# Neutrinos and nucleosynthesis of elements

Tobias Fischer  Gang Guo  Karlheinz Langanke  Gabriel Martínez-Pinedo  Yong-
Zhong Qian  Meng-Ru Wu<EMAIL_ADDRESS>Institute of Theoretical
Physics, University of Wroclaw, 50-204 Wroclaw, Poland  School of Mathematics
and Physics, China University of Geosciences, Wuhan, 430074, China  GSI
Helmholtzzentrum für Schwerioneneforschung, 64291 Darmstadt, Germany
Technische Universität Darmstadt, 64289 Darmstadt, Germany  School of Physics
and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, U.S.A.
Institute of Physics, Academia Sinica, Taipei 11529, Taiwan  Institute of
Astronomy and Astrophysics, Academia Sinica, Taipei 10617, Taiwan  Physics
Division, National Center for Theoretical Sciences, Taipei 106, Taiwan

###### Abstract

Neutrinos are known to play important roles in many astrophysical scenarios
from the early period of the big bang to current stellar evolution being a
unique messenger of the fusion reactions occurring in the center of our sun.
In particular, neutrinos are crucial in determining the dynamics and the
composition evolution in explosive events such as core-collapse supernovae and
the merger of two neutron stars. In this paper, we review the current
understanding of supernovae and binary neutron star mergers by focusing on the
role of neutrinos therein. Several recent improvements on the theoretical
modeling of neutrino interaction rates in nuclear matter as well as their
impact on the heavy element nucleosynthesis in the supernova neutrino-driven
wind are discussed, including the neutrino-nucleon opacity at the mean field
level taking into account the relativistic kinematics of nucleons, the effect
due to the nucleon-nucleon correlation, and the nucleon-nucleon
bremsstrahlung. We also review the framework used to compute the neutrino-
nucleus interactions and the up-to-date yield prediction for isotopes from
neutrino nucleosynthesis occurring in the outer envelope of the supernova
progenitor star during the explosion. Here improved predictions of energy
spectra of supernova neutrinos of all flavors have had significant impact on
the nucleosynthesis yields. Rapid progresses in modeling the flavor
oscillations of neutrinos in these environments, including several novel
mechanisms for collective neutrino oscillations and their potential impacts on
various nucleosynthesis processes are summarized.

###### keywords:

Core-collapse supernova, neutron star merger , neutrino , nucleosynthesis

††journal: Progress in Particle and Nuclear Physics

###### Contents

  1. 1 Introduction
  2. 2 Astrophysical sites
    1. 2.1 Core-collapse supernovae
      1. 2.1.1 Supernova progenitors
      2. 2.1.2 Stellar core collapse
      3. 2.1.3 Supernova core-bounce and post-bounce evolution
      4. 2.1.4 Neutrino-driven supernova explosions
      5. 2.1.5 Proto-neutron star deleptonization and the neutrino-driven wind
      6. 2.1.6 Presupernova and supernova neutrino detection
    2. 2.2 Binary neutron star mergers
      1. 2.2.1 Merger and dynamical ejecta
      2. 2.2.2 Secular evolution of remnant and its outflow
  3. 3 Neutrino interactions and nucleosynthesis
    1. 3.1 Neutrino-nucleon reactions in SN matter
      1. 3.1.1 Neutrino-nucleon opacity at mean field level
      2. 3.1.2 Effects of nucleon-nucleon correlation
      3. 3.1.3 Nucleon-nucleon bremsstrahlung
      4. 3.1.4 Additional degrees of freedom
    2. 3.2 Impact of neutrino-nucleon rates on PNS evolution and nucleosynthesis in the $\nu$-driven wind
    3. 3.3 Neutrino-nucleus interaction
    4. 3.4 Impact of neutrino-nucleus interaction on neutrino nucleosynthesis in supernovae
  4. 4 Neutrino flavor conversions and nucleosynthesis
    1. 4.1 Flavor conversions of neutrinos in astrophysical environments
      1. 4.1.1 Collective flavor oscillations
      2. 4.1.2 Flavor transformation due to varying matter profile
    2. 4.2 Impact on nucleosynthesis in supernovae and in neutron star mergers
      1. 4.2.1 Neutrino-driven wind in supernovae
      2. 4.2.2 SN explosive nucleosynthesis
      3. 4.2.3 Neutrino nucleosynthesis
      4. 4.2.4 $r$-process in neutron star mergers
  5. 5 Summary and outlook

## 1 Introduction

Finding the exact origins of nature’s different elements and their isotopes
has been a long quest since the early 20th century. The seminal papers in 1957
from Margaret Burbidge, Geoffrey Burbidge, William A. Fowler, and Fred Hoyle
(B2FH) and independently from Alistair Cameron established the foundation that
stars and the associated astrophysical explosions are the main sites that
produced nearly all different nuclei after the Big Bang nucleosynthesis during
the cosmic evolution [1, 2]. Several major nucleosynthesis processes and the
general requirements in nuclear physics and astrophysics conditions, including
the hydrogen burning, helium burning, $\alpha$ process, slow ($s$-) and rapid
($r$-) neutron capture processes, as well as $p$\- or $\gamma$-processes, were
discussed in [1, 2]. Tremendous advances were made since then to unravel the
complexities in astrophysics and nuclear physics that are involved in these
nucleosynthetic processes; see e.g., reviews [3, 4, 5, 6, 7]. Among these, the
charged particle reactions (CPR), in which case the Coulomb barrier needs to
be overcome, are essential for the understanding of the big bang
nucleosynthesis and the evolution of stars from the main sequence to later
stages. However, they cannot account for the formation of elements
significantly heavier than iron, such as barium, europium as well as the
actinides such as uranium and plutonium, which requires the capture of
neutrons by the seed nuclei formed via CPR ($r$-process) or by existing nuclei
inside stars ($s$-process). Meanwhile, other nucleosynthetic processes (e.g.,
neutrino- ($\nu$-) process [8, 9], the $\nu p$ process [10, 11], $i$-process
[12, 13]) were found to potentially be responsible for producing certain
isotopes whose traces are present in stellar observations or in the solar
system, together with $p$\- or $\gamma$-process [4, 5]. High-energy collisions
with cosmic rays are a production source of the light elements beryllium,
boron and also lithium [14].

Explosive astrophysical events such as a variety of core-collapse supernovae
(CCSNe) and the merger of two neutron stars (NS) or a neutron star with a
black hole (BH) have long been considered as sites where a number of the above
nucleosythetic processes can take place. In particular, the involved short
dynamical timescale of $\mathcal{O}(1)$ s as well as the associated neutron-
rich, hot, and dense environments provide ideal conditions for the $r$-process
to occur. Moreover, the extreme conditions of density and temperature allow
the production of neutrinos. These neutrinos not only affect or even determine
the dynamical evolution of the systems, but also play essential roles in
shaping the formation of nuclei in nucleosynthesis processes associated with
these events, including the $r$-process as well as the $\nu$\- and $\nu p$
processes. Consequently, knowledge of how neutrinos interact with nuclei and
nuclear matter and how flavor oscillations of neutrinos happen at relevant
astrophysical conditions are needed to robustly model these events and their
nucleosynthesis yields. The resulting observational signatures, which include
the direct ones: neutrinos, electromagnetic, and gravitational waves (GWs),
and the indirect ones: the fossil records of elemental abundances in stars as
well as the presence of the short-lived radioactive nuclei in the Milky Way or
nearby galaxies and/or their traces in meteorites and deep-sea crusts, can
then be combined altogether to advance our understanding.

In this article, we review the recent progress in elucidating the role of
neutrinos on the nucleosynthesis of elements. In Sec. 2, we introduce the two
main astrophysical sites, CCSNe and binary NS mergers (BNSMs) and discuss the
general evolution of these systems and the role of neutrinos therein. In Sec.
3, we review recent work on neutrino interaction with matter and nuclei as
well as their impact on CCSNe nucleosynthesis. We then discuss recent findings
of flavor oscillations of neutrinos in SNe and in BNSMs, and their
implications for different nucleosynthesis processes in Sec. 4. We conclude in
Sec. 5.

## 2 Astrophysical sites

The canonical sites where neutrinos are copiously produced and affect the
evolution of the system as well as nucleosynthesis therein are the CCSNe and
BNSMs. Both sites involve conditions of high temperature
$T\gtrsim\mathcal{O}(10)$ MeV and mass density
$\rho\gtrsim\mathcal{O}(10^{14})$ g cm-3. Under these conditions, weak
interactions that allow neutrino productions can proceed with time scale much
shorter than $\mathcal{O}(1)$ s, which allows massive production of neutrinos
in all flavors ($\nu_{e}$, $\nu_{\mu}$, $\nu_{\tau}$ and their antineutrinos).
The density of the environment can be so high that even neutrinos are
“trapped” for certain duration and only diffuse out slowly. Consequently, the
interaction of neutrinos with matter around the nuclear saturation density
sets the evolution of the system. Moreover, neutrino interaction with nucleons
and nuclei at these sites can determine or change the composition in the
matter ejected, and therefore are critical in shaping the nucleosynthesis
outcome of these events. In the following sections, we will review the current
understanding of CCSNe and BNSMs, with particular focus on the role of
neutrinos.

### 2.1 Core-collapse supernovae

#### 2.1.1 Supernova progenitors

Progenitors of CCSNe are massive stars with a zero-age main sequence (ZAMS)
mass heavier than about 8–10 $M_{\odot}$. Such stars spent about several
million years on the main sequence burning the hydrogen burning stage,
followed by the core helium burning that lasts about several hundred thousand
years. In contrast to low- and intermediate-mass stars, these massive stars
further enter the advanced nuclear burning stages, starting with core carbon
burning that lasts for several hundred years. Details about the more advanced
nuclear burning stages, e.g., oxygen and neon burning, which can last for
several months up to years, strongly depend on the stellar evolutionary track
determined by the ZAMS mass and metallicity (see also Ref. [15]). Those with
ZAMS mass ranging between 8–10 M⊙, which develop stellar cores composed of
mostly oxygen, neon and magnesium (c.f. Ref. [16, 17, 18] and references
therein) can undergo core collapse triggered by electron captures on neon and
magnesium nuclides and result in the so-called electron-capture supernovae.
Heavier stars can further enter the silicon burning phase, which lasts only
for a day or less, and develop iron cores before core collapse happens.

The time spans of different nuclear burning stages are mostly dictated by the
nuclear burning timescales involved, and are typically much longer than the
cooling as well as the dynamical timescales. Thus, hydrostatic simulations of
the evolution of such stars in spherically symmetric (1D) models have been
employed for several decades [19, 20, 21, 22, 23] to provide a variety of
progenitors as the initial conditions for modeling SN explosions. However, it
is also well known that the stellar structure can be convectively unstable,
featuring nuclear burning regions with nearly constant entropy per baryon,
resulting in composition mixing. This important multi-dimensional phenomenon
has been typically implemented into the stellar evolution models through the
mixing length theory [24, 25]. Other important multi-dimensional effects such
as the stellar rotation, magnetic fields, and mass loss have also been studied
but remain as the major sources of uncertainty [26, 27, 28, 29, 30, 31, 32,
33]. Among those, the mass loss rate determines the core evolution at the
onset of core carbon burning and hence has a direct impact on the later SN
evolution. Low mass loss rates result in a compact carbon core and lead to a
high density silicon and sulfur layer, which, in turn, will power a high mass
accretion rate onto the proto-neutron star formed later during the SN
evolution powering high neutrino luminosities. This aspect of CCSN
phenomenology has been analyzed in terms of the compactness parameter as a
useful, though ad hoc, diagnostic tool [34]. For stars residing in close
binary system, the binary interactions can also affect the associated SN
progenitor properties and have been under intense investigations [35, 36, 37,
38, 39, 40].

