUNO-like detector. This limit is tighter than the
current constraint of 0.8 (90% CL) from KATRIN [22]. We have also shown that
for a nearby or more luminous CCSN, this limit could be competitive with the
target limit from KATRIN of 0.2 (90% CL) [23], and with THEIA-100, this limit
could be far exceeded. At the very least, a high-statistics observation of
this event would allow a model-independent limit to be set on this value.

We have shown in this paper that the combination of a ToF analysis and liquid
scintillator detectors has exceptional potential, but it is not unique. For
instance, a similar analysis with CC argon reactions in DUNE [76] could derive
a limit on the
$\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{\\!e}}^{\mspace{-1.0mu}\scriptstyle{}}}$
mass using exactly the prescription detailed in this work. Complimentary
analyses across many detectors and locations could tighten the $\nu$ mass
limit even further. We are excited by the ever-growing supernova community
[77, 78]. With some luck, JUNO and other next-generation detectors will be
running when neutrinos from the next core-collapse supernova reach Earth, and
we can use these cosmic messengers to test the huge theoretical and
experimental advances we have made since SN1987A.

###### Acknowledgements.

We thank David Maksimovic and Vsevolod Orekhov for discussions throughout the
course of this work. We would also like to thank Hans-Thomas Janka and Daniel
Kresse for their help in understanding CCSN mechanisms and simulations, Evan
O’Connor for his help in understanding neutrino echoes, and Scott Oser for his
help with statistics. We are grateful to Yufeng Li and Mariangela Settimo for
carefully proofreading the manuscript. This work is supported by PRISMA+
Cluster of Excellence.

