# Supernova Neutrinos: flavour conversion mechanisms and new physics scenarios

Manibrata Sen<EMAIL_ADDRESS>Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany.

###### Abstract

A core-collapse supernova (SN) releases almost all of its energy in the form
of neutrinos, which provide a unique opportunity to probe the working
machinery of a SN. These sites are prone to neutrino-neutrino refractive
effects, which can lead to fascinating collective flavour oscillations among
neutrinos. This causes rapid neutrino flavour conversions deep inside the SN
even for suppressed mixing angles, with intriguing consequences for the
explosion mechanism as well as nucleosynthesis. We review the physics of
collective oscillations of neutrinos- both slow and fast, along with the well-
known resonant flavour conversion effects, and discuss the current state-of-
the-art of the field. Furthermore, we discuss how neutrinos from a SN can be
used to probe novel particle physics properties, extreme values of which are
otherwise inaccessible in laboratories.

##  I Introduction

The observation of neutrinos originating from the supernova SN1987A stands as
a pioneering moment in the realm of multi-messenger physics. This pivotal
event involved the detection of neutrino signals several hours before the
emergence of visible light. Despite the modest quantity of data, encompassing
roughly 30 neutrino events recorded by Kamiokande-II (KII) Hirata  _et al._
(1987, 1988), IMB Bionta  _et al._ (1987); Bratton  _et al._ (1988), and
Baksan Alekseev  _et al._ (1988), it has garnered considerable attention
within the fields of particle and astroparticle physics, serving as a valuable
resource for exploring fundamental questions in the discipline.

The working machinery of a core-collapse supernovae (SN) can be briefly
described as follows (for detailed reviews, see Refs. Bethe (1990); Mezzacappa
(2005); Raffelt (1996); Kotake  _et al._ (2006); Woosley and Janka (2005);
Janka  _et al._ (2007); Janka (2012); Mirizzi  _et al._ (2016); Burrows and
Vartanyan (2021)). At the end of nuclear burning, a massive star (more than
$8\,M_{\odot}$) ends up with an iron core with concentric shells of lighter
elements above it. As the iron core grows in mass due to accretion and crosses
the Chandrasekhar mass limit, electron degeneracy pressure fails to support
it, and a collapse is triggered. The collapse proceeds until nuclear densities
are reached, which is when the equation of state becomes stiff and resists
further infall, resulting in a core bounce. This launches an energetic
shockwave which travels outwards from the core, which is responsible for
exploding the star.

Simulations suggest that this shockwave is not powerful enough to blow up the
star, rather it loses energy and stalls at a radius of around 100-200 km from
the core Baade, W. and Zwicky, F. (1934); Brown  _et al._ (1982). As the shock
stalls, the compact protoneutron star at the centre begins to grow by
accretion of infalling stellar material. In this entire process, a lot of
neutrinos are released, which start free-streaming from the neutrinosphere-
the surface of last scattering- at a radius of around 30 km from the core
Burrows (1990). It is widely believed that the emitted neutrinos deposit their
energy in the region behind the shock and re-energise the shockwave causing a
successful explosion, also known as the delayed neutrino-heating mechanism of
explosion Colgate and White (1966); Bethe and Wilson (1985). This scenario is
highly sensitive to the neutrino flavour information and hence requires a
precise understanding of neutrino flavour oscillations. The question of SN
explosion also depended on the SN geometry used in simulations. Initial
studies focussed on simplistic spherically symmetric (1D) SN models seldom
ended in an explosion, however, we are now in an era of 3D simulations, which
can capture the multidimensional turbulent convection in the SN envelope in
detail. This has led to a better understanding of the role played by
convection as well as the breaking of spherical symmetry in SN explosions
Burrows and Vartanyan (2021).

It is widely acknowledged that investigating neutrinos emitted during the core
collapse of a galactic SN offers a unique opportunity to delve into the
dynamics of these explosive phenomena. These neutrinos originate from deep
inside the supernova and traverse the entire matter envelope before reaching
Earth. The composition of neutrino flavours is exquisitely sensitive to the
fermion density they encounter during their journey.

Before the turn of the millennium, prevailing scientific belief centred on
resonant flavour conversions within the SN, driven by the Mikheyev-Smirnov-
Wolfenstein (MSW) effect Mikheev and Smirnov (1985); Wolfenstein (1978); Kuo
and Pantaleone (1989a); Dighe and Smirnov (2000). This perspective, however,
overlooked a fundamental factor: the impact of neutrino self-interactions, a
critical element governing neutrino flavour dynamics within a dense medium, as
discussed by Pantaleone in two classic papers Pantaleone (1992, 1995). In
regions near the neutrinosphere, where neutrino density is substantial,
forward scattering of neutrinos on a background of other neutrinos prevails as
the dominant factor in the propagation Hamiltonian Kostelecky and Samuel
(1995); Samuel (1996); Duan  _et al._ (2006a, b, c). These self-interactions
manifest as a potential linked to the background neutrino density, denoted by
$n_{\nu}$. Furthermore, neutral current interactions among neutrinos of
different flavours, characterised by their indifference to flavour, also
contribute off-diagonal components to this forward-scattering potential.
Consequently, neutrinos were found to undergo collective flavour
transformation, unveiling a diverse range of phenomena Duan  _et al._ (2006a,
b, c).

Until the first decade of the millennium, the prevailing belief was that these
collective oscillations, termed as bipolar flavour conversions, would grow
with a rate roughly linked to $\sqrt{(\Delta m^{2})n_{\nu}}$, where $\Delta
m^{2}$ represents the neutrino mass-squared difference Duan  _et al._ (2006b);
Hannestad  _et al._ (2006); Duan  _et al._ (2007). Simplistic toy models
indicated that such flavour conversions occurred in the vicinity of the
stalled shockwave, at approximately 150-200 km from the core. This hinted at
their potential involvement in facilitating a shockwave-driven explosion.
However, as research progressed, it became evident that moving beyond the
initial single-angle approximation, which assumed uniform emission angles for
neutrinos of all flavours from a specific neutrinosphere, revealed these
flavour conversions to happen at radii too distant from the core to
significantly influence the SN explosion mechanism (for a detailed review, see
Duan  _et al._ (2010)).

More recently, researchers have recognised an entirely distinct phenomenon
known as “fast” flavour conversions, which can manifest in the depths of SNe
and neutron star mergers Sawyer (2005, 2009, 2016); Chakraborty  _et al._
(2016a); Dasgupta  _et al._ (2017). This stems from distinct angular
distributions of neutrinos and antineutrinos across different flavours. It can
thrive in regions inaccessible to other flavour transformation mechanisms.
Even for massless neutrinos, small perturbations can amplify with a rate
proportional to $n_{\nu}$, and the rate remains almost independent of neutrino
masses and mass hierarchy (see Dasgupta and Sen (2018); Shalgar and Tamborra
(2021a) for a detailed study of the dependence). It is believed that one of
the most likely outcomes of such rapid flavour conversions is flavour
depolarization, which can lead to equipartition of neutrino flavours inside a
SN. This can have drastic consequences for the explosion mechanism, as well as
nucleosynthesis. From a neutrino transport simulation standpoint, such a
flavour equipartition implies that neutrino oscillations will not play an
important role in the supernova dynamics, hence this can be a huge gain as
this partially obliterates the need to include neutrino oscillations into a
transport code, which is computationally very expensive. These developments
have led to intense research work to assess the effect of these flavour
conversions on the SN explosion mechanism (see Tamborra and Shalgar (2021);
Richers and Sen (2022) for a recent review). However, we are still far from
having a complete picture of the potential impact of these collective
oscillations.

The neutrino flux from a SN can be utilised to facilitate a SN as a unique
laboratory for scientific exploration Raffelt (1996); Horiuchi and Kneller
(2018). The extreme conditions prevailing within these objects provide a rare
opportunity to impose stringent constraints on neutrino properties, which
often reach extreme values not achievable in terrestrial laboratories. This is
exemplified by the limited number of neutrino events detected from SN1987A,
which have enabled the imposition of robust constraints on physics beyond the
Standard Model. Remarkably, even hypothetical “feebly” interacting particles
can be effectively constrained through luminosity arguments derived from
supernova observations.