Another interesting aspect is that the nuclear burning timescale during the
final oxygen and in particular the silicon burning phase becomes much closer
to the hydrodynamical timescale. The former can be approximated as

$\tau\propto 0.52~{}{\rm days}~{}\left(\frac{Q}{2\times 10^{17}~{}{\rm
erg~{}g}^{-1}}\right)\left(\frac{M_{\rm core}}{1.5~{}{\rm
M}_{\odot}}\right)\left(\frac{3.5\times L_{\odot}}{L}\right)~{},$ (2.1)

where $Q$ measures the energy output for the “schematic” process, ${}^{28}{\rm
Si}\,+\,^{28}{\rm Si}\longrightarrow\,^{56}{\rm Ni}$, $L=3.5\ L_{\odot}$ is a
typical value for the silicon-burning luminosity found in stellar evolution
calculations and 1.5 $M_{\odot}$ is a typical core mass. Recently, multi-
dimensional simulations of the silicon burning stage have been investigated
(for a recent review, see Ref. [41] and references therein). Simulations for a
short epoch of the evolution during the oxygen as well as the silicon-sulfur
burning stages [42, 43, 44, 45, 46, 47] demonstrated that the implemented
initial perturbations will grow to large scale sizes, which locally reduces
the density in the silicon-sulfur layer and hence leading to a lower ram
pressure at the later SN shock evolution. However, even though all these
models show a direct impact on the later SN dynamics, favoring the SN shock
revival, the choice of the initial perturbations, mimicking the presence of
convection at start from a spherically symmetric stellar model, are still
under debate.

During the final burning phases, in particular the silicon burning, the Fermi
energy of electrons reaches values large enough to trigger electron capture on
nuclei so that the core develops a neutron excess. Neutrino emission from
thermal processes and from nuclear electron captures become the main cooling
mechanisms at these stages, lowering the entropy of the stellar core and
causing the stellar core to deleptonize. The processes involved give rise to
the production of low energy, sub-MeV to few MeV neutrinos.

#### 2.1.2 Stellar core collapse

The aforementioned neutrino losses cause the continuous deleptonization of the
stellar core, which keeps the central entropy per baryon at a low value of few
$k_{\rm B}$. The associated decrease of the electron abundance, $Y_{e}\equiv
n_{e}/n_{b}$ where $n_{e}$ and $n_{b}$ are the number density of electrons and
baryons, results in the loss of pressure support, as the partly degenerate
electron gas provided the dominant contribution to the pressure and hence
stability of the stellar core. The subsequent adiabatic contraction of the
stellar core cause the continuous rise of the central temperature and density,
such that the nuclear composition is dominated by iron-group nuclei which are
in nuclear statistical equilibrium at $T\gtrsim 6$ GK $(0.5~{}{\rm MeV})$. At
these high temperatures, in addition to neutrino losses from electron capture
on nuclei, a second process contributes to destabilize the core, namely the
photodisintegration of heavy nuclei. The photodisintegration leads to the
presence of free nucleons and alpha nuclei. Since electron captures on free
protons are significantly faster than electron captures on heavy iron group
nuclei [48], this might accelerate the deleptonization of the contracting
stellar core. However, modern supernova simulations indicate that the
abundance of free protons stays low during the collapse [49].

Figure 2.1: (Figure reproduced from Ref. [50].) Neutrino emission spectra from
electron captures on heavy nuclei [48] (blue dashed lines) and from de-
excitations of heavy-nuclei [50], comparing two different approximated
treatments of the nuclear de-excitation function, from Ref. [51] (green solid
line) and assuming a Gaussian distribution (red dash-dotted line), at low and
intermediate temperatures and densities (top and middle panels) and at high
temperatures/densities (bottom panel). In the bottom panel the
$\nu\bar{\nu}$-emission rate from the nucleon-nucleon ($NN$) bremsstrahlung is
shown (magenta dotted line) instead of nuclear electron captures. Note that
the scales (right axis) for $\nu_{e}$ from electron captures are different
from the pair production spectrum (left axis). All spectra shown here are for
a single neutrino species.

Nuclear de-excitation by neutrino pair emission has been proposed as an
additional channel of neutrino losses in Ref. [51]. This idea was revisited in
Ref. [50] and applied to stellar core collapse simulations, implementing the
rates of Ref. [51] and a simplistic rate based on Gaussian strength functions.
The latter also takes into account the first forbidden transitions. Fig. 2.1
compares these neutrino pair emission rates with electron captures on nuclei
at low and intermediate temperatures and densities (top and middle panels) as
well as to the nucleon–nucleon bremsstrahlung at high density (bottom panel).
The de-excitation process greatly enhances the neutrino luminosities of
$\bar{\nu}_{e}$ as well as the heavy lepton flavor neutrinos $\nu_{\mu/\tau}$
and their antineutrinos during the stellar collapse phase [50], for which the
only source otherwise is the electron-position annihilation. However, the
entire collapse phase is still dominated by nuclear electron captures, whose
rates are significantly higher (as shown in Fig. 2.1), and hence the electron
neutrino luminosity is larger by about 4 orders of magnitude.

#### 2.1.3 Supernova core-bounce and post-bounce evolution

The stellar core collapse halts due to nuclear repulsion when the core density
exceeds the saturation density at $n_{\rm sat}\simeq 0.15~{}{\rm
fm}^{-3}~{}(\rho_{\rm sat}\simeq 2.4\times 10^{14}~{}{\rm g}~{}{\rm
cm}^{-3})$. During the collapse, a phase transition similar to the gas-liquid
transition, from inhomogeneous matter – conditions at which heavy nuclei and
unbound nucleons coexist – to homogeneous nuclear matter is completed around
$n_{\rm sat}$. While the precise density value of this Mott transition is
still an active subject of research, common approaches modeling this
transition include the first-order phase transition with the Maxwell
constructions [52], the geometric excluded volume approach [53, 54, 55] and
recently the development of an effective model taking into account medium
modified nuclear binding energies, consistent with quantum statistical
calculations for light nuclear clusters based on scattering phase shifts [56].

During the collapse, the inner core contracts subsonically and becomes
disconnected from the supersonic outer core at the sonic point. Consequently,
when the nuclear repulsion halts the collapse of the inner core, the outer
part is still falling inwards. This leads to the formation of a stagnation
front, which first develops into a sound wave and later, when reaching the
sonic point, becomes a hydrodynamic shock wave. This phenomenon is known as
the core bounce. Following the core bounce, mass accretion continues with a
rate reaching several $M_{\odot}$ per second for $\mathcal{O}(10)$ ms and
results in the formation of a compact central object known as the proto-
neutron star (PNS)111The exact mass accretion rate and duration can
sensitively depend on progenitor mass and metallicity..

The neutrinos emitted during the collapse can no longer escape the core freely
when the density reaches $\rho\sim 10^{12}$ g cm-3, corresponding to
temperatures of a few MeV, and become trapped by the nuclear absorption
processes as well as their coherent scatterings with nuclei. This effect of
neutrino trapping is visible as the $\nu_{e}$ luminosity slightly dips right
before the core bounce shown in Fig. 2.2. In general, there is a variety of
weak processes operating to produce neutrinos of different flavors, in
addition to those associated with nuclear electron captures and nuclear de-
excitations. In particular, the neutrino pair reactions, dominantly the
electron-positron annihilation and at higher baryon density also the nucleon-
nucleon bremsstrahlung, produce heavy lepton flavor neutrinos and
antineutrinos. The $\bar{\nu}_{e}$ luminosity rises slightly slower than
$\nu_{\mu/\tau}$ due to the final state blocking of $\nu_{e}$, which
suppresses their production via the pair processes. Notice that muonic weak
reactions, including the charged current and purely leptonic processes,
already give rise to the non-negligible abundance of $\mu^{-}$ and $\mu^{+}$
at core bounce, once high energy $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ ($\gtrsim
50-100$ MeV) are present. However, their abundances remain small on the order
of $\sim 10^{-4}$ and do not affect the dynamical evolution at this stage
[57].

Figure 2.2: (Figure reprinted from Ref. [58].) Evolution of neutrino
luminosities and average energies as a function of the post-bounce time,
$t_{\rm pb}$ obtained in a spherically symmetric SN simulation with an 18
$M_{\odot}$ progenitor in Ref. [59].

The bounce shock expands rapidly outwards, suffering energy loss of about 8
MeV per baryon from the dissociation of infalling heavy iron-group nuclei into
nucleons. Furthermore, when propagating across the neutrinospheres, a large
number of electron neutrinos due to electron captures on protons made by
photo-dissociation are released producing the so-called $\nu_{e}$ burst, as
shown in Fig. 2.2 which lasts about 20–50 ms. As a result, the electron
fraction drops rapidly in that region from initially about $Y_{e}\simeq 0.3$
to $Y_{e}\simeq 0.15$ after the launch of the $\nu_{e}$ burst, which is often
referred to as deleptonization or neutronization burst. These energy losses
cause the dynamic shock wave to turn into an accretion shock front, which
stalls at around $\mathcal{O}(100)$ km about $\sim 50$ ms post bounce. The
later evolution is determined by mass accretion from the still infalling outer
layers of the progenitor stars, as well as neutrino heating and cooling of the
material located in the post shock layer, all of which would occur in the
presence of convection and the possible existence of (magneto) hydrodynamic
instabilities [60].

Figure 2.3: Top panel: Radial profile of rest mass density $\rho$ (solid line)
and temperature $T$ (dash-dotted line), showing in addition the neutrino
energy dependent neutrinospheres at the top for $E_{\nu_{e}}=0.5-300$ MeV
(color coded). Bottom panel: mass fractions $X_{i}$ of the most abundant
nuclear species (blue and red lines) and the electron abundance $Y_{e}$ (black
solid line). (Figure reprinted from Ref. [61])

The complex phenomenon of neutrino decoupling during this phase is illustrated
in Fig. 2.3, where the radial profile of mass density and temperature taken
from a spherically symmetric simulation based on Boltzmann neutrino transport
at about 300 ms post bounce are shown. Due to the energy dependence of the
various neutrino reaction cross sections, neutrinos of different energies
decouple at different radii, shown by the vertical lines in Fig. 2.3 for
$\nu_{e}$ as an example, defined as where the corresponding optical depths
equal to $2/3$ [62].

Low energy $\nu_{e}$, of about 0.5–1.5 MeV, decouple deep inside the PNS, at
high densities and temperatures on the order of few times $10^{13}$ g cm-3 and
$T\simeq 8-12$ MeV, while high energy $\nu_{e}$, of about 200–300 MeV,
decouple only behind the accretion shock (i.e. the sharp density gradient
located at about 100 km in Fig. 2.3), at low densities and temperatures, on
the order of $\rho\simeq 10^{8}-10^{9}$ g cm-3 and $T\simeq 1-3$ MeV. Outside
the shock wave, neutrinos are freely streaming and at the PNS interiors
neutrinos are trapped.