## References

  * Super Kamiokande Collaboration  _et al._ [1998] Super Kamiokande Collaboration, Y. Fukuda, T. Hayakawa, E. Ichihara, K. Inoue, and et al, Phys. Rev. Lett. 81, 1562 (1998).
  * SNO Collaboration  _et al._ [2002] SNO Collaboration, Q. R. Ahmad, and et al, Phys. Rev. Lett. 89, 011301 (2002).
  * Zatsepin [1968] G. T. Zatsepin, Pisma Zh. Eksp. Teor. Fiz. 8, 333 (1968).
  * Stodolsky [2000] L. Stodolsky, Phys. Lett. B 473, 61 (2000).
  * Fan [2016] Phys. Lett. B 757, 505 (2016).
  * Fleury [2016] P. Fleury, Phys. Lett. B 760, 350 (2016).
  * Beacom  _et al._ [2000] J. F. Beacom, R. N. Boyd, and A. Mezzacappa, Phys. Rev. Lett. 85, 3568 (2000).
  * Beacom  _et al._ [2001] J. F. Beacom, R. N. Boyd, and A. Mezzacappa, Phys. Rev. D Part. Fields 63 (2001).
  * Hirata  _et al._ [1987] K. Hirata, T. Kajita, M. Koshiba, M. Nakahata, and Y. t. Oyama, Phys. Rev. Lett. 58, 1490 (1987).
  * Bio [1987] Phys. Rev. Lett. 58, 1494 (1987).
  * Alexeyev  _et al._ [1988] E. N. Alexeyev, L. N. Alexeyeva, I. V. Krivosheina, and V. I. Volchenko, Phys. Lett. B 205, 209 (1988).
  * Arnett and Rosner [1987] W. D. Arnett and J. L. Rosner, Phys. Rev. Lett. 58, 1906 (1987).
  * Loredo and Lamb [2002] T. J. Loredo and D. Q. Lamb, Phys. Rev. D Part. Fields 65 (2002).
  * Pagliaroli  _et al._ [2010] G. Pagliaroli, F. Rossi-Torres, and F. Vissani, Astropart. Phys. 33, 287 (2010).
  * Abe  _et al._ [2016] K. Abe, Y. Haga, Y. Hayato, M. Ikeda, and K. t. S. Iyogi, Phys. Rev. D. 94 (2016).
  * Lu  _et al._ [2015] J.-S. Lu, J. Cao, Y.-F. Li, and S. Zhou, J. Cosmol. Astropart. Phys. 2015 (05), 044.
  * An [2016] F. An, J. Phys. G Nucl. Part. Phys. 43, 030401 (2016).
  * Scholberg [2018] K. Scholberg, J. Phys. G Nucl. Part. Phys. 45, 014002 (2018).
  * Choudhury and Choubey [2018] S. R. Choudhury and S. Choubey, J. Cosmol. Astropart. Phys. 2018 (09), 017.
  * Planck Collaboration  _et al._ [2020] Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, and t. Baccigalupi, C, Astron. Astrophys. 641, A6 (2020).
  * Di Valentino  _et al._ [2015] E. Di Valentino, A. Melchiorri, and J. Silk, Phys. Rev. D 92 (2015).
  * The KATRIN Collaboration  _et al._ [2022] The KATRIN Collaboration, M. Aker, A. Beglarian, J. Behrens, A. Berlev, and t. Besserer, U, Nat. Phys. 18, 160 (2022).
  * Angrik  _et al._ [2005] J. Angrik _et al._ (KATRIN), (2005).
  * Esfahani  _et al._ [2017] A. A. Esfahani, D. M. Asner, S. Böser, and R. t. Cervantes, J. Phys. G Nucl. Part. Phys. 44, 054004 (2017).
  * Adams  _et al._ [2013a] S. M. Adams, C. S. Kochanek, J. F. Beacom, M. R. Vagins, and K. Z. Stanek, Astrophys. J. 778, 164 (2013a).
  * Rozwadowska  _et al._ [2021] K. Rozwadowska, F. Vissani, and E. Cappellaro, New astron. 83, 101498 (2021).
  * O’Connor and Ott [2011] E. O’Connor and C. D. Ott, Astrophys. J. 730, 70 (2011).
  * Neustadt  _et al._ [2021] J. M. M. Neustadt, C. S. Kochanek, K. Z. Stanek, C. Basinger, T. Jayasinghe, C. T. Garling, S. M. Adams, and J. Gerke, Mon. Not. R. Astron. Soc. 508, 516 (2021).
  * Janka  _et al._ [2012] H.-T. Janka, F. Hanke, L. Huedepohl, A. Marek, B. Mueller, and M. Obergaulinger, (2012).
  * Burrows [2013] A. Burrows, Rev. Mod. Phys. 85, 245 (2013).
  * Janka [2017] H.-T. Janka, in  _Handbook of Supernovae_ (Springer International Publishing, Cham, 2017) pp. 1095–1150.
  * Li  _et al._ [2021] S. W. Li, L. F. Roberts, and J. F. Beacom, Phys. Rev. D. 103 (2021).
  * Walk  _et al._ [2020] L. Walk, I. Tamborra, H.-T. Janka, A. Summa, and D. Kresse, Phys. Rev. D. 101 (2020).
  * Blondin  _et al._ [2003] J. M. Blondin, A. Mezzacappa, and C. DeMarino, Astrophys. J. 584, 971 (2003).
  * [35] The Garching Core-Collapse Supernova Archive, https://wwwmpa.mpa-garching.mpg.de/ccsnarchive/.
  * Super Kamiokande Collaboration and Ikeda [2021] Super Kamiokande Collaboration and M. Ikeda, J. Phys. Conf. Ser. 2156, 012150 (2021).
  * Proto-Collaboration, Hyper-Kamiokande  _et al._ [2018] Proto-Collaboration, Hyper-Kamiokande, , and K. a. Abe, (2018).
  * JUN [2022] Progress in Particle and Nuclear Physics 123, 103927 (2022).
  * Askins  _et al._ [2020] M. Askins, Z. Bagdasarian, N. Barros, E. W. Beier, and E. B. et al, The European Physical Journal C 80, 10.1140/epjc/s10052-020-7977-8 (2020).
  * Zsoldos  _et al._ [2022] S. Zsoldos, Z. Bagdasarian, G. D. Gann Orebi, A. Barna, and S. Dye, Eur. Phys. J. C 82, 1151 (2022), arXiv:2204.12278 [hep-ex] .