The few neutrino events observed from SN1987A were sufficient to confirm our
basic understanding of the working machinery of a core-collapse SN Fiorillo
_et al._ (2023a); Li  _et al._ (2023). On average, galactic SNe are expected
to occur once every 30-50 years, which implies that we need to be prepared for
another one anytime in the next few decades. We are in a period of enormous
advancement in supernova neutrino detection experiments. Among the currently
running experiments, Super-Kamiokande (Japan), with the addition of Gd Abe
_et al._ (2022a), can detect the highest number of SN neutrino events. Among
the liquid scintillator detectors, KamLAND (Japan) Abe  _et al._ (2022b) offer
the best bets for detection of these neutrinos. The IceCube detector at the
South Pole does not have enough sensitivity to detect individual events, but
it can observe a SN burst as a correlated rise of background noise Köpke and
IceCube Collaboration (2011). Experiments like the Deep Underground Neutrino
Experiment (USA) Abi  _et al._ (2020), Hyper-Kamiokande (Japan) Abe  _et al._
(2018) and Jiangmen Underground Neutrino Observatory (China) An  _et al._
(2016) offer immense potential in terms of deducing the unknowns of SN
neutrino spectra. With improving technologies, even coherent elastic neutrino-
nucleus scattering experiments can play a major role in SN neutrino detection
Pattavina  _et al._ (2020). For a more detailed discussion on the aspects of
detection, please refer to Mirizzi  _et al._ (2016) and references therein

In this brief review, we summarise the current status of the field, focusing
on neutrino oscillation physics, as well as the different probes of new
physics that can be achieved from the observation of supernova neutrinos. We
would also like to point interested readers to a list of other comprehensive
reviews on various aspects of this topic Mirizzi  _et al._ (2016); Chakraborty
_et al._ (2016b); Horiuchi and Kneller (2018); Fuller  _et al._ (2023); Volpe
(2023); Duan  _et al._ (2010); Tamborra and Shalgar (2021); Richers and Sen
(2022); Capozzi and Saviano (2022). This article is organised as follows. In
Sec. II, we discuss the basic picture of neutrino emission from a SN, followed
by a discussion of the governing equations of motion in Sec. III. Then we
expand on the aspect of neutrino flavour propagation inside a SN, focussing on
recent developments in collective oscillations, in Sec. IV. Finally, we
describe the various kinds of new physics scenarios which can be tested from
the observation of neutrinos from a SN in Sec. V before concluding in Sec. VI.

##  II Supernova neutrino: emission phases and spectra

It is anticipated that approximately $10^{58}$ neutrinos are emitted from a SN
explosion. This emission occurs over a timescale of approximately 10 seconds,
which corresponds to the typical diffusion timescale of neutrinos within the
core. The emission of neutrinos is expected to occur in four distinct phases
Colgate and White (1966); Bethe and Wilson (1985); Burrows, A. (1984); Raffelt
(1996):

  1. 1.

During the initial collapse phase, preceding the core bounce, a small amount
of electron neutrinos is primarily emitted due to processes like beta decay
and electron capture.

  2. 2.

Following the core bounce, approximately 25 milliseconds later, there is a
rapid increase in the flux of electron neutrinos known as the “neutronization
burst” phase. During this phase, small amounts of electron antineutrinos and
other neutrino flavours are also emitted, although their contribution is
negligible compared to the flux of electron neutrinos.

  3. 3.

Subsequently, the “accretion phase” ensues, lasting for approximately a few
hundred milliseconds. As the shockwave stalls, matter accumulates onto the
core over several hundred milliseconds. During this phase, the star undergoes
cooling by emitting neutrinos and antineutrinos of all flavours. Typically,
there is an excess of electron neutrinos compared to other species, with a
discernible hierarchy in average energies: $\langle
E_{\nu_{\mu,\tau}}\rangle>\langle E_{\bar{\nu}_{e}}\rangle>\langle
E_{\nu_{e}}\rangle$.

  4. 4.

Finally, the “Kelvin-Helmholtz cooling phase” begins, wherein the proto-
neutron star at the core cools down by emitting neutrinos over a period of
around ten seconds. In this phase, the hierarchy of average energy among
neutrino flavours is observed to be less pronounced.

SN simulations fit the neutrino spectra with a “quasi-thermal” spectra, known
in the literature as the “alpha-fit” Keil  _et al._ (2003); Tamborra  _et al._
(2012a):

$f_{\nu_{\alpha}}(E)=\frac{1}{\langle
E_{\nu}\rangle}\frac{(\alpha+1)^{(\alpha+1)}}{\Gamma(\alpha+1)}\left(\frac{E}{\langle
E_{\nu}\rangle}\right)^{\alpha}{\rm e}^{-(\alpha+1)\frac{E}{\langle
E_{\nu}\rangle}}\,.$ (1)

Here, $\Gamma(z)$ is the Euler gamma function, and $\alpha$ is the pinching
parameter which gives the deviation from a thermal spectrum, given by

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

A value of $\alpha=2$ corresponds to a Maxwell-Boltzmann spectrum, while
$\alpha>2$ signifies a pinched spectrum, characterised by suppressed high and
low energy tails. Conversely, $\alpha<2$ denotes a broadened spectrum.
Commonly used simulations typically yield values of $\alpha$ falling within
the range $2\leq\alpha\leq 5$ Keil  _et al._ (2003). It is generally expected
that for neutrinos, $\langle E_{\nu_{e}}\rangle\in(10\,{\rm MeV},12\,$MeV),
for antineutrinos $\langle E_{\bar{\nu}_{e}}\rangle\in(12\,{\rm MeV},15\,{\rm
MeV})$, while for the non-electron flavour neutrinos, we have $\langle
E_{\nu_{\mu,\tau}}\rangle\in(15\,{\rm MeV},18\,{\rm MeV})$, although the exact
values depend on the simulation. The observed hierarchy in the average
energies of different neutrino flavours primarily stems from variations in
their interaction cross-sections. For example, due to larger cross-section and
charged-current interaction with neutrons, which are more abundant than
protons, $\nu_{e}$ have a lower average energy than $\bar{\nu}_{e}$. On the
other hand, since $\nu_{\mu,\tau}$ interacts less than $\nu_{e}$ and
$\bar{\nu}_{e}$, their average energy is expected to be higher, and hence the
hierarchy. While initial studies ignored the presence of muons and taus in SN
simulations, and assumed a similar spectra for $\nu_{\mu,\tau}$, more recent
simulations have shown that substantial amount of muons can be present due to
the high electron chemical potential, and hence cannot be neglected Bollig
_et al._ (2017). It has been shown that the addition of muons considerably
softens the equation of state of the core and thus faciliates the contraction
of the protoneutron star, leading to larger luminosities and average energies
for neutrinos.

##  III Neutrino propagation: equations of motion

The straightforward approach of applying the Schrödinger equation to address
neutrino mixing encounters limitations when dealing with the dynamics of a
statistical ensemble of neutrinos engaged in both mixing and scattering within
a medium. A more effective approach necessitates the utilization of the
density matrix formalism, which allows for the comprehensive consideration of
these effects. Within this framework, it becomes possible to depict the mixing
of quantum states among neutrinos, accounting for potential coherence loss
resulting from actual collisions. In this section, we will outline the
equations of motion using an effective two-flavour setup, consisting of
$\nu_{e}$ and $\nu_{x}$, where $\nu_{x}$ denotes a linear combination of the
$\nu_{\mu}-\nu_{\tau}$ flavours. Generalisation to three flavours is
straightforward.

An ensemble of neutrinos and antineutrinos can be characterised by the
$2\times 2$ density matrices $\varrho_{{\bf p},{\bf r},t}$ (
$\overline{\varrho}_{{\bf p},{\bf r},t}$ for antineutrinos)Raffelt  _et al._
(1992, 1993); Dolgov (1981); Barbieri and Dolgov (1991); Sigl and Raffelt
(1993),

$\displaystyle\small{\varrho_{{\bf p},{\bf
r},t}=\left(\begin{array}[]{cc}\varrho_{ee}&\varrho_{ex}\\\
\varrho_{xe}&\varrho_{xx}\\\ \end{array}\right)}~{},$ (5)

where the diagonal elements of $\varrho_{{\bf p},{\bf r},t}$ are proportional
to the occupation numbers for the corresponding flavours, while the off-
diagonal ones give the phase information. At this point, it is crucial to
mention that these entries are essentially expectation values of the bilinears
of the neutrino fields. This so-called mean-field approximation neglects any
possible entanglement among the neutrinos. We will later discuss the
implications of relaxing these assumptions.