During the post bounce evolution, neutrinos of all flavors are produced by
various weak reactions (summarized in Table 3.1 below). However, since the
post-shock layers have temperatures above 1 MeV, the abundance of heavy nuclei
and hence processes involving them are strongly suppressed. Furthermore, the
dissociated nuclei give rise to free baryons through accretion from the
infalling outer layers of the progenitor, which accumulate at the PNS surface,
forming a thick layer shown in Fig. 2.3 behind the shock around the region of
neutrino decoupling. There, the electronic charged current processes (1)–(3)
in Table 3.1 dominate the production of $\nu_{e}$ and $\bar{\nu}_{e}$, while
both charged current and the neutrino-nucleon scattering contribute to their
decoupling [63]. Their luminosities can be approximated as [59],

$L_{\nu_{e}}\sim 1.5\times 10^{52}~{}\left(\frac{M}{1.5~{}{\rm
M}_{\odot}}\right)\left(\frac{80~{}{\rm
km}}{R}\right)\left(\frac{\dot{M}}{0.03~{}{\rm M}_{\odot}/{\rm
s}}\right)~{}{\rm erg}~{}{\rm s}^{-1},$ (2.2)

for typical values of the PNS mass $M$ and radius $R$ as well as for a mass
accretion rate $\dot{M}$. In contrast, heavy lepton flavor neutrinos are
dominantly produced by pair processes, they decouple at generally higher
temperatures and densities. Hence, their luminosity is determined by
diffusion.

#### 2.1.4 Neutrino-driven supernova explosions

During the phase of shock stalling, the energy-dependent decoupling of
neutrinos described above results in the development of regions between the
shock and the PNS surface where net heating (with the specific heating rate
$\dot{Q}>0$) and net cooling ($\dot{Q}<0$) from various reactions are
obtained. Fig. 2.4 shows a snapshot of $\dot{Q}$ as a function of the rest
mass density ($\rho$) from the charged current interactions, neutrino
electron/positron scattering ($\nu e^{\pm}$) and pair processes, for
$\nu_{e}$, $\bar{\nu}_{e}$ and $\nu_{\mu/\tau}$ as representative for the
heavy lepton flavors, taken from the 1D CCSN simulations of Ref. [57]. These
panels show that generally the $\nu_{e}$ and $\bar{\nu}_{e}$ charged current
reactions dominate the cooling and heating in this region. Since the electron
and positron capture rates on nucleons decrease much faster with declining
density (and therefore temperature and radius) than the $\nu_{e}$ and
$\bar{\nu}_{e}$ capture rates (see e.g., Ref. [60]), net cooling (heating)
develops at higher (lower) densities. As a result, there exists a gain radius
(in 1D simulations) above (below) which the net $\dot{Q}>0$ ($\dot{Q}<0$) from
all reactions is achieved for neutrinos to deposit (take) energy into (from)
the medium.

Figure 2.4: Neutrino heating ($\dot{Q}>0$) and cooling rates ($\dot{Q}<0$) for
all flavors, separated into charged-current reactions, denoted as “cc” (red
solid lines), neutrino-electron scattering $\nu e^{\pm}$ (blue dash-dotted
lines) and pair processes (gray dashed lines), corresponding to a typical post
bounce situation at about 250 ms. Thin vertical lines mark the locations of
the shock (solid lines) and the corresponding neutrino spheres (dash-dotted
lines). (CCSN simulation data from Ref. [57])

The heating of material in the post shock region by neutrinos results in the
continuous rise of the entropy per baryon. This energy deposition is the
source of the shock revival, turning the accretion front back into a dynamic
shock wave which expands to increasingly larger radii. This concept is the
physical foundation for the neutrino-driven SN explosion scenario, first
reported in Refs. [64, 65]. If successful, this determines the onset of the
CCSN explosion, leaving behind the nascent hot and lepton rich PNS as remnant.

In spherically symmetric CCSN simulations, self-consistent neutrino-driven
explosions could be obtained only for the low-mass progenitors with the ZAMS
masses of about 8–10 $M_{\odot}$ [66]. With these low-mass progenitors, the
shock revival due to neutrino heating occurs shortly after shock stalling, on
the order of less than 100 ms after bounce, due to a sharp density gradient
separating the stellar core from the low-density envelope. The resulting SN
explosions generally have low explosion energies in 1D or in multi-dimensional
simulations, about $10^{50}$ erg with ejected amount of nickel possibly
ranging between $\mathcal{O}(10^{-3}-10^{-1})$ $M_{\odot}$ [67, 68, 69, 70].
However, quantitative and important differences between multi-dimensional
simulations and spherically symmetric cases exist, particularly concerning the
explosive nucleosynthesis. In multi-dimensional simulations, it may allow for
the existence of neutron-rich convective lumps which are only mildly exposed
to neutrinos that are absent in the corresponding 1D simulations [71]. The
explosive nucleosynthesis in these inner layers have recently been studied
based on two or three dimensional CCSN simulations [68, 72], with the
production of elements up to zirconium including possibly some $p$-nuclei such
as 92Mo. However, the final answer about the contribution of innermost CCSN
ejecta to the nucleosynthesis will require 3-dimensional simulations with
accurate neutrino transport, given the significantly different dynamics
observed in 2D and 3D models [73].

Neutrino-driven explosions of more massive iron-core progenitors require
multi-dimensional simulations, where the neutrino heating efficiency increases
in the presence of convection and the development of hydrodynamic
instabilities, such as the standing accretion shock instability (SASI) [74],
exposing material in the heating region to neutrino heating more efficiently.
The success of multidimensional SN simulations has been demonstrated in the
past decades for neutrino-driven explosion models. These neutrino radiation
hydrodynamics models typically implement different approximated but
sophisticated Boltzmann neutrino transport schemes to capture the important
role of neutrino heating and cooling. In the following, we briefly describe
(in alphabetical order) the assumptions employed in several state-of-the-art
multi-dimensional SN simulation codes developed by different research groups.

  * 1.

The 3DnSNe three-dimensional, magnetohydrodynamics simulation code of Ref.
[75] is based on the spherical coordinates. It employs the Isotropic Diffusion
Source Approximation (IDSA) of Ref. [76], which treats separately trapped and
freely streaming neutrinos at the level of spectral distributions, carefully
matched via a source term. The module of 3DnSNe implements a complete set of
neutrino-matter interaction rates [77]. The intermediate computational costs
of the IDSA enable detailed qualitative comparison of the spatial
dimensionality dependence of the SN simulations [78].

  * 2.

The 3DGRMHD fully general relativistic three-dimensional neutrino-radiation
magneto-hydrodynamics model in Cartesian coordinates [79]. It is based on a
spectral two-moment M1 approach, where both the neutrino radiation energy
density (zeroth angular moment) and the momentum density (first angular
moment) are evolved, with M1 tensor closure for the second and third moments
of the neutrino radiation fields [81]. 3DGRMHD enabled the systematic
quantification of the impact of magnetic fields, in combination with stellar
rotation, in neutrino-driven SN explosions [80, 82].

  * 3.

The neutrino transport of the CHIMERA Oak Ridge code extends the spherically
symmetric multi-group flux limited diffusion scheme of Ref. [83] in multiple
spatial dimensions (see Ref. [84] and references therein, where also the
details about the flux limiter are provided). CHIMERA has a flexible
hydrodynamics module that solves the Newtonian equations, w/o approximate
general relativistic monopole component, in Cartesian, polar or spherical
coordinates, where as the transport module implements the ray-by-ray+
treatment and hence requires spherical coordinates. A representative set of
CHIMERA axially symmetric and three-dimensional simulations can be found in
Refs. [85, 86].

  * 4.

The neutrino transport of the FLASH-M1 open-source code implements the
$\nu$GR1D energy-dependent M1 scheme [87], via an analytic closure using the
Eddington factor, that interpolates between the optically thick and optically
thin regimes.

  * 5.

The FORNAX Princeton simulation code solves multi-group two-moment, velocity-
dependent transport equations keeping terms up to order $\mathcal{O}(v/c)$ in
co-moving frame in spherical coordinates, with analytic M1 tensor closure for
the second and third radiation moments [88]. The neutrino transport has been
compared quantitatively with the ray-by-ray+ treatment in core-collapse SN
simulations in axial symmetry [89]. FORNAX yields a comprehensive catalogue of
self-consistent axially symmetric and three-dimensional core-collapse SN
explosions (see Refs. [90, 91, 92] and references therein).

  * 6.

The VERTEX Garching simulation code implements the ray-by-ray+ approach [93],
featuring a two-moment neutrino transport scheme in spatial angular bins with
minimal coupling. The system of equations is closed by a variable Eddington
factor obtained by solving model Boltzmann equations iteratively given
hydrodynamic profiles along each radial ray and being coupled locally to the
spherical solution of that ray. Besides this advantage over other multi-group
transport schemes, implementing an approximate angular dependence, the ray-by-
ray+ approach neglects transverse fluxes and lateral transport, however, non-
radial advection of neutrinos as well as the non-radial neutrino pressure is
coupled to the radiation hydrodynamics equations. Axially symmetric and three-
dimensional VERTEX simulation results of SN explosions for a variety of non-
rotating massive stars can be found in Ref. [94, 95], whereas the role of
rotation has been studied in Ref. [96]. The VERTEX ray-by-ray+ results are
compared with full multi-dimensional, multi-group neutrino transport, based on
an analytical closure scheme of the moments equations, finding quantitative
good agreement in 3D simulations besides natural stochastic variations known
to occur due to multi-dimensional phenomena in turbulent environments [97].

We note that in the above models, CHIMERA, FLASH-M1, FORNAX and VERTEX
implement general relativistic contributions into the transport schemes in an
approximate fashion through an effective GR potential [98]. Also noted is that
a qualitative comparison of the aforementioned transport schemes in 1D SN
simulations, including in addition results of AGILE-BOLTZTRAN, which is based
on general relativistic neutrino radiation hydrodynamics in 1D and implements
Boltzmann neutrino transport [99, 100, 101, 102], has been presented in Ref.
[103]. Moreover, implementation of full Boltzmann neutrino transport in axial
symmetry has been developed by Nakamura et al. [104, 105]. Although at present
it is applied only to hydrostatic background and under special relativity due
to the high computational cost [106], it enables the quantitative comparison
with the aforementioned approximate multi-dimensional neutrino transport
schemes [107].

A general caveat that all present successful multi-dimensional neutrino-driven
CCSN explosion models face is the obtained low explosion energies, more
precisely the diagnostic explosion energy (see e.g., definition in Sec. 2.4 of
Ref. [108]) on the order of a few times $10^{50}$ erg. This is somewhat in
tension with the observed SN explosions for which canonically explosion
energies on the order of a few times $10^{51}$ erg are deduced [109]. This
deficit might be related to the insufficient simulation duration that current
SN simulations can consider due to computational limitations; c.f. Refs. [73,
72], or alternatively the lack of some incompletely understood input physics
in the models; see, e.g., Sec. 4. A possible boost of the explosion energy, in
agreement with observations, could also be obtained when considering
alternative explosion mechanism, which will be discussed below.

Weak processes involving muons have to be considered once the conditions of
temperature and density inside the PNS are such that a substantial amount of
$\nu_{\mu}$ have energies in excess of the rest mass of muons,
$m_{\mu}=105.66$ MeV, and/or the chemical potential of the electrons exceeds
this value. The involved processes not only include the muonic charged current
processes and (anti)neutrino-muon scattering but also the set of purely
leptonic reactions, such as the muon decay and its inverse as well as lepton
flavor exchange and conversion processes (see Table 1 in Ref. [57]).
Simulations including this set of muonic weak reactions into the collision
integral of the neutrino transport equation show that muons are produced from
the absorption of high energy muon-(anti)neutrinos on (protons)neutrons,
shortly before bounce. During the post-bounce evolution, the presence of
(anti)muons replaces (positrons)electrons, which has a softening effect but
leaves a small impact on the evolution of the PNS, featuring slightly higher
central rest mass density and a smaller radius. However, it can result in an
enhancement of the neutrino heating efficiency in the gain region in multi-
dimensional simulations, enabling neutrino-driven shock revival in some cases
[110].