  * Woosley and Heger [2007] S. Woosley and A. Heger, Phys. Rep. 442, 269 (2007).
  * Woosley  _et al._ [2002] S. E. Woosley, A. Heger, and T. A. Weaver, Rev. Mod. Phys. 74, 1015 (2002).
  * Rampp and Janka [2002] M. Rampp and H.-T. Janka, (2002).
  * Marek  _et al._ [2005] A. Marek, H. Dimmelmeier, H.-T. Janka, E. Mueller, and R. Buras, (2005).
  * Keil  _et al._ [2003] M. T. Keil, G. G. Raffelt, and H.-T. Janka, Astrophys. J. 590, 971 (2003).
  * Tamborra  _et al._ [2012] I. Tamborra, B. Müller, L. Hüdepohl, H.-T. Janka, and G. Raffelt, Phys. rev. 86 (2012).
  * Wolfenstein [1978] L. Wolfenstein, Phys. Rev. D Part. Fields 17, 2369 (1978).
  * Wolfenstein [1979] L. Wolfenstein, Phys. Rev. D Part. Fields 20, 2634 (1979).
  * Dighe and Smirnov [2000] A. S. Dighe and A. Y. Smirnov, Phys. Rev. D Part. Fields 62 (2000).
  * Takahashi  _et al._ [2001] K. Takahashi, M. Watanabe, K. Sato, and T. Totani, Phys. Rev. D Part. Fields 64 (2001).
  * Kawagoe  _et al._ [2010] S. Kawagoe, T. Yoshida, T. Kajino, H. Suzuki, K. Sumiyoshi, and S. Yamada, Phys. rev. 81 (2010).
  * Lu  _et al._ [2016] J.-S. Lu, Y.-F. Li, and S. Zhou, Phys. Rev. D. 94 (2016).
  * Esteban  _et al._ [2020] I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz, and A. Zhou, J. High Energy Phys. 2020 (9).
  * Strumia and Vissani [2003] A. Strumia and F. Vissani, Phys. Lett. B 564, 42 (2003).
  * [55] K. Scholberg, Snowglobes: Supernova observatories with globes, GitHub, https://github.com/SNOwGLoBES/snowglobes.
  * Pagliaroli  _et al._ [2009] G. Pagliaroli, F. Vissani, M. L. Costantini, and A. Ianni, Astropart. Phys. 31, 163 (2009).
  * O’Connor and Ott [2013] E. O’Connor and C. D. Ott, Astrophys. J. 762, 126 (2013).
  * Asimov [1990] I. Asimov,  _Franchise_ (Broadway Books, New York, 1990).
  * Cowan  _et al._ [2011] G. Cowan, K. Cranmer, E. Gross, and O. Vitells, Eur. Phys. J. C Part. Fields 71, 1 (2011).
  * Borexino Collaboration  _et al._ [2009] Borexino Collaboration, G. Alimonti, C. Arpesella, H. Back, and M. B. et al, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 600, 568 (2009).
  * SNO+ Collaboration  _et al._ [2021] SNO+ Collaboration, V. Albanese, R. Alves, M. Anderson, S. Andringa, and L. A. et al, Journal of Instrumentation 16 (08), P08059.
  * Wurm  _et al._ [2012] M. Wurm, J. F. Beacom, L. B. Bezrukov, D. Bick, and J. B. et al, Astroparticle Physics 35, 685 (2012).
  * KamLAND RCNS Group  _et al._ [2004] KamLAND RCNS Group, F. Suekane, T. Iwamoto, H. Ogawa, O. Tajima, and H. Watanabe, An overview of the kamland 1-kiloton liquid scintillator (2004).
  * Abe [2023] 130, 10.1103/physrevlett.130.051801 (2023).
  * Abe  _et al._ [2008] S. Abe, T. Ebihara, S. Enomoto, K. Furuno, and Y. G. et al, Physical Review Letters 100, 10.1103/physrevlett.100.221803 (2008).
  * Hunt  _et al._ [2016] J. A. S. Hunt, J. Bovy, and R. G. Carlberg, Astrophys. J. Lett. 832, L25 (2016).
  * Adams  _et al._ [2013b] S. M. Adams, C. S. Kochanek, J. F. Beacom, M. R. Vagins, and K. Z. Stanek, Astrophys. J. 778, 164 (2013b).
  * Firestone [2014] R. B. Firestone, Astrophys. J. 789, 29 (2014).
  * Borriello  _et al._ [2012] E. Borriello, S. Chakraborty, A. Mirizzi, P. D. Serpico, and I. Tamborra, Phys. rev. 86 (2012).
  * Liao [2016] W. Liao, Phys. Rev. D. 94 (2016).
  * Walk  _et al._ [2018] L. Walk, I. Tamborra, H.-T. Janka, and A. Summa, Phys. Rev. D. 98 (2018).
  * Wang  _et al._ [2021] J.-S. Wang, J. Tseng, S. Gullin, and E. P. O’Connor, Phys. Rev. D. 104 (2021).
  * Gullin  _et al._ [2022] S. Gullin, E. P. O’Connor, J.-S. Wang, and J. Tseng, Astrophys. J. 926, 212 (2022).
  * O’Connor and Ott [2010] E. O’Connor and C. D. Ott, Class. Quantum Gravity 27, 114103 (2010).
  * O’Connor [2015] E. O’Connor, Astrophys. J. Suppl. Ser. 219, 24 (2015).
  * DUNE Collaboration and Acciarri [2015] DUNE Collaboration and R. Acciarri, (2015), arXiv:1512.06148 [physics.ins-det] .
  * Kharusi  _et al._ [2020] S. A. Kharusi, S. Y. BenZvi, J. S. Bobowski, W. Bonivento, and V. t. Brdar, (2020), arXiv:2011.00035 [astro-ph.HE] .
  * Al Kharusi  _et al._ [2021] S. Al Kharusi, S. Y. BenZvi, J. S. Bobowski, W. Bonivento, and V. t. Brdar, New J. Phys. 23, 031201 (2021).
  * Abud [2021] A. A. Abud, (2021).
  * Yang  _et al._ [2018] M. S. Yang, Z. Y. Yu, J. Cao, X. L. Sun, B. X. Yu, and G. P. An, (2018).