The evolution of the density matrices $\rho_{{\bf p},{\bf r},t}$ for momentum
${\bf p}$ at position ${\bf r}$ and time $t$ is governed by the equations of
motion (EoMs) Sigl and Raffelt (1993)

$\left(\partial_{t}+{\bf v}_{\bf p}\cdot\nabla_{\bf r}\right)\rho_{{\bf
p},{\bf r},t}=-i[\Omega_{{\bf p},{\bf r},t},\rho_{{\bf p},{\bf
r},t}]\,\,+C[\rho_{{\bf p},{\bf r},t}]\,,$ (6)

where the left-hand side accounts for time dependence and spatial drift caused
by free-streaming neutrinos. The Hamiltonian consists of three terms, i.e.,
$\Omega_{{\bf p},{\bf r},t}=\Omega_{\rm vac}+\Omega_{\rm
MSW}+\Omega_{\nu\nu}$. These contributions comprise the vacuum term,

$\Omega_{\rm vac}=\frac{\omega}{2}\begin{pmatrix}-1&0\\\ 0&1\end{pmatrix}\,,$
(7)

where $\omega=\Delta m^{2}/2\,E$; the Mikheyev-Smirnov-Wolfenstein matter
potential term Wolfenstein (1978); Mikheev and Smirnov (1985)

$\Omega_{\rm MSW}=\sqrt{2}G_{F}n_{e}\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}\,,$
(8)

where $n_{e}$ denotes the background electron number density and $G_{F}$ is
the Fermi constant; and the $\nu-\nu$ self-interaction potential Pantaleone
(1992, 1995),

$\Omega_{\nu\nu}=\sqrt{2}G_{F}\int\frac{d^{3}{\bf q}}{(2\pi)^{3}}(1-{\bf
v}_{\bf p}\cdot{\bf v}_{\bf q})({\rho}_{{\bf q},{\bf r},t}-{\bar{\rho}}_{{\bf
q},{\bf r},t})\,.$ (9)

The self-interaction is usually described in terms of an overall normalisation
$\mu=\sqrt{2}G_{F}n_{\nu}$, where $n_{\nu}$ is the net neutrino density. A
similar equation exists for antineutrinos with a relative change in sign for
${\Omega}^{\rm vac}$ and ${\Omega}^{\rm ref}$. In addition to this, neutrinos
can also undergo inelastic number-changing/preserving collisions, which are of
order $G_{F}^{2}$, and only relevant deep inside the SN. These collisional
processes are denoted symbolically through $C[\rho_{{\bf p},{\bf r},t}]$.

The solutions to the EoMs govern the flavour evolution of neutrinos inside a
SN. These equations constitute a complex set of non-linear, interrelated
partial differential equations that encompass three spatial dimensions, three
momentum dimensions, and one temporal dimension. As such, they pose a
formidable challenge for analytical solutions. Even numerical solutions are
arduous to obtain for the entire problem, necessitating simplifications. In
the next section, we will delve into the dynamics of the flavour evolution of
neutrinos that can be gauged by solving these equations under simplifying
assumptions and for toy setups.

##  IV Neutrino flavour conversions inside a supernova

In this section, we discuss the various flavour conversion mechanisms relevant
to SN neutrinos. We begin with resonant adiabatic MSW flavour conversions,
which were initially believed to be the major mechanism of flavour conversions
inside a SN. Then we discuss the collective flavour oscillations which were
realised at the beginning of the millennium. Finally, we shed some light on
fast flavour conversions - a unique type of flavour conversion mechanism
operating deep inside the SN.

###  IV.1 Matter induced resonant flavour conversions

Resonant flavour conversions of neutrinos arise from coherent forward
scattering interactions with background fermions. These interactions primarily
involve one power of the Fermi coupling $G_{F}$, dominating over inelastic and
loop processes, which typically entail larger powers of $G_{F}$ Wolfenstein
(1978). In ordinary matter, which comprises electrons, protons, and neutrons
but not muons or taus generally, forward scattering processes lead to charged
current (CC) interactions between electron neutrinos and electrons, as well as
neutral current (NC) interactions among all neutrino flavours and neutrons,
protons, and electrons. In this section, we operate in the limit where the
neutrino-neutrino self-interaction term $\Omega_{\rm MSW}\gg\Omega_{\nu\nu}$,
which typically happens at larger radii inside a SN, where we observe a
decline in neutrino density.

The coherent effects of these processes on neutrinos can be likened to
neutrino propagation within an effective potential Wolfenstein (1978); Mikheev
and Smirnov (1985); Kuo and Pantaleone (1989b). The effective potential due to
CC interactions with electron neutrinos is expressed as,

$V_{\rm CC}(r)=\sqrt{2}G_{F}n_{e}(r)\,,$ (10)

where $n_{e}(r)$ represents the local electron density in the medium.
Similarly, for NC interactions involving neutrinos and nucleons, the effective
potential is denoted as,

$V_{\rm NC}(r)=-\frac{G_{F}n_{n}(r)}{\sqrt{2}}\,,$ (11)

where $n_{n}(r)$ denotes the local nucleon density in the medium. It is
important to note that since NC interactions are flavour-blind, this potential
remains the same for all neutrino flavours and can therefore be eliminated by
a phase rotation. For antineutrinos, both these effective potentials acquire a
relative minus sign.

This alters the dispersion relation of neutrinos in a medium. Within a simple
two-flavour framework, it can be shown that for a constant matter density, the
oscillation probability can reach a maximum when Mikheev and Smirnov (1985)

$\frac{\Delta m^{2}}{2E}\,\cos 2\vartheta=V_{\rm CC}\,,$ (12)

where $\Delta m^{2}$ is the mass-squared difference between the two mass
eigenstates in vacuum, $E$ is the neutrino energy, and $\vartheta$ is the two-
flavour mixing angle. This is the MSW (Mikheyev-Smirnov-Wolfenstein) resonance
condition and can lead to a maximal mixing in matter, causing large flavour
conditions even when $\vartheta$ is small. For a varying matter density,
flavour propagation is more complicated and depends on whether the matter
density changes adiabatically or in a non-adiabatic manner Kuo and Pantaleone
(1989a).

Numerous investigations have underscored the sensitivity of neutrino spectra
to the underlying density profile of a supernova, with a particular emphasis
on the MSW resonances Kuo and Pantaleone (1989a); Dighe and Smirnov (2000).
Within a SN, the mass-squared differences relevant to solar ($\Delta
m^{2}_{\odot}$) and atmospheric ($\Delta m^{2}_{\text{atm}}$) neutrinos
exhibit distinct resonance behaviours, termed the “L-resonance” and
“H-resonance” respectively. Since $\Delta m^{2}_{\odot}>0$, the “L-resonance”
is known to take place for neutrinos for typical SN matter densities around
$\rho_{L}\approx(10-100)\,\text{g/cc}$. On the other hand, the H-resonance,
manifesting at matter densities approximately
$\rho_{H}\approx(10^{3}-10^{4})\,\text{g/cc}$, pertains to atmospheric
neutrinos in the normal mass ordering (NMO) and antineutrinos in the inverted
mass ordering (IMO), owing to the sign of $\Delta m^{2}_{\text{atm}}$. Global
analyses of neutrino data suggest that both resonances are adiabatic, given
the substantial mixing angles nuf ; Esteban  _et al._ (2017). Extensive
investigations have explored these resonant flavour conversions within
supernovae, contributing to our understanding of neutrino properties and
supernova dynamics Dighe and Smirnov (2000); Fuller  _et al._ (1987);
Lunardini and Smirnov (2003); Dighe  _et al._ (2003a, b, 2004); Barger  _et
al._ (2005); Fogli  _et al._ (2002).

The matter effects also offer insights into the propagation of the SN
shockwave at later stages Schirato and Fuller (2002); Takahashi  _et al._
(2003); Lunardini and Smirnov (2003); Fogli  _et al._ (2003, 2005); Tomas  _et
al._ (2004); Dasgupta and Dighe (2007); Galais  _et al._ (2010). Typically,
non-adiabaticity in neutrino propagation can surface when abrupt density
changes occur at the shockfront, leaving a distinctive mark on flavour
transitions. This phenomenon becomes a means to reconstruct the real-time
progression of the shockwave. A characteristic manifestation of non-
adiabaticity due to shockwave propagation can be observed in the form of
energy-dependent dips in neutrino survival probabilities Tomas  _et al._
(2004). A few seconds post the core bounce in a supernova, the potential
emergence of a reverse shockwave hinges on whether a supersonic neutrino-
driven wind intersects with the ejecta. The sensitivity of this phenomenon to
the supersonic or subsonic nature of the neutrino-driven wind has been
thoroughly explored recently Friedland and Mukhopadhyay (2022).