If the shock revival fails, then mass accretion onto the PNS will continue
until the maximum mass, given by the nuclear equation of state (EoS), is
reached and the PNS collapses into a BH, swallowing eventually the entire
progenitor star. The failed SN explosion scenario has long been studied in the
context of neutrino emission, featuring a sudden termination of neutrino
emission at the moment of BH formation [111, 59, 34]. However, the actual
moment of neutrino termination is defined when the neutrinospheres fell
through the event horizon (more precisely the apparent horizon), which has
recently been demonstrated in Ref. [112].

There are additional proposals for alternative SN mechanisms. If a rotating
collapsing star boosts an extremely strong magnetic field, the magnetic
pressure along the rotational axis supports the shock expansion leading to a
jet-like explosion in what is called a magneto-rotational SN [113, 114, 115,
116, 117, 118]. Another alternative scenario invokes a first-order phase
transition of hadronic matter to a quark-gluon plasma at high densities [119].
Related to this transition, a second shock wave is produced moving with ultra-
relativistic velocities triggering the explosion [120, 121, 122, 123, 124,
125]. For heavy progenitor stars with rotations, which lead to BH formation on
collapse, jets may be powered by accretion disk and result in energetic
explosions named collapsars as a plausible mechanism for Type Ic SNe
associated with long gamma-ray bursts [126, 127, 128, 129]. Other interesting
possibilities such as the jittering jets explosion mechanism [130, 131] or the
collapse induced thermonuclear explosions [132] have also been proposed.
Notably, these alternative scenarios may result in different nucleosynthesis
yields from those obtained in neutrino-driven SNe and are potential sites of
$r$-process nucleosynthesis [118, 133, 134, 135, 136, 137]. We note here that
the nucleosynthesis yields in magneto-rotational SNe are very sensitive to the
magnitude of the magnetic-field enhancement during collapse due to the
magneto-rotational instability. High magnetic fields favor a fast ejection of
material that is barely exposed to neutrinos and allow for the production of
heavy $r$-process elements [138].

#### 2.1.5 Proto-neutron star deleptonization and the neutrino-driven wind

In the case of a successful shock revival and the subsequent continuous shock
expansion to increasingly larger radii, the mass accretion onto the central
remnant PNS eventually ceases. Neutrino emissions from the nascent PNS lead to
the subsequent PNS cooling and deleptonization for $\mathcal{O}(10)$ s, which
is consistent with the detection of $\bar{\nu}_{e}$ from SN1987A [139].

An important aspect in this phase is the energy deposition of the decoupling
neutrinos in the surface layer of the PNS, which drives a continuous low-mass
outflow, known as the neutrino-driven wind [140, 141, 142, 143, 144]. Roughly
speaking, it requires about 7–10 (anti)neutrino absorption processes for
(anti)neutrinos with average energies between 10–15 MeV, to overcome the
gravitational binding energy of a baryon of about 104 MeV located at the
surface of a PNS with typical mass of 1.5 $M_{\odot}$. The neutrino-driven
wind can also expand with supersonic velocities and collide with the slower
moving material ejected by the SN explosion. If this happens, a reverse shock
known as the neutrino-driven wind termination shock forms and can affect the
thermodynamical properties in the wind [145, 146, 147]. Fig. 2.5 shows the
time evolution of several selected mass tracers of the neutrino-driven
wind222These are typically selected in a post-processing fashion in
spherically symmetric models, while they have to be predefined in multi-
dimensional simulations. from an 1D model [108] to illustrate the general wind
properties, including the dynamical timescale, shown here via the velocities
in panel (f), the entropy of the expanding wind in panel (c), the $Y_{e}$ in
panel (d), together with the radial evolution in panel (a) as well as the
corresponding density and temperature in panel (b) and (e). The solid and
dashed lines in panel (a) mark the locations of the bounce shock, which
expands continuously to increasingly larger radii, and the reverse shock,
respectively.

Figure 2.5: (Figure reprinted from Ref. [108]) Evolution of dynamic and
thermodynamic properties for a selection of mass tracers during the neutrino-
driven wind phase of a parametrized neutrino-driven CCSN explosion launched
from 11.2 $M_{\odot}$ progenitor. Credit: Fischer et. al., A&A, 517, A80, 2010
[108], reproduced with permission $\copyright$ ESO.

The early studies of the neutrino-driven wind based on neutrino radiation
hydrodynamics [141, 142] simulations or by the steady-state wind models [143,
144] suggested that heavy-element nucleosynthesis via the $r$-process may be
obtained due to the moderate neutron richness of $Y_{e}\sim 0.38-0.5$ and high
entropy per nucleon $\sim 150-400$ $k_{B}$ in the wind. However, more recent
long term 1D simulations carried out to more than 10 seconds post bounce with
accurate Boltzmann neutrino transport [148, 108] as well as multi-dimensional
simulations lasting for several seconds with approximated neutrino transport
scheme [149, 95, 150, 72] found that the physical conditions in the wind do
not enable the $r$-process nucleosynthesis. Instead, the composition in the
wind may be proton-rich [$Y_{e}\gtrsim 0.5$; see e.g., Fig. 2.5(d)] and may
allow the $\nu p$ process [10, 11] to occur, which will be discussed further
in Sec. 3.2.

The modeling of neutrino decoupling also plays a key role in determining the
wind properties and the associated nucleosynthesis conditions. During the PNS
deleptonization, the PNS contracts due to the neutrino cooling, which leads to
a steeper density profile around the PNS surface (see e.g., Fig. 1 in Ref.
[62]). In comparison to the post-bounce mass accretion phase prior to the
shock revival, shown in the top panel of Fig 2.3, the energy-dependent
neutrinosphere locations shift to higher densities around $10^{12}-10^{13}$ g
cm-3. The subsequent evolution of the neutrino luminosities and average
energies are shown in Fig. 2.2, during the PNS deleptonization phase up to
about 10 s. It shows the continuously decreasing fluxes and average energies
for all flavors as neutrinos decouple at higher densities (lower temperatures)
at later times [62]. Comparing the earlier accretion phase to the later
deleptonization phase, an important change is that a scattering atmosphere,
which existed for heavy lepton flavor neutrinos at earlier phase [63, 151],
also develops for $\nu_{e}$ and $\bar{\nu}_{e}$. This is because the
increasingly dominant role of (nearly) elastic neutrino–nucleon scattering
over the inelastic charged current processes, as the latter is further
suppressed by final state blocking [62]. Since neutrino–nucleon scattering is
flavor blind, the neutrino spectra and fluxes of all flavors become
increasingly similar during the PNS deleptonization, as shown in Fig. 2.2.
Although this general trend is robust and insensitive to the detailed
treatment of neutrino interactions with matter around the neutrinosphere,
precise prediction of neutrino luminosities and average energies are needed to
accurately model the nucleosynthesis condition and yields in the neutrino-
driven wind, as will be the focuses of Secs. 3.1.1 and 3.2.

Another important point to note here is related to the multidimensional nature
of the SN explosion. Recent multidimensional simulations carried over to
several seconds post the core bounce found the mass accretion can persist for
seconds after the onset of explosion, especially for explosions that are more
aspherical with heavier progenitors, which might alter the appearance of a
spherically symmetric neutrino-driven wind [149, 95, 152, 150, 72] assumed in
1D models. Consequently, the clean separation of the explosion and PNS cooling
phase obtained in 1D models, which is reflected in the rapid drop of the
neutrino fluxes shown in Fig. 2.2 becomes no longer possible in those cases,
and the relation between the thermodynamical condition of the wind and the
remnant property can be affected [72]. On the other hand, for light
progenitors in the ZAMS mass range of 8–10 M⊙ and depending on metallicity,
where explosions are not too aspherical, spherical neutrino-driven wind
solutions remain a relatively good approximation to results obtained in
multidimensional simulations [153, 154, 155, 92].

The asymmetric nature of the SN explosions may further leave imprints on the
morphology and chemical composition distribution in SN remnants. Attempts to
model the very late-time evolution of SN explosions until the shock break-out
or even the remnant evolution in multi-dimensions have been performed with
various assumptions [156, 157, 158, 159, 160, 161, 162, 163, 164, 154, 150].
These studies revealed potentially intriguing dependence of observables of
lightcurves and nuclear $\gamma$ lines on the progenitor model and structure,
as well on the interaction of SN shock with the circumstellar medium, relevant
for the observation of Cassiopeia A and SN1987A.

#### 2.1.6 Presupernova and supernova neutrino detection

As mentioned above, massive stars are profuse sources of neutrinos even before
they become SNe. During later stages of their evolution, their central
temperature and density increase dramatically, and $\nu\bar{\nu}$ pair
production by photo-neutrino emission, plasmon decay, and $e^{\pm}$ pair
annihilation become the dominant mechanism of energy loss (see e.g., [165,
166]). Likewise, $\nu_{e}$ and $\bar{\nu}_{e}$ production by weak nuclear
processes become more and more significant as such stars evolve. These
neutrinos not only provide cooling that drives the evolution of massive stars,
but also serve as potential signatures of their evolution and advance warning
for the ensuing SNe. With the capability of detecting low-energy
$\bar{\nu}_{e}$ of several MeV by the gadolinium-loaded Super-Kamiokande (SK)
detector [167] and by the next generation of detectors such as the Jiangmen
Underground Neutrino Observatory (JUNO) [168], there is growing interest in
detecting these presupernova neutrinos [169]. It is plausible to detect
presupernova $\bar{\nu}_{e}$ emitted during the late oxygen burning or the
silicon burning stages a few days before the explosion of a massive star
within a few kpc [170, 171, 172, 173, 174, 175, 176, 177]. At a relatively
close distance of $222^{+48}_{-34}$ pc [178], Betelgeuse with a mass of
$20_{-3}^{+5}\,M_{\odot}$ [179] would be a candidate star. However, its
current evolutionary status is uncertain. It is most likely undergoing core
Helium burning according to Refs. [180, 181], but it may already be in the
late stage of core carbon burning as suggested by a recent study [182]. In the
latter case, Betelgeuse would be a promising source of presupernova neutrinos
for detection by JUNO.