##  Appendix A Other Channels

A complimentary study for electron neutrinos,
$\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{\\!e}}^{\mspace{-1.0mu}\scriptstyle{}}}$,
could be performed in the Deep Underground Neutrino Experiment (DUNE), is a
liquid argon time-projection chamber with a fiducial mass of 40 [79]. The main
detection channel on argon is the charged-current (CC) reaction with
$\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{\\!e}}^{\mspace{-1.0mu}\scriptstyle{}}}$,

$\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{\\!e}}^{\mspace{-1.0mu}\scriptstyle{}}}+{}^{40}\textrm{Ar}\rightarrow\mathit{{e}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{-}}}+{}^{40}\textrm{K}^{*}.$

This
$\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{\\!e}}^{\mspace{-1.0mu}\scriptstyle{}}}$
sensitivity is unique to DUNE, and will allow a detailed look at the
neutronization burst of the CCSN neutrino curve [76]. The complimentary
analysis in DUNE could use the neutronization burst as a similarly clear
signature as black hole formation to carry out a ToF analysis.

Although it is less suitable for studies involving ToF effects, JUNO presents
a unique opportunity to detect $\nu_{x}$ from CCSN. Proton-elastic scattering
$\nu+p\rightarrow\nu+p$ is a neutral-current interaction only available to
low-threshold scintillator detectors [80], giving JUNO another perspective on
the engine of core-collapse.