Nevertheless, the story does not end here. Supernova progenitors harbour an
array of hydrodynamic instabilities that materialise behind the shockfront,
generating turbulence within the ejecta. This turbulence, often present in
aspherical supernova simulations, introduces stochastic fluctuations in matter
density Radice  _et al._ (2018). These fluctuations, in turn, exert a non-
trivial influence on neutrino flavour evolution. Extensive research has been
conducted to investigate neutrino flavour evolution in turbulent media,
especially in the context of predicting neutrino spectra from future galactic
supernovae Fogli  _et al._ (2006); Friedland and Gruzinov (2006); Kneller and
Volpe (2010); Borriello  _et al._ (2014); Lund and Kneller (2013); Kneller and
Kabadi (2015); Patton  _et al._ (2015); Yang and Kneller (2018); Kneller and
de los Reyes (2017); Abbar (2021); Mukhopadhyay and Sen (2024). The impact of
turbulence on neutrino survival probability has been studied assuming power-
law fluctuations in the density profile Borriello  _et al._ (2014). As the
turbulence amplitude increased, a power-law behaviour was observed in the
electron neutrino survival probability. This led to an analytical estimate
explaining the power-law phenomenon within the framework of the simple
turbulence model Friedland and Gruzinov (2006). Subsequent works employed
various techniques, including the rotating wave approximation Patton  _et al._
(2015); Yang and Kneller (2018); Kneller and de los Reyes (2017), to predict
the final survival probability and study the dependence of neutrino flavour
transition on turbulence amplitude and spectral index Mukhopadhyay and Sen
(2024).

Finally, the neutrinos are also expected to undergo matter effects as they
traverse through the Earth to the detector. These effects, aptly dubbed as
“Earth-matter” effects, can be quite relevant for SN neutrino neutrinos
Lunardini and Smirnov (2001a, b); Dighe  _et al._ (2003b); Mirizzi  _et al._
(2006); Dasgupta  _et al._ (2008a); Borriello  _et al._ (2012). It was found
that the major outcome of this effect is an oscillatory modulation of the
(anti)neutrino spectra. This effect increases with the neutrino energy and can
be larger than $\mathcal{O}(20\%)$ for $E_{\nu}\gtrsim 20\,{\rm MeV}$. The
relevant Earth-matter effects can be gauged by comparing the neutrino spectra
at different detectors. However, due to the inherent uncertainty concerning
the SN neutrino spectra, this effect is expected to be subdominant, and only
relevant in large upcoming detectors. In fact, it was shown that the Earth-
matter effects can be invoked to explain the discrepancy in the SN1987A
neutrino events detected by Kamiokande and IMB, as well as explaining the lack
of any neutrino events beyond 40 MeV Lunardini and Smirnov (2001a). Earth-
matter effects of SN neutrinos have also been shown to play an important role
in Earth tomography with neutrinos Hajjar  _et al._ (2023).

###  IV.2 Slow collective flavour conversions

In the first decade of the millennium, the impact of the neutrino self-
interaction term on flavour propagation started getting investigated. The non-
linear nature of the governing equations necessitated the introduction of
simplifications. One such simplification was the adoption of the single-angle
approximation Duan  _et al._ (2006a, b, c); Hannestad  _et al._ (2006); Duan
_et al._ (2007), where neutrinos and antineutrinos escape from the
neutrinosphere with a single angle. This reduces the evolution to one degree
of freedom (namely, the line of sight), and hence either temporal or spatial
evolution was considered.

Figure 1: An example setup of interacting neutrino beams showing bipolar
collective oscillations. The survival probability of one of the beams is
plotted as a function of time, in units of $\omega$.

Under these approximations, a simple system of interacting neutrino beams was
shown to undergo collective effects in flavour evolution. For $\mu\gg\omega$,
the system undergoes “synchronized oscillations”, where neutrinos of different
energies oscillate with a common frequency with an amplitude proportional to
the suppressed mixing angle (initial phase in Fig. 1). However, as $\mu$
decreases, large amplitude flavour oscillations are seen to happen even with
tiny mixing angles. The phenomenon, known as “bipolar oscillations”, involves
pair conversions between
$\nu_{e}\bar{\nu}_{e}\leftrightarrow\nu_{x}\bar{\nu}_{x}$. In this scenario,
all neutrino energies oscillate with an averaged uniform frequency,
approximately proportional to $\sqrt{\omega\mu}$, where $\omega$ has been
defined following Eq. 7. For monochromatic neutrino beams, the system can be
mathematically likened to a pendulum in flavour space, akin to the equivalence
between ordinary neutrino oscillations in vacuum or matter and the precession
of a spin Hannestad  _et al._ (2006); Duan  _et al._ (2007). Depending on the
neutrino mass ordering, the gravitational force acting on this flavour
pendulum can be directed either upward or downward, rendering specific flavour
configurations unstable, much like an inverted pendulum. Bipolar oscillations
involve the pendulum’s initial position being in an unstable, inverted state,
slightly offset by a small mixing angle, and swinging through the lowest point
to the opposite side. Just like the time period of an inverted pendulum
depends logarithmically on the initial tilting angle from the vertical, the
period of the flavour pendulum oscillations also exhibits a similar
logarithmic dependence on the vacuum mixing angle.

The introduction of multi-angle effects also led to the presence of additional
instabilities in flavour evolution Sawyer (2009); Raffelt  _et al._ (2013).
Nevertheless, during the accretion phase, the influence of multi-angle matter
effects was demonstrated to suppress collective oscillations Chakraborty  _et
al._ (2011). However, the introduction of temporal pulsations was also shown
to negate this multi-angle matter suppression Dasgupta and Mirizzi (2015).

While bipolar oscillations are periodic, in a real SN the neutrino number
density decreases adiabatically. This was shown to lead to a spectral split,
where in IMO, all $\nu_{e}$ and $\nu_{x}$ within a certain energy range are
swapped, whereas in NMO, the swapping occurs beyond a certain energy Duan  _et
al._ (2006b) (see Fig. 2). Such a flavour exchange is called a “swap”, whereas
the sharp boundaries at either side of the swaps are called “splits”. A simple
analytical understanding of the swaps was offered in terms of the neutrino
spectra defined using the variable $\omega$ Raffelt and Smirnov (2007a, b).
Later works revealed the presence of multiple spectral splits in the cooling
phase spectra Fogli  _et al._ (2007, 2008), which were explanined analytically
in terms of of the development of spectral swaps in the difference spectrum
$g_{\omega}$, defined as Dasgupta  _et al._ (2009)

$\displaystyle g_{\omega}$ $\displaystyle\propto$ $\displaystyle
f_{\nu_{e}}(\omega)-f_{\nu_{x}}(\omega)\qquad{\rm for~{}}\omega>0\;,$ (13)
$\displaystyle\propto$ $\displaystyle
f_{\bar{\nu}_{x}}(\omega)-f_{\bar{\nu}_{e}}(\omega)\qquad{\rm
for~{}}\omega<0\,.$

The development of a spectral swap is related to the presence of “spectral
crossings”. A crossing is defined as “positive” (“negative”) if $g_{\omega}$
changes sign from negative (positive) to positive (negative). The emergence of
spectral swaps around a crossing is intricately linked to the conservation of
flavour lepton number during bipolar oscillations Dasgupta  _et al._ (2009).
According to the equations of motion (EoMs), the quantity $\int d\omega
g_{\omega}$ remains conserved, hence attaining a value of zero across the
spectral swap. Consequently, spectral swaps can only materialise in proximity
to a spectral crossing, which is also a necessary and sufficient condition for
the flavor pendulum to be unstable. Additionally, three-flavour effects were
shown to induce additional splits, which could largely be understood as a
sequence of two-flavour splits Dasgupta  _et al._ (2008b); Dasgupta and Dighe
(2008); Friedland (2010); Choubey  _et al._ (2010).

Figure 2: Spectral swap in the two mass orderings. Here “$e$” refers to
$\nu_{e}$ whereas “$x$” refers to $\nu_{x}$.