The main presupernova neutrino signals in JUNO result from $\bar{\nu}_{e}+p\to
n+e^{+}$ (inverse $\beta$-decay, IBD) and $\nu+e^{-}\to\nu+e^{-}$ (electron
scattering, ES). Considering neutrinos produced by $e^{\pm}$ pair annihilation
from a star at 222 pc during the last day prior to its core collapse and
assuming the normal ordering (NO) of neutrino masses, Ref. [183] estimated
about 123.8, 243.5, 416.0 and 497.1 IBD events with neutrino energy of 1.8–4
MeV for progenitor models of 12, 15, 20, and 25 $M_{\odot}$, respectively. The
corresponding numbers of ES events with recoil electron kinetic energy of
0.8–2.5 MeV are 117.2, 212.9, 380.9, and 479.8, respectively. The ratio of the
expected number of IBD events for the inverted ordering (IO) to that for the
NO is $N_{\rm IBD}^{\rm IO}/N_{\rm IBD}^{\rm NO}\approx 0.29$, and the
corresponding ratio for the ES events is $N_{\rm ES}^{\rm IO}/N_{\rm ES}^{\rm
NO}\approx 1.23$ [183]. The high sensitivity of the IBD events to the yet-
unknown neutrino mass ordering is due to the Mikheyev-Smirnov-Wolfenstein
(MSW) effect [184, 185], which governs the flavor transformation during
neutrino propagation through the stellar interior. Note that while for each
progenitor model, the number of events of each kind has substantial
uncertainties (within a factor of $\sim 2$) due to uncertainties in treatment
of stellar evolution, the ratios $N_{\rm IBD}^{\rm NO}/N_{\rm ES}^{\rm NO}$,
$N_{\rm IBD}^{\rm IO}/N_{\rm ES}^{\rm IO}$, $N_{\rm IBD}^{\rm IO}/N_{\rm
IBD}^{\rm NO}$, and $N_{\rm ES}^{\rm IO}/N_{\rm ES}^{\rm NO}$ are nearly
independent of the stellar models. Because $N_{\rm IBD}^{\rm NO}/N_{\rm
ES}^{\rm NO}\approx 4.2N_{\rm IBD}^{\rm IO}/N_{\rm ES}^{\rm IO}$, it is
possible to distinguish the neutrino mass ordering not only without knowing
the exact mass of the SN progenitor but also allowing for significant
uncertainties in the predicted numbers of IBD and ES events. An important
issue to consider is the backgrounds. For IBD events, the largest backgrounds
come from reactor $\bar{\nu}_{e}$ and geo-$\bar{\nu}_{e}$, which contribute
about 15.7 and 1.1 events, respectively, within the same time and energy
windows as for the signals. These backgrounds will be well measured and are
below the expected number of IBD events for a star like Betelgeuse
(17–$25\,M_{\odot}$) even considering the uncertainties in stellar models.
Unlike the IBD events, which can be identified by coincident detection of the
positron and the neutron produced in the reaction, ES events only cause single
hits in JUNO and suffer from much higher backgrounds of different origins. The
dominant background, due to $\beta^{+}$ decay of the cosmogenic 11C produced
by $(\gamma,n)$ spallation following a muon shower, is at the level of a few
$10^{4}$ events per day [168]. It is feasible to determine the neutrino mass
ordering from the nearly model-independent ratio of the IBD and ES event
numbers as discussed above when this background is suppressed by a factor of
$\sim 2.5$–10 depending on the mass of the stellar source [183]. Such
background reduction might be achieved by three-fold coincident detection of
the muon initiating the shower, the neutron from the spallation production of
11C, and the positron from the 11C decay [186].

Compared to presupernova neutrinos, neutrinos from the ensuing SN have much
higher average energies and orders of magnitude higher luminosities. To date
only neutrinos from a single event, SN 1987A at a distance of $\approx 50$
kpc, have been detected [187, 188, 189, 190]. The approximately twenty
neutrino events observed have been extensively studied to understand SN
neutrino emission and neutrino properties (see e.g., [191] for a review of
earlier works and e.g., [192] and [193] for detailed analysis methodologies).
While parametric models of neutrino emission were commonly used to extract
several parameters from the relatively sparse SN 1987A data for comparison
with the results from numerical SN models, it is more straightforward to
compare the SN models directly with the data. Specifically, Bayesian
techniques can be used to rank the SN models in the latter approach. Such
analyses were carried out in Refs. [194, 195] using three baseline models from
the Garching group [196] with progenitors of 9.6, 20, and $27\,M_{\odot}$.
These models are representative of current 1D SN simulations and cover a range
of neutrino emission. Variation of the overall flux normalization and simple
scenarios of neutrino oscillations were also explored. With the data on the 11
events observed in a time span of $\approx 12$ s in the Kamiokande II
detector, it was found that models with a brief accretion phase of neutrino
emission are the most favored [194, 195]. This result is not affected by
varying the overall flux normalization or considering neutrino oscillations.
In addition, $p$-value tests of the predicted event distributions in energy
and arrival time showed no evidence that any of the best-fit models are
incompatible with the data [194, 195]. Another study [197] focused on the
first $\sim 0.5$–1.5 s of neutrino emission and examined a number of 1D, 2D,
and 3D simulations. Given that the exact model for SN 1987A is not known and
that even the state-of-the-art simulations need further improvements, perhaps
it is not surprising that those simulations examined by Ref. [197] generally
disagree with the data. Yet another study [198] showed that the relatively
short cooling timescales of $\sim 5$–9 s resulting from current treatment of
convection in the PNS are not easily compatible with the duration of the SN
1987A neutrino emission except for rare fluctuations in the sparse data. Based
on the above discussion, while some discrimination among the neutrino emission
models could be provided by the SN 1987A data, no definitive conclusions can
be drawn.

More useful will be a future neutrino signal from a Galactic SN, which is
expected to produce a greater number of neutrino events in current and planned
detectors (e.g., [199]). Focusing on an early phase of neutrino emission
spanning 500 ms, which is closely related to the explosion mechanism, the
authors of Ref. [200] performed a detailed study of the feasibility of
distinguishing between five SN simulations (1D [201, 202, 203], 2D [204], 3D
[205]) using hypothetical neutrino data in the planned Hyper-Kamiokande (HK)
detector (e.g., [206]). Taking into account flavor oscillations due to the MSW
effect, they performed detailed reconstruction of 100 and 300 simulated
events, which were dominated by IBD contributions (ES and charged-current
reactions of $\nu_{e}$ and $\bar{\nu}_{e}$ on 16O were also included). They
concluded that all the five simulations could be distinguished with 300
events, which are expected from a SN at a distance of $\sim 60$–100 kpc based
on these simulations. Further analyses similar to the above were carried out
in Ref. [207] to study the feasibility of distinguishing between four
simulations (1D [202]) that differ specifically in the mass or initial
metallicity of the progenitor. A complementary study was carried out in Ref.
[208] using seven 1D SN simulations from the Garching group [196], none of
which were considered in Refs. [200, 207]. Each of these simulations covers
neutrino emission for at least 9 s and therefore, includes both the accretion
and PNS cooling phases. These simulations correspond to progenitor masses of
9.6–$27\,M_{\odot}$, and predict $\sim 2500$–7150 ($\sim 3550$–9940) IBD
events in a detector similar to SK (an idealized SK with perfect detection
efficiency, perfect energy resolution, and no background) for a SN at 10 kpc.
It was found that for each of the three neutrino oscillation scenarios
considered (no oscillations or standard MSW effect with the NO or IO), all the
neutrino emission models can be distinguished from each other with the IBD
signal in a detector similar to SK from a SN at a known distance up to $\sim
25$ kpc or at an unknown distance up to $\sim 10$ kpc. In addition, provided
that the emission model is known, for example, from observations of the SN
progenitor, the three oscillation scenarios can be distinguished from each
other with an idealized SK and a SN at a known distance of 10 kpc. Based on
the analyses of Ref. [208], it should be quite possible to distinguish between
1D SN neutrino emission models using the IBD signal in HK from a future
Galactic SN, at least when the same neutrino oscillation scenario is assumed
for all the models.

### 2.2 Binary neutron star mergers

Figure 2.6: Schematic plot showing the major mass ejection phases of BNSMs.
Left: Dynamical mass ejection during the merging of two NSs from tidal
disruption and from the contact interface. Right: Secular mass ejection from
the accretion disk during the remnant phase, where the central object can be a
HMNS (upper panel) or BH (lower panel). Red arrows indicate the possible
evolution paths while black arrows indicate the (angular) velocity of the
object(s) or ejecta. Neutrinos play important roles in both phases in
affecting the $Y_{e}$ of the ejecta (see text for details).

The mergers of two NSs or one NS with one BH were proposed to be promising
sites of heavy element production through the $r$-process nucleosynthesis
several decades ago [209, 210, 211]. With the advances in numerical
simulations, it became increasingly clear that they may even be the most
dominating sites of $r$-process [212, 213, 214, 215, 216, 217] and can produce
unique electromagnetic emissions powered by the energy generation from the
decay of unstable $r$-process nuclei, called kilonovae [218, 219, 220]. This
association was further confirmed by the detection of kilonovae associated
with short gamma-ray bursts [221, 222, 223] as well as the seminal discovery
of GW170817 and the associated kilonova AT2017gfo directly linked to the BNSM
[224, 225]. Meanwhile, studies also revealed that neutrinos can play
substantial roles in the evolution of the BNSM remnant and in determining the
nucleosynthesis yields in the associated outflows. In this section, we briefly
outline the general picture of BNSM evolution and focus on recent findings
related to neutrinos therein. Readers interested in detailed reviews on BNSMs,
their GWs and kilonovae can refer to Refs. [6, 226, 227, 228, 229].

#### 2.2.1 Merger and dynamical ejecta

NS binaries (NSBs) can originate from binary star systems that remain
gravitationally bound after undergoing two SN explosions or from dynamic
captures in dense stellar regions. For an isolated NSB, its orbital separation
shrinks over time due to the loss of angular momentum and energy by the
emission of GWs. The typical inspiral timescale from an initial separation of
$\sim\mathcal{O}(1)$ A.U. to their eventual merge is $\sim\mathcal{O}(100)$
Myr, depending on the system’s initial eccentricity. For most systems, the
orbits become quasi-circular before mergers occur.

Figure 2.7: Left panel (adapted from Ref. [230]): The energy luminosity of
different neutrino species during the early post-merger time of BNSMs from
simulations done in [231] with two NSs of $1.35$ $M_{\odot}$ each and two
nuclear EoSs – DD2 [53, 54, 232] and SFHo [54, 233]. Right panel (courtesy of
O. Just): The energy luminosity of $\nu_{e}$ (solid), $\bar{\nu}_{e}$
(dashed), and $\nu_{x}$ (dotted) during BNSM remnant evolution phase from the
model sym-n1-a6 of Ref. [234].

Shortly before the merger, the strong gravitational forces between the two NSs
lead to the tidal disruption of materials at the NS surfaces. A part of these
tidally disrupted material can become unbound and get ejected from the system
while some part of them fall back to form an accretion disk (see Fig. 2.6).
The ejected mass is typically $\sim\mathcal{O}(10^{-4}-10^{-3})$ $M_{\odot}$
and distributed close to the merger plane. At merger, the violent collision of
the two NSs (with relative velocity of $\sim 0.3$ c) leads to shock formation
and can also efficiently eject materials from the contact interface of similar
or even larger amount (see also Fig. 2.6). Moreover, the temperature of the
remnant rises significantly due to the collision, up to $\sim\mathcal{O}(50)$
MeV. Consequently, the heated matter seeks to reach a new thermodynamical
equilibrium, during which the weak interactions (electron and positron
captures) work to raise $Y_{e}$ and produce $\nu_{e}$ and $\bar{\nu}_{e}$
while $\nu_{x}$ (denoting $\nu_{\mu}$, $\nu_{\tau}$ and their antineutrinos
collectively) are also generated via pair processes. The resulting total
neutrino luminosity can reach $L_{\nu,\rm{tot}}\sim\mathcal{O}(10^{53})$ erg
$s^{-1}$, with $L_{\bar{\nu}_{e}}\gtrsim L_{\nu_{e}}\sim L_{\nu_{x}}$ (see
e.g., the left panel of Fig. 2.7 and Ref. [235] for a compilation of neutrino
emission property from BNSMs). The dominant emission of $\bar{\nu}_{e}$ over
other species is expected since the remnant must protonize for some period to
reach the newly established equilibrium. This fact also has important
consequence for the flavor conversion of neutrinos, which will be addressed in
Sec. 4. Consequently, charged-current neutrino absorption on nucleons further
raise the $Y_{e}$ of the dynamical ejecta, particularly for the component
closer to the polar direction (perpendicular to the merger plane) [236, 237,
238].