It quickly becomes evident that solving this complicated system of non-linear
equations necessitates recourse to numerical simulations. However, gaining an
intuitive grasp of the inception of these oscillations can be facilitated by
interpreting them as an instability within flavour space and subjecting them
to a linearised stability analysisSawyer (2009); Banerjee  _et al._ (2011). In
this case, the density matrix is first expanded in the weak interaction basis
as

$\varrho_{\mathbf{p,r}}(t)=\frac{f_{\nu_{e}}+f_{\nu_{x}}}{2}+\frac{f_{\nu_{e}}-f_{\nu_{x}}}{2}\,\begin{pmatrix}s_{\mathbf{p}}(t,\mathbf{r})&S_{\mathbf{p}}(t,\mathbf{r})\\\
S^{*}_{\mathbf{p}}(t,\mathbf{r})&-s_{\mathbf{p}}(t,\mathbf{r})\end{pmatrix}\,,$
(14)

where $f_{\nu_{\alpha}}$ represents the occupation number of $\nu_{\alpha}$.
The real field $s_{\mathbf{p}}$ measures the evolution of the occupation
number, whereas $S_{\mathbf{p}}$ measures the flavour coherence and hence is
an indicator of the oscillation dynamics. Coherence dictates that
$s^{2}+|S|^{2}=1$.

Deep inside a SN, the matter density is high enough to assume that
$s_{\mathbf{p}}\ll|S_{\mathbf{p}}|$ holds initially. As a result, Eq. 6 can be
expanded to linear order in $S_{\mathbf{p}}$. Using these linearised
equations, one looks for plane wave solutions of the form
$S_{\mathbf{p}}(t,\mathbf{r})=Q_{\mathbf{p,k}}\,e^{-i(k_{0}t-\mathbf{k}\cdot\mathbf{r})}$,
where $(k_{0},\mathbf{k})$ denote the temporal and the spatial frequency of
the flavour wave respectively. Using this in Eq. 6 leads to a dispersion
relation relating $k_{0}$ and $\mathbf{k}$, given by Izaguirre  _et al._
(2017); Capozzi  _et al._ (2017)

$\rm{det}[\Pi^{\mu\nu}]=0\,,$ (15)

where

$\Pi^{\mu\nu}=\eta^{\mu\nu}+\int_{0}^{+\infty}\frac{E^{2}dE}{2\pi^{2}}\int\frac{d\mathbf{v}}{4\pi}\,g_{E,\mathbf{v}}\,\frac{v^{\mu}v^{\nu}}{v_{\alpha}\,k^{\alpha}}\,.$
(16)

Here, $v^{\mu}=(1,\mathbf{v})$ denotes the velocity 4-vector of the neutrinos
with $\mathbf{v}=\mathbf{p}/|\mathbf{p}|$, $k^{\mu}=(k^{0},\mathbf{k})$, and
$g_{E,\mathbf{v}}$ is the angle-dependent difference spectrum, similar to what
is defined in Eq. 13.

If the dispersion relation is solved with an imaginary wave vector
$\mathbf{k}$ for some real wave number $k_{0}$, the system exhibits spatial
instability. Conversely, if the dispersion relation yields an imaginary
temporal wave number $k_{0}$ for some real spatial wave vector $\mathbf{k}$,
the system displays temporal instability. Although this method identifies
whether a system is unstable, it does not provide insight into the extent of
flavour conversions that may occur.

Instabilities are classified based on the growth rate of unstable modes,
denoted by the imaginary part of the wave vector, ${\rm Im}(k)$. Instabilities
with ${\rm Im}(k)$ proportional to $\sqrt{\omega\mu}$ are categorised as slow
instabilities, while those with ${\rm Im}(k)$ proportional to $\mu$ are termed
fast instabilities. Fast instabilities can further be classified as absolute
or convective instabilities Capozzi  _et al._ (2017, 2019a); Yi  _et al._
(2019). Fast instabilities can reach ${\rm Im}(k)\sim O(1)$ cm-1 in regions
deep within the supernova core, where $\mu$ can be on the order of $O(10^{5})$
km-1. Consequently, fast instabilities are often considered to have probable
impacts on the stellar dynamics, provided they are initiated deep insdie the
SN.

###  IV.3 Fast collective flavour conversions

As discussed, flavour conversions are categorised as “fast” if they happen
deep inside the supernova, at a distance of $\mathcal{O}({\rm cm})$ from the
neutrinosphere. These rapid flavour conversions are mainly governed by the
neutrino self-interaction strength $(\mu)$, and grow with the neutrino
density. Therefore, they can take place even for $\Delta m^{2}\rightarrow 0$,
requiring it only as numerical seeds, and hence are independent of the
neutrino mass ordering. These solutions of the non-linear equations of motion
were first discussed by Sawyer in a series of works Sawyer (2005, 2009, 2016).
This was later confirmed in Chakraborty  _et al._ (2016a); Dasgupta  _et al._
(2017), and this rekindled a wave of efforts in understanding the flavour
propagation of neutrinos in dense media.

Figure 3: Schematic diagram of the neutrino angular distributions near the
neutrinosphere, showing different flavours having different distributions.
Neutrino flavours which decouple earlier have a more forward-peaked angular
distribution, while those that decouple later are more isotropic. This leads
to a zero-crossing in the ELN. Figure adapted from Ref. Dasgupta  _et al._
(2017).

From the linear stability analysis, one can deduce that the condition required
for the onset of the fast instability is that there exists at least one
angular direction where the flux of $\bar{\nu}_{e}$ is greater than that of
the $\nu_{e}$ (under the assumption that
$\nu_{\mu,\tau}=\bar{\nu}_{\mu,\tau}$). This can also happen in scenarios
where there is a backward flux of neutrinos. Hence, this is inherently a
multi-angle phenomenon. The key to understanding the conditions for these fast
conversions relies on the fact that different neutrino flavours start free-
streaming at different radii, and hence near the neutrinosphere, they are
expected to have different angular distributions (see Fig. 3).

This can be formally defined by integrating Eq. 13 to give the angular
distribution of the flavour lepton number. Initial works in this direction
defined the electron lepton number (ELN) as Izaguirre  _et al._ (2017)

$G_{\bf
v}=\sqrt{2}G_{F}\,\int_{0}^{\infty}\frac{dE\,E^{2}}{2\pi^{2}}\left[f_{\nu_{e}}(E,{\bf
v})-f_{\bar{\nu}_{e}}(E,{\bf v})\right]\,$ (17)

under the assumption that the non-electron flavours have the same angular
distribution. In terms of this, it has been demonstrated, numerically as well
as analytically, that a necessary and sufficient condition for the presence of
fast modes, in an effective two-flavour setup, is the requirement of a zero-
crossing in the ELN, i.e., the ELN changes sign for some angular modes Abbar
and Duan (2017); Dasgupta (2022); Morinaga (2021).

The assumption of equal angular distributions for the non-electron flavours
was relaxed in Chakraborty and Chakraborty (2020); Capozzi  _et al._ (2020a),
which incorporated a stability analysis to study three flavour effects. This
led to the definition of a corresponding muon lepton number $(\mu{\rm LN})$
and tau lepton number $(\tau{\rm LN})$, demonstrating that fast modes
essentially depend on the zero-crossing between any two flavour lepton number,
and not just the ELN alone. As a result, a positive zero-crossing in the ELN
can be removed by another positive zero crossing in the $\mu{\rm LN}$ or the
$\tau{\rm LN}$. Therefore, fast modes can have very different behaviour in the
three-flavour setup, as compared to a two-flavour setup. Follow-up works
confirmed that this difference persists not only at the linear level but also
at the non-linear regime, thereby emphasizing the necessity of a three-flavour
study Shalgar and Tamborra (2021b); Capozzi  _et al._ (2022); Shalgar and
Tamborra (2021a).

The discovery of fast flavour conversions deep inside a SN-like toy setup
immediately led to a dilemma - if flavour conversions indeed exist deep inside
the SN medium, is it still valid to decouple neutrino inelastic collisions
from flavour oscillations? It is a well-known fact that collisions can cause
scattering-induced decoherence leading to damping of flavour oscillations.
This was first investigated using a simple two-neutrino beam model in 1+1
dimension Capozzi  _et al._ (2019b) and it was shown that although collisions
are necessary to create the angular crossings required for fast oscillations,
they are not strong enough to damp the flavour conversions. This was followed
by a series of works, where the importance of the interplay of neutrino
flavour conversions, inelastic collisions and advection was emphasised Shalgar
and Tamborra (2021c); Martin  _et al._ (2021); Sasaki and Takiwaki (2022);
Sigl (2022); Shalgar and Tamborra (2023a); Hansen  _et al._ (2022); Padilla-
Gay  _et al._ (2022a); Akaho  _et al._ (2024); Fiorillo  _et al._ (2024a);
Shalgar and Tamborra (2023b). More recently, a new class of instabilities,
known as “collisional instabilities”, has been found, where inelastic
collisions are found to cause flavour instabilities Johns (2023); Johns and
Xiong (2022); Xiong  _et al._ (2023a); Lin and Duan (2023); Padilla-Gay  _et
al._ (2022a); Xiong  _et al._ (2023b); Liu  _et al._ (2023); Kato  _et al._
(2023). Inelastic collisions can trigger instabilities if the collisional rate
changes with energy. Since neutrinos interact more than antineutrinos, this
condition seems to be fulfilled in supernova. On the other hand, collisional
instabilities have a much lower rate than fast instabilities (of the order of
the collisional rate itself), so unless they are triggered in a region where
fast instabilities are not, they would generally play a subdominant role;
whether this can happen is an open question. While considerable progress has
been made in the study of the impact of inelastic collisions on fast flavour
conversions, the final verdict is yet to be delivered.

Figure 4: The depolarisation factor $f^{D}_{v}$ as a function of
${\mathbf{v}}$. It varies between 0 (no-depolarisation) and 0.5 (complete
depolarisation), whereas $0.5\leq f^{D}_{v}\leq 1$ indicates a net flavour
conversion. Figure adapted from Ref. Bhattacharyya and Dasgupta (2021).

The other major question eluding SN neutrino enthusiasts is regarding the
outcome of fast conversions and its impact on SN neutrino spectra. This
requires going beyond a linearised analysis and studying the solutions to the
equations of motion in the non-linear region Sawyer (2009). This problem is
hard to solve analytically and requires various simplifications to proceed. An
attempt at an analytical solution in the non-linear regime was developed in
Dasgupta and Sen (2018), where it was shown that a system of four intersecting
neutrino beams undergoing fast oscillations resembles a particle oscillating
in a quartic potential well. Ref. Abbar and Volpe (2019) studied the evolution
of fast modes in the non-linear regime numerically and deduced that it is not
necessary for the final survival probability to approach a complete quasi-
steady flavour equilibration. Ref. Bhattacharyya and Dasgupta (2020, 2021)
demonstrated through analytical as well as numerical techniques that at late
times, a system undergoing fast oscillations reaches a steady state and the
most likely outcome is a rapid mixture of all flavours, leading to _fast
flavour depolarization_ with the conservation of flavour lepton number (see
Fig. 4 which depicts the depolarisation factor $f^{D}_{v}$ as a function of
${\mathbf{v}}$). This was subsequently confirmed in Wu  _et al._ (2021);
Richers  _et al._ (2021a, b); Bhattacharyya and Dasgupta (2022); Zaizen and
Nagakura (2023a). This has been further investigated recently in Ref.
Cornelius  _et al._ (2024), which studied the impact of imposing boundary
conditions on the final fate of fast flavour conversions and demonstrated that
flavour equipartition can be absent if the underlying boundary condition in a
given simulation setup is not periodic. A number of other papers have also
explored the non-linear regime of fast flavour conversions of neutrinos to
understand the final fate of these flavour conversions Johns  _et al._
(2020a); Sigl (2022); Xiong and Qian (2021); Sasaki and Takiwaki (2022); Abbar
and Capozzi (2022); Richers  _et al._ (2022); Nagakura and Zaizen (2022);
Zaizen and Nagakura (2023b); Fiorillo and Raffelt (2023a); Abbar  _et al._
(2024a, b); Fiorillo  _et al._ (2024b); Fiorillo and Raffelt (2024); Xiong
_et al._ (2024). Finally, studies have also developed the analogy with that of
a flavour pendulum, much like in the bipolar (slow-collective) oscillations
case Johns  _et al._ (2020b); Padilla-Gay  _et al._ (2022b); Bhattacharyya and
Dasgupta (2022); Fiorillo and Raffelt (2023a, b). In this case, the final fate
of fast flavour conversions was shown to be related to the relaxation of the
flavour pendulum. Note that, however, these analogies are strictly valid for
purely homogeneous systems. The presence of any inhomogeneity can get
exponentially enhanced due to fast instabilities. In such a case, the final
outcome is expected to be a depolarized state, with flavour changes on very
small scales.

The potential impact of fast flavour conversions on supernova dynamics
prompted numerous investigations into the occurrence of rapid instabilities in
various SN models. An initial exploration in this direction was undertaken in
Tamborra  _et al._ (2017), where a specialised examination of the angular
distributions of neutrino radiation fields in several spherically symmetric
(1D) SN simulations failed to identify any crossings in the electron lepton
number (ELN) near the neutrinosphere. However, the presence of a large-scale
dipole asymmetry in the ELN emission, described as Lepton-Emission Self-
sustained Asymmetry (LESA) Tamborra  _et al._ (2014), has led to the
possibility of the presence of an ELN crossing in multi-dimensional SN
simulations. This has motivated numerous studies in the search for the
conditions of fast conversions in state-of-the-art, self-consistent
multidimensional SN models Dasgupta  _et al._ (2018); Glas  _et al._ (2020);
Nagakura and Johns (2021a); Abbar (2020a); Abbar  _et al._ (2021); Capozzi
_et al._ (2020b); Nagakura and Johns (2021b); Nagakura  _et al._ (2021);
Ehring  _et al._ (2023a); Abbar and Nagakura (2024); Froustey  _et al._
(2024). Most of the works rely on the fact that SN simulations generally do
not provide the full angular distributions of neutrino flavours, but the
moments of the angular distributions. Hence the studies focus on optimizing
the use of these angular moments to look for conditions conducive to the
presence of fast conversions.

Figure 5: Schematic diagram denoting different regions of flavour conversion
inside a SN envelope. Figure adapted from Ref. Sen (2018).

Flavour conversions of neutrinos deep inside a SN represent a significant area
of uncertainty in a comprehensive, first-principle modelling of the SN
dynamics. Our studies over the past decades have unearthed different types of
flavour conversion mechanisms operating at different distances from the core,
as depicted in Fig. 5. Significant progress is being made on the impact of
flavour conversions on the explosion dynamics- depending on the extent of
flavour conversions, explosion could be either hindered or assisted Dasgupta
_et al._ (2012); Ehring  _et al._ (2023a, b) as well as the nucleosynthesis of
heavy elements Fujimoto and Nagakura (2022); Xiong  _et al._ (2020); Fischer
_et al._ (2024); Balantekin  _et al._ (2023); Friedland  _et al._ (2023).
However, we are still far from a complete picture and the final verdict on the
outcome is yet to be out.

###  IV.4 A comment on mean-field approximation _vs._ many-body effects

Discussions on the dynamics of collective oscillations have primarily relied
on a mean-field approximation, neglecting all quantum correlations among
neutrinos. Despite the wide use of this approach, its validity has been a
matter of debate over the past years Friedland and Lunardini (2003a, b);
Balantekin and Pehlivan (2007); Pehlivan  _et al._ (2011); Volpe  _et al._
(2013); Väänänen and Volpe (2013); Serreau and Volpe (2014). Hence, it is
crucial to assess the applicability of this approximation across various
scenarios to ensure the accurate interpretation of results and to delve into
the concept of entanglement among the neutrinos within the many-body system.
Over the last few years, there has been growing effort in this direction, with
the aim of understanding the efficacy of a mean-field treatment of neutrinos
in such a dense environment Birol  _et al._ (2018); Patwardhan  _et al._
(2019); Cervia  _et al._ (2019); Rrapaj (2020); Roggero (2021); Patwardhan
_et al._ (2021). We will not go into the details of this discussion, but refer
the interested readers to Patwardhan  _et al._ (2023) (and references therein)
for more detailed review.

##  V New physics from supernova neutrinos

A galactic core-collapse SN can act as the ultimate particle physics
laboratory to probe physics beyond the Standard Model Raffelt (1996, 1999).
The extreme environment allows for probing a wide range of new physics, which
cannot otherwise be accessible in terrestrial laboratories. From a
phenomenological point of view, new physics inside a supernova can have either
of the following effects:

  1. 1.

It can affect the net luminosity and average energy of the emitted neutrinos,
and hence the duration of the neutrino emission phase.

  2. 2.

It can affect the shape/normalisation of the emitted neutrino spectra.