The resulting $Y_{e}$ distribution obtained in recent simulations that
included nucleonic weak interactions in dynamical ejecta typically ranges in
between $0.05\lesssim Y_{e}\lesssim 0.5$ with an average value $\langle
Y_{e}\rangle\sim 0.2-0.3$ [236, 237, 238]. This is in sharp contrast to the
low $Y_{e}\lesssim 0.1$ obtained in earlier work without including weak
interactions [213, 214]. As a result of the broad $Y_{e}$ distribution, the
$r$-process in the dynamical ejecta can produce a wide range of nuclei from
the first peak up to the third peak and actinides [217, 236, 237, 238]. The
exact $Y_{e}$ distribution and the nucleosynthesis yield for a given set of
binary parameter still depend on a number of factors that need to be further
clarified, including the treatment of neutrino transport [239], the EoS (that
affects the post-merger temperature and the amount of ejecta) [240], as well
as the flavor oscillations of neutrinos (see Sec. 4.2.4). Moreover, potential
impact due to the existence of muon [241] and/or pions [242, 243] in BNSMs
remain to be clarified.

Similar to studies that included the possible occurrence of hadron-quark phase
transition in CCSNe (see the end of Sec. 2.1.4), recent works also found that
a phase transition happening early in the post-merger phase can leave
interesting imprints on the emitted GW signals [244, 245, 246]. These works
seem to suggest that it does not significantly affect the amount of dynamical
ejecta and the associated $r$-process nucleosynthesis [247, 246].

For BH-NS mergers, it requires a relatively low mass BH with $M_{\rm
BH}\lesssim 10$ $M_{\odot}$ with relative high BH spin with dimensionless spin
parameter $\chi\gtrsim 0.5$ to tidally disrupt the NS before it is being
swallowed by the BH [248, 249]. When these conditions are satisfied, mass
ejection and $r$-process nucleosynthesis in both the dynamical ejecta as well
as the post-merger secular winds can occur. Although the fraction of BH-NS
mergers that contribute to the $r$-process production remains largely
uncertain, it will be better constrained by the ongoing and future GW
telescopes. For the dynamical ejecta of BH-NS mergers, neutrinos do not play a
significant role for clear reason that there is no central neutrino emitting
source. However, at the post merger phase, neutrino emission from the BH-NS
post-merger remnants could have similar impact as in that of BNSM remnant case
that will be discussed below.

#### 2.2.2 Secular evolution of remnant and its outflow

After the early phase of dynamical mass ejection, the remnant of a BNSM
settles into a nearly axisymmetric system consisting of a hypermassive NS
(HMNS), or a BH in the case of prompt collapse, surrounded by an accretion
disk/torus whose mass is $\sim 10^{-3}-10^{-1}$ $M_{\odot}$. The lifetime of
the HMNS can range from $\mathcal{O}(10-100)$ ms, depending on the system’s
initial mass before collapsing to a BH or further settles into a supermassive
NS (SMNS) after the differential rotation has been removed [250, 234]. Both
the HMNS and the accretion disk are also strong neutrino emitters for
$\mathcal{O}(100)$ ms; see e.g., the right panel of Fig. 2.7. Earlier studies
of the long term evolution of BH–torus systems revealed that the viscous
heating inside the torus can inflate the disk when neutrino emission becomes
inefficient at the time scale of $\mathcal{O}(1)$ s. The viscous heating,
together with the further energy generation from the associated nuclear
recombination of 4He, is capable of unbinding $\sim 20-50\%$ of the disk mass
within a few seconds. Thus, the amount of disk outflow can dominate that of
the dynamical ejecta and is the primary candidate that accounts for the
observed kilonova associated with GW170817 [227].

Although hydrodynamic and magneto-hydrodynamic simulations indicated that the
detailed disk outflow properties such as their velocity and the mass outflow
rate can sensitively depend on the initial magnetic field configuration inside
the disk [251], the predicted $Y_{e}$ distribution of the ejecta appears to be
robust. For cases where the central remnant consists of a BH, the resulting
$Y_{e}$ in disk outflow ranges between $0.1\lesssim Y_{e}\lesssim 0.5$, with
an average value $\langle Y_{e}\rangle\sim 0.2-0.3$, similar to the dynamical
ejecta. The density and temperature condition inside the torus (where
neutrinos are trapped initially) allow weak interactions to set the torus
$Y_{e}$ close to values given by reaction equilibrium [215, 252]. Neutrino
absorption on nucleons plays an important role for a small amount of the early
ejecta along the polar direction, which is exposed to larger amount of
neutrino flux [215, 216]. These neutrino-driven/aided ejecta mainly contribute
to the high $Y_{e}$ portion. For the majority of the outflow, neutrino
absorption does not affect much their $Y_{e}$ evolution.

Figure 2.8: The $Y_{e}$ distribution of disk outflow from the BNSM remnant,
which initially consists of a HMNS that later collapses to a BH taken from
Ref. [234] (courtesy of O. Just). Different colors represent post-merger
models adopting different viscosity parameters, which result in different
times for the BH formation.

However, the role of neutrino absorption on the remnant outflow may be
different for cases where a HMNS exists for a substantial amount of time after
the merger. This is mainly because before the HMNS collapses to a BH, its
neutrino emission can dominate that from the disk and can thus strongly affect
the outflow $Y_{e}$ [253]. Ref. [254] first investigated how the lifetime of
the HMNS and the associated neutrino emission impact the outflow and found
that the presence of the HMNS can lead to much higher averaged $Y_{e}$ values
than cases where BHs form promptly. Subsequent work further confirmed that the
secular ejecta from a remnant with delayed formation of the BH, i.e., having a
HMNS whose lifetime is $\sim\mathcal{O}(100-1000)$ ms indeed have
$Y_{e}\gtrsim 0.25$ (see Fig. 2.8) and produce much reduced amount of
lanthanides than cases with prompt formation of BH [250, 255, 234]. In
particular, the high $Y_{e}$ conditions in the outflow are obtained not only
during the phase before the formation of the BH, but also in the epoch after
the BH formation, since the thermodynamic conditions inside the disk are
altered during the phase where a HMNS still exists.

Besides the impact on disk cooling and the nucleosynthesis in the ejecta,
neutrinos can also play non-negligible roles in jet-launching in post-merger
environments responsible for short gamma-ray bursts. Although studies suggest
that the energy deposition from neutrino interaction and annihilation is
likely not enough to launch a relativistic jet [256], the inclusion of
neutrino processes can affect the lifetime of the HMNS as well as help clear
out the baryon loading in the vicinity of the BH, thereby affecting the time
of jet-launching powered mainly by the Blandford–Znajek mechanism [257].

## 3 Neutrino interactions and nucleosynthesis

As neutrinos play crucial roles in various phases of CCSNe and BNSMs, accurate
modeling of the production and decoupling of neutrinos are among the central
focuses of efforts made over the last couple of decades to improve the
accuracy in modeling the evolution of CCSNe and BNSMs as well as their
associated nucleosynthesis yields. These include the improved treatment of
neutrino-nucleon interaction taking into account various modifications in the
presence of the nuclear medium, which will be reviewed in Sec. 3.1. The impact
on the nucleosynthesis condition and yields in the neutrino-driven wind will
be discussed in Sec. 3.2. Beyond neutrino-nucleon interaction, calculations of
weak interaction rates involving nuclei are critical in determining the
precise evolution during the collapse phase of CCSN. Moreover, the profuse
amount of SN neutrinos also interact with existing nuclei in the stellar
mantle, which can reshape the abundances of certain key isotopes and make
important contribution to their abundances at present day through the neutrino
nucleosynthesis. We briefly review the advances made in evaluating the
neutrino-nucleus interaction in Sec. 3.3, and discuss in Sec. 3.4 recent
progresses in understanding the neutrino nucleosynthesis in CCSNe.

Table 3.1 lists the most relevant neutrino interactions with leptons,
nucleons, and nuclei that are discussed in this review in determining the
production and decoupling of neutrinos. Note that the neutrino-nucleus
interactions discussed in Secs. 3.3 and Sec. 3.4 are not included in this
Table.

Table 3.1:  Major neutrino reactions most relevant for modeling the CCSN
evolution. References that list the detailed expression of the interaction
rates are given.

|  | Weak process | References
---|---|---|---
1 | charged current | $l^{-}+p\rightleftarrows\nu_{l}+n$ | [83, 258, 259, 260, 57]
2 | charged current | $l^{+}+n\rightleftarrows\bar{\nu}_{l}+p$ | [83, 258, 259, 260, 57]
3 | charged current – neutron decay | $n\rightleftarrows p+\bar{\nu}_{e}+e^{-}$ | [260]
4 | nuclear electron captures | $e^{-}+(A,Z)\rightleftarrows\nu_{e}+(A,Z-1)$ | [48]
5 | nuclear de-excitations | $(A,Z)^{*}\rightleftarrows(A,Z)+\nu+\bar{\nu}$ | [51, 50]
6 | elastic neutrino–nucleon scattering | $\nu+N\rightleftarrows N+\nu^{\prime}$ | [83, 99]
7 | coherent neutrino–nucleus scattering | $\nu+(A,Z)\rightleftarrows\nu^{\prime}+(A,Z)$ | [83, 99]
8 | inelastic neutrino–lepton scattering | $\nu+l^{\pm}\rightleftarrows\nu^{\prime}+l^{\pm}$ | [83, 101, 259, 57]
9 | electron–positron annihilation | $e^{-}+e^{+}\rightleftarrows\nu+\bar{\nu}$ | [83]
10 | nucleon–nucleon bremsstrahlung | $N+N\rightleftarrows N+N+\nu+\bar{\nu}$ | [261, 262, 263]
11 | electron neutrino pair annihilation | $\nu_{e}+\bar{\nu}_{e}\rightleftarrows\nu_{\mu/\tau}+\bar{\nu}_{\mu/\tau}$ | [264]
12 | lepton flavor exchange | $\nu_{e}+\mu^{-}\rightleftarrows\nu_{\mu}+e^{-}$ | [259, 57]
13 | lepton flavor exchange | $\bar{\nu}_{e}+\mu^{+}\rightleftarrows\bar{\nu}_{\mu}+e^{+}$ | [259, 57]
14 | lepton flavor conversion | $\nu_{e}+e^{+}\rightleftarrows\nu_{\mu}+\mu^{+}$ | [259, 57]
15 | lepton flavor conversion | $\bar{\nu}_{e}+e^{-}\rightleftarrows\bar{\nu}_{\mu}+\mu^{-}$ | [259, 57]
16 | muon decay | $\mu^{-}\rightleftarrows e^{-}+\nu_{\mu}+\bar{\nu}_{e}$ | [259]

Note: $\nu=\\{\nu_{e},\bar{\nu}_{e},\nu_{\mu/\tau},\bar{\nu}_{\mu/\tau}\\}$,
$N=\\{n,p\\}$ and $l=\\{e,\mu\\}$ unless stated otherwise

### 3.1 Neutrino-nucleon reactions in SN matter

The neutrino-nucleon interaction is governed by the current-current
interaction from the standard model of particle physics. In vacuum, the
neutrino-nucleon reactions across all known energy scales have been studied,
considering the full kinematics, the complete weak interaction terms as well
as different degrees of freedom of constituents relevant at different energy
scales [265]. The situation is very different for neutrino interactions with
the hot and dense nuclear matter, where nuclear medium effects could play an
important role and alter the reaction rates significantly. However, for
strongly interacting nuclear system, a complete and consistent treatment of
the full hadronic interaction terms, the relativistic kinematics, and more
importantly, the nuclear medium effects, is too involved and theoretically
impossible (see e.g., [266, 267] for reviews). In practice, various
approximations have been adopted.