Using a combination of one or both of these effects, strong constraints can be
imposed on new physics from a galactic supernova.

The most well-known of the bounds coming from SN1987A is the anomalous cooling
bound Ellis and Olive (1987); Raffelt and Seckel (1988); Turner (1988);
Burrows  _et al._ (1990); Raffelt (1996). Neutrinos from SN1987A can be used
to constrain the amount of energy that can be transported by any new degree of
freedom, weakly coupled to the SM. Such particles, if produced inside a
supernova, can lead to new modes of energy loss and hence additional cooling
channels. Based on this, a criterion for the maximum luminosity of the new
particle was proposed. If the luminosity of the new particle,
$\mathcal{L}\gtrsim 3\times 10^{52}\,{\rm erg\,s}^{-1}$ for $\rho=3\times
10^{14}\,{\rm g/cc}$ and $T=30\,{\rm MeV}$, then the duration of the neutrino
burst is reduced from the observed $\sim 10\,{\rm s}$ of SN1987A. This
argument leads to a bound on the coupling of the new particles. However, the
bound cannot be applied for arbitrarily large couplings. For couplings
stronger than a certain value, the new particles get trapped and cannot escape
the SN core, leading to an exclusion window in the coupling-mass parameter
space. On the other hand, for couplings which are too weak, the particles may
not be efficiently produced to alter the dynamics of a SN.

This argument has been used to constrain new weakly coupled particles like the
axion Raffelt and Seckel (1988); Dolan  _et al._ (2017); Chang  _et al._
(2018); Carenza  _et al._ (2019, 2021); Lucente  _et al._ (2020); Calore  _et
al._ (2021); Caputo and Raffelt (2024), dark gauge bosons charged under new
symmetries Rrapaj and Reddy (2016); Chang  _et al._ (2017); Sung  _et al._
(2021); Caputo  _et al._ (2021); Cerdeño  _et al._ (2023), majorons and other
light scalars coupled to neutrinos Farzan (2003); Brune and Päs (2019);
Heurtier and Zhang (2017); Diamond  _et al._ (2023); Antel  _et al._ (2023),
and so on. More recently, the cooling argument was refined to include physical
effects which were ignored previously and a detailed prescription was laid
down for limits from more updated models of SNe Caputo  _et al._ (2022a, b).
However, note that in cases where there exists significant self-interactions
in the dark sectors, these cooling arguments may not apply Fiorillo and
Vitagliano (2024). Modern SN codes nowadays can perform a complete 6-flavour
neutrino (antineutrino) transport, which has demonstrated the relevance of
muons in simulations Bollig  _et al._ (2017). This has led to additional
constraints on _muon-philic_ new degrees of freedom Bollig  _et al._ (2020);
Croon  _et al._ (2021); Caputo  _et al._ (2022a). The dynamics of a core-
collapse can also be altered by the decay of bosonic particles into neutrinos
Fiorillo  _et al._ (2023b), or photons Caputo  _et al._ (2022c) deep inside
the SN progenitor.

Similar cooling arguments have also been put forth to constrain the mass and
mixing of additional species of sterile neutrinos $(\nu_{s})$. It is well
known that sterile neutrinos with masses $\sim\mathcal{O}(100\,{\rm MeV})$ can
be produced from different processes inside the SN core Mastrototaro  _et al._
(2020); Carenza  _et al._ (2024); Brdar  _et al._ (2023). These sterile
neutrinos, if produced, can stream out of the SN, thereby leading to
additional cooling, and hence can be constrained. Lower mass (keV) sterile
neutrinos are excellent dark matter candidates. The $\nu_{s}$ can be easily
produced inside a SN through adiabatic flavour conversion at MSW resonances,
or through a scattering-induced decoherence Raffelt and Zhou (2011); Argüelles
_et al._ (2019); Syvolap  _et al._ (2022); Suliga  _et al._ (2019); Chen  _et
al._ (2022); Suliga  _et al._ (2020). The usual cooling arguments apply here
as well. However, since the $\nu_{s}$ is produced from an active neutrino,
this in turn affects the forward scattering potential that the propagating
neutrinos experience. This leads to feedback, which when taken into account
consistently can loosen the bounds on the sterile mass-mixing parameter space
Suliga  _et al._ (2019, 2020). However, secret interactions of neutrinos can
spoil the party, and enhance the production of $\nu_{s}$, which can
potentially overcome this feedback effect Chen  _et al._ (2022). Lighter mass
$\nu_{s}$ (eV scale) can be produced from active neutrinos directly via
resonant MSW transitions at densities that can affect the heating of low mass
SNe and affect their explosion Tamborra  _et al._ (2012b); Franarin  _et al._
(2018); Tang  _et al._ (2020). For larger mass SNe, the production happens at
densities which can affect the active neutrino flux, and hence have important
consequences for nucleosynthesis Tamborra  _et al._ (2012b).

Interactions of neutrinos beyond the SM can also be probed effectively inside
a SN. Non-standard interactions (NSI) of neutrinos with matter can lead to
additional MSW resonances inside the SN (like the I-resonance Esteban-Pretel
_et al._ (2007, 2010)), thereby altering the neutrino flavour evolution. These
interactions can change the neutron-proton ratio, leading to intriguing
consequences for nucleosynthesis yields. Similarly, a core-collapse SN is one
of the few laboratories where non-standard neutrino self-interactions (NSSI)
can be tested Blennow  _et al._ (2008); Das  _et al._ (2017); Dighe and Sen
(2018). In fact, a smoking-gun signature of the presence of flavour-violating
NSSI will be the presence of spectral splits in the neutronisation burst
spectra due to collective oscillations, which do not take place within the SM
in this time phase Das  _et al._ (2017). Furthermore, the presence of
neutrinophilic bosons can lead to $2\nu\rightarrow 4\nu$ processes inside the
SN, which reduces the net energy deposited behind the shock, thereby hindering
explosion Shalgar  _et al._ (2021), however the inclusion of the inverse
reaction can counter this effect Fiorillo  _et al._ (2024c). Secret
interactions of neutrino have also been argued to extend the duration of the
SN neutrino burst Chang  _et al._ (2023), but these claims were recently
countered in Fiorillo  _et al._ (2024c). Secret interactions of neutrinos can
also be constrained from their interactions with the cosmic neutrino
background, which leads to a dissipation of the neutrino energy below the
expected value Shalgar  _et al._ (2021). Neutrinos from a SN can also be used
to probe neutrino-dark matter interactions through their interaction with the
DM in the galactic halo Carpio  _et al._ (2023); Das and Sen (2021); Lin  _et
al._ (2023); Das  _et al._ (2024).

The other direction where SN neutrinos can contribute immensely is towards
constraining neutrino parameters. The large neutrino flux, combined with the
propagation through extreme conditions, allows one to put stringent
constraints on neutrino properties. For example, using SN1987A data, time of
flight constraints were used to set neutrino mass bounds to $m_{\nu}<20\,{\rm
eV}$ Kernan and Krauss (1995). Future experiments like DUNE can use the
neutronization burst phase to narrow it down to $\mathcal{O}(1)\,{\rm eV}$
Pompa  _et al._ (2022). Of course, this requires a good estimation to the
distance of the next galactic SN, which can be achieved via triangulation
Hansen  _et al._ (2020); Linzer and Scholberg (2019).

The neutronization burst is an excellent tool to probe the neutrino mass-
ordering, which is one of the major unknowns in neutrino physics. Deep inside
the supernova, neutrinos are generated within regions of exceedingly high
local matter densities. Conventional matter effects dictate that the $\nu_{e}$
from the burst phase align closely with the instantaneous Hamiltonian
eigenstate associated with the highest instantaneous Hamiltonian eigenvalue
upon production. Subsequent evolution treats the distinct eigenstates as
incoherent, with the adiabatic approximation proving effective owing to the
spatial variation of matter density and known neutrino mass-squared
differences. Consequently, the neutronization-burst neutrino flux at Earth,
when disregarding the minor initial $\nu_{x}$ population, can be well
approximated by Dighe and Smirnov (2000):

$f_{\nu_{e}}(E,t)=\frac{1}{4\pi R^{2}}|{\rm U_{eh}}|^{2}\Phi_{\nu_{e}}(E,t)$
(18)

where $h$ denotes the heaviest neutrino mass eigenstate, with $h=3$ for the
NMO and $h=2$ for the IMO, ${\rm U_{eh}}$ represents the relevant element of
the leptonic mixing-matrix, while $R$ signifies the distance between Earth and
the supernova. Given the negligible production of other flavours during this
phase, the $\nu_{e}$ flux for the NMO experiences a relative suppression
factor of approximately $\sim|{\rm U_{e3}}|^{2}/|{\rm U_{e2}}|^{2}\simeq 0.1$.
This facilitates the determination of the neutrino mass ordering during the
neutronization-burst phase Dighe and Smirnov (2000). The mass-ordering can
also be deduced from timing information in upcoming detectors Serpico  _et
al._ (2012); Brdar and Xu (2022).