#### 3.1.1 Neutrino-nucleon opacity at mean field level

Figure 3.1: Schematic illustration of the neutron-rich situation, given large
differences between the chemical potentials of neutrons $\mu_{n}$ and protons
$\mu_{p}$, typically encountered in astrophysical situations feature hot and
dense nuclear matter, where neutrons and protons are treated within the mean-
field approximation given by their respective effective masses $m_{N}^{*}$ and
single-particle potentials $U_{N}$ both of which enter the particle’s
dispersion relations $E_{N}$.

In the mean field approximation, nucleons in nuclear medium can be viewed as
independent quasiparticles, with modified dispersion relations parametrized in
terms of effective masses $m_{n,p}^{*}$ and mean field potentials $U_{n,p}$ as
$E_{n,p}=\sqrt{m_{n,p}^{*2}+p_{n,p}^{2}}+U_{n,p}\equiv E_{n,p}^{*}+U_{n,p}$.
The effective masses and potentials can be deduced from relativistic mean
field theory, based on which the EoS of nuclear matter can be derived [54,
268, 53, 269, 270, 271, 272, 232, 233]. Under typical neutron rich conditions
encountered in CCSNe, the neutron chemical potential ($\mu_{n}$) exceeds that
of protons ($\mu_{p}$) by a large amount as illustrated in Fig. 3.1. Since
most commonly used nuclear EoS are based on the relativistic mean field
approach, relativistic dispersion relations are usually employed. The nuclear
EoS can also be computed using nonrelativistic formalism along with
nonrelativistic parameterizations of nuclear potentials, such as the widely
used Lattimer and Swesty (LS) EoS [52]. Correspondingly, nucleons take
nonrelativistic energy-momentum relation as
$E_{\rm{NR}}=p^{2}/(2m^{*})+U_{\rm{NR}}$. Taking the relativistic form,
$E\approx E_{\rm{NR}}+m\approx\sqrt{m^{*2}+p^{2}}+m-m^{*}+U_{\rm{NR}}$ with
$m$ being the bare nucleon mass. For noninteracting nuclear medium,
$m_{n,p}^{*}$ becomes the bare nucleon masses and $U_{n,p}\to 0$.

Neglecting the residual correlation between the dressed nucleons at the mean
field level, the differential rates for neutrino-nucleon scattering,
$\nu(K_{1})+N(P_{2})\to\nu(K_{3})+N(P_{4})$, and neutrino absorption on
nucleon, $\nu(K_{1})+N(P_{2})\to l(K_{3})+N(P_{4})$, can be obtained from the
Fermi’s golden rule as [258]

$\displaystyle{d^{2}\Gamma\over d\omega d\cos\theta_{13}}={(1-f_{3})k_{3}\over
8\pi^{2}E_{1}}\int\frac{d^{3}\bm{p}_{2}}{\left(2\pi\right)^{3}}\int\frac{d^{3}\bm{p}_{4}}{\left(2\pi\right)^{3}}\frac{\left|\mathcal{M}\right|^{2}}{4E_{2}^{*}E_{4}^{*}}\times\left(2\pi\right)^{4}\delta^{(4)}\\!\left(K_{1}+\\!P_{2}-\\!K_{3}-\\!P_{4}\right)f_{2}\left(1-f_{4}\right),$
(3.1)

where $\omega$ is the energy transfer to the nuclear medium, $\theta_{13}$ is
the angle between the three-momentum of leptons,
$\left|\mathcal{M}\right|^{2}$ is the squared scattering amplitude summed over
final state spins and averaged over initial state spins, and $f_{i}$ is the
standard Fermi-Dirac distribution function. By integrating over $\omega$ and
$\cos\theta_{13}$, the total scattering rate or the inverse mean free path of
neutrino as a function of $E_{\nu}$ can be computed. Note that the same
expression shown in Eq. (3.1) can also be derived equivalently within the
framework of finite temperature field theory [273].

The nuclear EoS only has a modest impact on neutral current neutrino-nucleon
scattering rates at the mean field level, but can affect directly the charged
current rates. In what follows, we focus on the charged current channel.
Taking into account the effective nucleon masses $m_{N}^{*}$ and the mean-
field potential $U_{N}$, both the transition matrix elements and the phase
spaces of nucleons in Eq. (3.1) are modified. The evolution of SN dynamics as
well as nucleon abundances depend sensitively on the EoS taken, which are also
related to the neutrino-nucleon interaction rates. Constraints on the EoS at
low density, in particular up to nuclear saturation density, are given by
chiral effective field theory for pure neutron matter, which is shown in the
left panel of Fig. 3.2 (gray band) from Refs. [274, 275], in comparison to a
representative set of relativistic and non-relativistic nuclear EoSs [52, 54,
232, 233] and a quark bag model [119, 120], all of which have been commonly
used in astrophysical simulations of SNe as well as BNSMs. This comparison
indicates that there is some tension among most models, although there are a
few exceptions that exhibit qualitative agreement. The EoS at supersaturation
densities can be constrained by the observation of massive pulsars of about 2
$M_{\odot}$ [276, 277], but the existing calculations still differ greatly.
Recently, it has become possible to further constrain the EoS using the GW
data obtained from the first BNSM event GW170817 [225, 278, 279], through the
determination of the tidal deformability which, in turn, provided constraints
for the radius of neutron stars of 8.9–13.2 km for intermediate mass neutron
stars of 1.2–1.6 $M_{\odot}$. Some tension exists between these results and
those obtained from the NICER (Neutron star Interior Composition ExploreR)
NASA X-ray satellite mission, where rather large radii were found from the
analysis of two independent groups [280, 281, 282, 283].

These variations of the EoSs are also reflected in the mean field potentials
and the effective masses of nucleons, which become more evident at
supersaturation densities, as shown in the right panel of Fig. 3.2. Note,
however, the relevant density range for the neutrino decoupling from matter,
i.e. the location of the neutrinospheres, corresponds to densities below 0.1
fm-3, where the differences between the different EoSs are somewhat less
pronounced due to the low-density constraints, not only from chiral effective
field theory but also for the EoS at nuclear saturation density which is
confirmed experimentally, such as the nuclear symmetry energy and the
corresponding slope. Further details and a detailed discussion comparing these
EoSs in simulations of CCSNe as well as the corresponding complete set of
nuclear matter properties can be found in Ref. [61]. It is important to note
here that the nuclear symmetry energy, $S(\rho)$, determines the mean field
potential difference, since $\mu_{n}-\mu_{p}=4\beta S(\rho)\propto
U_{n}-U_{p}$, with the asymmetry parameter $\beta=1-2Y_{p}$ for a given
proton-to-nucleon ratio $Y_{p}$. Some EoSs, such as NL3, TM1, TMA, and LS220,
lead to a steeply rising $U_{n}-U_{p}$ with density, as shown in the right
panel of Fig. 3.2, which feature a stiff symmetry energy and a too high slope
parameter compared to that derived from experiments [284], while other EoSs
indicate a negative slope towards high densities. It would be interesting,
that a priori it is not possible at current stage, to determine whether the
slope remains positive at all densities or not, which requires more
experimental and observational constraints in the future.

Figure 3.2: Left panel (Figure adapted from Ref. [285]): Energy per nucleon
for neutron matter from chiral effective field theory interactions computed up
to N3LO (gray band), including uncertainty estimates due to the many-body
calculation, low-energy constants and the regularization cutoffs in the 3N and
4N forces [274, 275]. The colored lines show the energy per nucleon for a
selection of RMF EoS from the catalogue of Ref. [232] as well as two commonly
used Skyrme models (LS180, LS220) [52], and a quark bag (QB) model EoS [59,
120]. Right panel: Mean-field potential difference, $U_{n}-U_{p}$, between
neutrons and protons and effective masses, $m^{*}/m$, for the same selection
of commonly used nuclear matter EoS, for a fixed electron fraction of
$Y_{e}=0.3$ and a temperature of $T=7$ MeV.

The neutrino-nucleon rate or opacity also depends on how the integral in Eq.
(3.1) is evaluated using different approximations. Taking nonrelativistic
energy-momentum relation for nucleons and keeping only the leading terms of
the matrix elements in the heavy nucleon limit (i.e., the terms proportional
to $g_{V}^{2}$ and $g_{A}^{2}$), the differential rates taking into account
the thermal motion and the final state blocking can be solved analytically in
terms of the density and spin dynamical structure factors [258]. Taking
further the so-called elastic approximation where no energy is transferred to
the heavy nucleons, the neutrino-nucleon opacity has a simple analytical form,
see Eqs. (34) and (35) of [258]. The nucleon recoil and weak magnetism can be
included in the elastic opacity approximately [286]. It was shown that weak
magnetism can enhance/suppress the neutrino/antineutrino rates and affects the
neutrino spectra and SN dynamics [286, 77].

A relativistic treatment of both full kinematics (nucleon thermal motion,
recoil, and blocking) and weak magnetism for charged current neutrino
reactions has been performed [273, 287], and its impact on CCSN dynamics,
neutrino signals as well nucleosynthesis was explored based on 1D simulation
[287]. Figure 3.3 (left panel) compares the charged-current rates in the full
kinematics treatment at the mean field level, including weak magnetism
contributions, with the elastic approximation of Ref. [83] and including in
addition approximately inelastic contributions and corrections due to weak
magnetism following Ref. [286]. The latter has been commonly used in most CCSN
studies. From this comparison it becomes evident that the full kinematics
treatment cannot be adequately captured solely by a multiplicative factor
varying with neutrino energy given in Ref. [286]. This is particularly
relevant in the energy domain when the $\bar{\nu}_{e}$ rates drop to zero due
to the energy threshold for this process, requiring
$E_{e^{+}}=E_{\bar{\nu}_{e}}-(E_{n}-E_{p})>0$. At the mean-field level, this
corresponds roughly to
$E_{\bar{\nu}_{e}}\gtrsim(m_{n}^{*}-m_{p}^{*})+(U_{n}-U_{p})$, as is
illustrated in Fig. 3.3, modulo momentum transfer contributions which even
further suppress the rates towards lower energies. Note that a threshold does
not exist for $\nu_{e}$ since the condition
$E_{e^{-}}=E_{\nu_{e}}+(E_{n}-E_{p})$ is always fulfilled. Furthermore, it is
interesting to note that the inverse neutron decay, which was previously
largely neglected, contributes significantly to the $\bar{\nu}_{e}$ opacity
(green lines in Fig. 3.3). For this reaction the threshold is given by
$E_{e^{-}}=-E_{\bar{\nu}_{e}}+(E_{n}-E_{p})$, i.e. the medium modifications
enhance the rate substantially with increasing density.