Finite neutrino masses imply that neutrinos can decay. Neutrinos from a SN can
be used to put tight constraints on invisible, visible as well as radiative
decays of neutrinos Raffelt (1999). While data from SN1987A constrains the
invisible decays to $\tau/m>10^{5}\,{\rm s/eV}$ Frieman  _et al._ (1988);
Iváñez Ballesteros and Volpe (2023) , future galactic SN can narrow it down to
$\tau/m\lesssim 10^{6}\,{\rm s/eV}$ for a SN happening at 10 kpc de Gouvêa
_et al._ (2020). Radiative decays of neutrinos can lead to a coincident
$\gamma$ flare - this has been used to constrain radiative decays to
$\tau/m\gtrsim 10^{15}\,{\rm s/eV}$ Raffelt (1999).

The strong magnetic field existing inside a SN can also be used to probe
neutrino electromagnetic properties. For Dirac neutrinos, a non-zero intrinsic
magnetic moment can give rise to a spin flip for a certain flavour $\alpha$,
i.e. $\nu_{\alpha L}\rightarrow\nu_{\alpha R}$, while a transition magnetic
moment also flips the flavour, i.e., $\nu_{\alpha L}\rightarrow\nu_{\beta R}$
Pal and Wolfenstein (1982). This conversion process,
$\nu_{L}\rightarrow\nu_{R}$, leads to faster cooling of the SN, and can be
used to constrain the magnetic moment $\mu_{\nu}\leq 10^{-11}\,\mu_{B}$
Barbieri and Mohapatra (1988). Furthermore, in the presence of a magnetic
field, coherent scattering of neutrinos off the background can resonantly
enhance these spin-flavour transitions, known as resonant spin-flavour
precession (RSFP) Lim and Marciano (1988); Akhmedov (1988). Such RSFP can lead
to intriguing signatures in the neutronisation spectra Ando and Sato (2003);
Akhmedov and Fukuyama (2003) and can be used to constrain non-zero transverse
magnetic moments much better than current laboratory constraints Jana  _et
al._ (2022). Neutrino magnetic moments have also been shown to be relevant for
collective oscillations de Gouvea and Shalgar (2012); Abbar (2020b). One can
also constrain the effective neutrino electric charge $(q_{\nu})$ from the
observation of neutrinos from SN1987A. The presence of galactic and
extragalactic magnetic fields may elongate the trajectory of millicharged
neutrinos, thereby delaying their arrival time at the Earth beyond the
observed period of a few seconds. This constrains $q_{\nu_{e}}<2\times
10^{-15}\,e$ for an intergalactic field $B=10^{-3}\,\mu G$ and a distance of
$R=50\,{\rm kpc}$, and $q_{\nu_{e}}<2\times 10^{-17}\,e$ for a galactic field
$B=1\,\mu G$ and a SN at a distance of $R=10\,{\rm kpc}$ Giunti and Studenikin
(2015).

The nature of neutrinos is another mystery that has not been solved yet.
Depending on whether lepton number is conserved or not, neutrinos can be Dirac
or Majorana. Supernova neutrinos can also be used to pin down the nature of
neutrinos, _in the presence of new physics_. For example, studies have shown
that for a future galactic SN, using a combination of upcoming experiments
like DUNE and HK, one can distinguish between a decaying Dirac and a decaying
Majorana neutrino from the neutronization burst flux de Gouvêa  _et al._
(2020). Similarly, in the presence of non-zero magnetic moments, Dirac and
Majorana can have different signatures in the neutrino flux Jana  _et al._
(2022). However, there also exists the possibility of soft lepton number
violation (LNV). The degree of LNV can be gauged by the relative smallness of
the Majorana mass term compared to the Dirac mass term for neutrinos. In such
a scenario, neutrinos exhibit pseudo-Dirac (or quasi-Dirac) behaviour
Wolfenstein (1981). The soft nature of LNV ensures that although neutrinos are
Majorana, they effectively behave akin to Dirac neutrinos in practical terms.
In this scenario, active-sterile neutrino oscillations are driven by a minute
mass-squared difference, denoted as $\delta m^{2}$, between the mass
eigenstates. However, this mass-squared difference is only accessible over
astronomically large baselines. Strong constraints can be set on this mass-
squared difference from the neutrino events from SN1987A $\delta m^{2}\lesssim
10^{-20}~{}{\rm eV}^{2}$ Martinez-Soler  _et al._ (2022); Sen (2022).

This is just an incomplete list of the different kinds of particle physics
constraints one can probe from the observation of supernova neutrinos. Studies
have shown that the few neutrino events from SN1987A can also be used to probe
the deviations of the equivalence principle Guzzo  _et al._ (2002), the
gravitational memory effect Mukhopadhyay  _et al._ (2020), the effect of
neutrino long-range forces Reddy and Zhou (2022); Ghosh  _et al._ (2023),
consequences of deviation from Lorentz symmetry in the neutrino sector
Chakraborty  _et al._ (2013), the QCD phase transition Dasgupta  _et al._
(2010), and several other exotic ideas which are otherwise difficult to test.
This is a testament to the immense potential offered by a future galactic SN
to test physics beyond the SM. In conclusion, we would also like to point to
the potential of the diffuse supernova neutrino background - a sea of
neutrinos from all possible core-collapse supernovae (and failed supernovae)
in the Universe - to probe fundamental physics (see Beacom (2010); De Gouvêa
_et al._ (2020); Ando  _et al._ (2023) and references therein for a detailed
discussion of this aspect).

##  VI Conclusion

The study of neutrinos from supernovae represents a captivating and
multifaceted field of research requiring significant contributions from
particle physics and astrophysics. Through decades of theoretical modelling,
and computational efforts, our understanding of the intricate processes
governing neutrino emission, propagation, and interaction within the
surroundings has advanced significantly. We have strong reasons to believe
that neutrinos play a crucial role in the supernova explosion mechanism, as
well as in the formation of heavy elements inside stars. Furthermore, these
neutrinos offer the best bet to probe the physics of dense matter existing
inside a supernova, which cannot be achieved inside a terrestrial laboratory.
As a result, a careful study of these neutrinos is necessary to utilise the
full potential of the next galactic supernova.

In this brief review, we discussed the major advancements made in the field
and where we stand. While we have made remarkable progress in the
understanding of neutrino flavour oscillations in dense environments, we still
do not know the exact impact of these effects. The numerical complexity
associated with the complete problem necessitates the introduction of
simplifications, which can sometimes hide additional flavour instabilities.
Previously, the relaxation of such symmetries has led to the discovery of new
instabilities associated with neutrino non-radial evolution inside a
supernova. A recent example is that of fast pairwise flavour conversions,
which was missed due to the assumption of equal angular distributions for
neutrinos and antineutrinos. These rapid flavour conversions, occurring deep
inside a supernova, are almost independent of the neutrino mass and mixing,
and hence unsuppressed due to matter effects. Recent studies indicate that
these fast oscillations eventually lead to an almost flavour equilibration
deep inside the supernova, however further studies are required to confirm
this idea. As a result, there has been a surge in research efforts aimed at
evaluating the impact of these flavour conversions on the mechanism of
supernova explosions.

We also highlighted how the neutrino flux from a supernova allows for imposing
stringent constraints on neutrino properties, which are typically unattainable
in terrestrial laboratories. This advocates a core-collapse supernova as the
ultimate laboratory for particle physics. Therefore, the study of supernova
neutrinos remains a vibrant and fruitful avenue of exploration, offering
invaluable insights into the fundamental nature of matter and the dynamics of
stellar evolution.

###### Acknowledgements.

We would like to thank Basudeb Dasgupta, Georg Raffelt and Irene Tamborra for
a careful reading of the manuscript and insightful comments. We also
acknowledge the hospitality of the Network for Neutrinos, Nuclear
Astrophysics, and Symmetries (N3AS) Physics Frontier Center of UC Berkeley
where part of this review was completed.

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