Figure 3.3: Charged current rates for $\nu_{e}$ and $\bar{\nu}_{e}$ at typical
conditions. Left panel (Figure adapted from Ref. [260]): Comparing the full
kinematics treatment at the mean-field level including weak magnetism (solid
lines) with the elastic approximation (dashed lines) of Ref. [83] and
including approximately inelastic contributions and weak magnetism corrections
(dash-dotted lines) following Horowitz (2002) [286]. The green lines
correspond to the inverse neutron decay. The corresponding ratios of the
results with full kinematics and weak magnetism (solid lines) and the results
considering the correction factors following Horowitz (2002) [286] (dashed
lines) to those using the elastic approximation without including the weak
magnetism are also shown in the lower figures for $\nu_{e}$ and
$\bar{\nu}_{e}$, respectively. Right panel: Comparing the charged-current
rates for selected relativistic mean field EoS, DD2 ($U_{n}-U_{p}=2.1$,
$m^{*}/m=0.9861$), NL3 ($U_{n}-U_{p}=1.2$, $m^{*}/m=0.9902$) and SFHo
($U_{n}-U_{p}=4.5$, $m^{*}/m=0.9855$) with full kinematics and weak magnetism
considered.

The right panel of Fig. 3.3 compares in addition the dependence of the
charged-current rates for $\nu_{e}$ (top panel) and $\bar{\nu}_{e}$ (bottom
panel) for three selected EoSs: DD2 [53, 54, 232], NL3 [54, 232], and SFHo
[54, 233]. Since each EoS has different single-particle properties as well as
a different density dependence, as was discussed before, there is a certain
degeneracy of the charged current weak rates. The largest impact is due to the
threshold for the $\bar{\nu}_{e}$ absorption on protons as well as for the
inverse neutron decay. Small (large) values of the neutron-proton mean field
potential differences result in the suppression of the $\bar{\nu}_{e}$
absorption rate at low (high) energy, while the opposite is found for the
$\bar{\nu}_{e}$ opacity from the inverse neutron decay, such that the
integrated $\bar{\nu}_{e}$ opacity becomes less sensitive to the value of
$U_{n}-U_{p}$. In other words, the symmetry energy impact on the charged
current rates entering directly the rate expressions at the mean field level
via the mean field potentials, is somewhat less strong than originally
anticipated [288, 289]. Spectral differences between $\nu_{e}$ and
$\bar{\nu}_{e}$ are rather driven by the different abundances of neutrons and
protons, which might differ greatly for the EoSs commonly used, at
intermediate densities relevant for the neutrino decoupling. This abundance
difference arises from the different descriptions of the nuclear medium,
particularly concerning the abundances of light and heavy nuclear clusters at
nuclear subsaturation densities. A detailed discussion about this aspect can
be found in Refs. [285, 56]. It depends sensitively on two aspects: (i) the
description of the nuclear clusters where commonly the modified nuclear
statistical equilibrium of Ref. [54] is employed based on thousands of nuclear
species with tabulated and partly calculated nuclear masses versus the quantum
statistical approach, implemented in particular for light clusters including
Pauli blocking as well as two- and effective three-body scattering phase
shifts, and (ii) the Mott transition already mentioned in Sec. 2.1.3.
Different treatments of the Mott transition give significantly different
nucleon abundances in the density region of $\rho\simeq 10^{12}-10^{14}$ g
cm-3, and accordingly the charged current rates differ substantially [56].
Nevertheless, definite conclusions cannot be drawn about the impact of
neutrino rates using different EoSs on SN dynamics and the neutrino signals by
simply looking at the neutrino opacity. The system is too complex since the
underlying equations are highly non-linear. For instance, the different EoSs
exhibit softer or stiffer high-density behaviors, resulting in different SN
dynamics, different abundance for each species, as well as different locations
of the neutrinospheres, all of which might compensate and entangle with the
direct impact on the neutrino opacity [55, 285]. In addition to studies
associated with known EoSs, neutrino-nucleon rates at the mean field level
have also been computed using effective nucleon masses and potentials obtained
from virial expansion at low densities [290], and from pseudo-potentials
constructed from nucleon scattering phase shifts [291], realistic nucleon-
nucleon potentials [292] and those derived from chiral effective field theory
[293], and Skyrme interactions [294, 295, 296].

Due to a large rest mass, the production of $\mu^{\pm}$ and their role in SN
dynamics were largely ignored in the previous literature. Their relevance has
recently been explored in 2D simulations [110], indicating that the presence
of $\mu^{\pm}$ softens the EoS, induces faster core contraction, and thus
leads to higher luminosities and averaged energies of neutrinos, thereby
stimulating the neutrino-driven explosions. A follow-up study considered the
muonization of SN matter in more details and focused on the possible impact on
neutrino signals [57]. Interestingly, muonization could lead to a burst of
$\nu_{\mu}$ shortly after core bounce when the shock passes the $\nu_{\mu}$
neutrinosphere [57]. Note that the accumulation of muons is closely related to
the charged current reactions of $\nu_{\mu}$ ($\bar{\nu}_{\mu}$), which are
produced in large amounts via electron-positron pair annihilation and nucleon-
nucleon bremsstrahlung early on before core bounce. To obtain accurate charged
current $\nu_{\mu}$-nucleon rates, Ref. [259] included the pseudoscalar term
and considered the $q^{2}$-dependent nucleon form factors in the hadronic
current at the mean field level, which are important due to large energy and
momentum transfer. Fig. 3.4 compares the effects of weak magnetism (WM), the
pseudoscalar term (PS), and the form factors (FF) on the neutrino opacities
for $\nu_{\mu}+n\to\mu^{-}+p$ at one selected condition with $T=38.3$ MeV and
$\rho=10^{14}$ g/cm3 from a 2D CCSN simulation [110] using the EoS LS200 [52],
where the relevant muonic reactions have been implemented. As shown in the
figure, weak magnetism enhances (suppresses) the absorption rates of
$\nu_{\mu}$ ($\bar{\nu}_{\mu}$) by $\sim$ 20% at $E_{\nu}=20$ MeV and by
30%–50% at $E_{\nu}=150$ MeV. The interference between the axial-vector term
and the pseudoscalar term is always negative. Therefore, inclusion of the
pseudoscalar term suppresses the opacities for both $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ absorption on nucleons. Similarly, taking into account the
form factor effect leads to reduced couplings for all interaction terms and
thus reduces the opacities. We note that the enhancement for $\nu_{\mu}$ due
to weak magnetism is largely canceled by the effects of pseudoscalar term and
form factor, while for $\bar{\nu}_{\mu}$, all effects contribute negatively
and the absorption opacity is significantly quenched compared to the result
obtained at lowest order.

Figure 3.4: Opacities for $\nu_{\mu}+n\to\mu^{-}+p$ and
$\bar{\nu}_{\mu}+p\to\mu^{+}+n$ and the effects of weak magnetism (WM),
pseudoscalar term (PS), and form factors (FF). The relative ratios of the
opacities to that considering only the leading-order terms (LO, the black
lines), i.e., terms proportional to $g_{V}^{2},g_{A}^{2}$, and $g_{V}g_{A}$,
are shown in the inset as functions of $E_{\nu}$. Note that for all the
results presented, full kinematics including nucleon recoil and blocking are
always considered. The condition is taken from a 2D simulation [110]
considering the muonic rates using LS200 at $r\approx 13.6$ km and at 0.4 s
post core bounce, with $T=38.3$ MeV and $\rho=10^{14}$ g/cm3. Figure adapted
from Ref. [259].

#### 3.1.2 Effects of nucleon-nucleon correlation

The nucleon-nucleon correlation due to the nuclear force affects neutrino-
nucleon opacities significantly. Because of the non-perturbative nature as
well as the complexity of the nuclear force, a complete treatment of the
nucleon correlations is still lacking. In what follows, we briefly review the
methods adopted in the literature to study the correlation effects on neutrino
opacities in hot and dense nuclear matter and highlight some progress made
during the past few years.

For low energy neutrinos interacting with nonrelativistic nucleons, the
reaction rates are dominated by the density (vector) dynamical structure
factor, $S_{V}(\omega,q)$, and the spin (axial) dynamical structure factor,
$S_{A}(\omega,q)$, where $\omega$ and $q$ are the energy and momentum transfer
to the nuclear medium [297, 298]. The energy exchange $\omega$ is of order
${\rm max}[q\sqrt{T/m_{N}},q^{2}/m_{N}]$ for neutrino-nucleon scattering,
which is much smaller than $q$ for heavy nucleons. When integrating over a
range of $\omega$ to obtain the total rate, the dynamical structure factors
can be approximately replaced by $S_{V,A}(q,\omega)\to\delta(\omega)\int
d\omega S_{V,A}(\omega,q)\equiv 2\pi\delta(\omega)S_{V,A}(q)$, where
$S_{V,A}(q)$ are the static structure factors. This corresponds to the elastic
approximation in which energy exchange to the nuclear medium can be neglected.
In some approaches discussed below, only the static structure factors can be
computed.

Nucleon correlation effects on neutrino opacities beyond mean field level are
often studied based on the random phase approximation (RPA) for both the
neutral current and the charged current reactions [299, 300, 301]. To account
for the screening of coupling between nucleons and the external probe due to
particle-hole excitations, an infinite series of ring or bubble diagrams are
summed in RPA when calculating the polarization function, whose imaginary part
gives the dynamical structure factor from the fluctuation-dissipation theorem.
Compared to previous studies using simple and energy-independent interactions
in terms of the Landau parameters [299, 301, 292], more complete particle-hole
interactions derived from the Skyrme interactions have been employed to study
the RPA correlations [302, 303, 304, 305, 294, 306, 296]. In addition to those
nonrelativistic studies, RPA calculations were also performed within the
framework of a relativistic mean field theory [301, 307]. The net effect of
RPA correction is to redistribute the strength of the response function to a
broad energy region and generally to suppress the neutrino-nucleon opacities
[299, 300]. Remarkably, nuclear interactions from chiral effective field
theory have recently been employed to study the RPA effects beyond the mean
field level [308]. Different from the studies in [299, 300], Ref. [308] found
that nucleon correlation enhances the density response and reduces the spin
response for neutral current scattering. For charged current reactions, the
RPA tends to redistribute the strength to higher energy for neutrino
absorption and lower energy for antineutrino absorption, leading to suppressed
$\nu_{e}$ absorption rates at all relevant energy regions and enhanced
(suppressed) $\bar{\nu}_{e}$ absorption rates at low (high) energy [308]. As a
consequence, the spectral difference between $\nu_{e}$ and $\bar{\nu}_{e}$ is
reduced, which tends to reduce the neutron richness in the neutrino-driven
wind from the PNS surface.

At low density and/or high temperature, the virial expansion in terms of the
fugacity $z\equiv e^{\mu/T}$, where $\mu$ is the relativistic chemical
potential of nucleons, can provide a model-independent description of the EoS
of nuclear matter [309, 310], as well as the static neutrino response
functions in the long wavelength limit, i.e., $S_{V,A}(q=0)$, for neutrino-
nucleon neutral current scattering [298]. The expansion is up to the second
order using the second virial coefficients from the scattering phase shifts
and is only valid for $z\lesssim 0.5$. The virial expansion indicates that
$S_{V}(q=0)$ is enhanced while $S_{A}(q=0)$ is suppressed due to nucleon
correlation, consistent with the findings of [308].

For CCSN simulations, the neutrino interactions at the neutrinosphere is
highly relevant. The neutrinosphere is typically neutron-rich with densities
$\sim 10^{12-13}$ g/cm3 and temperatures $\sim 10$ MeV. With a large neutron-
neutron scattering length and short effective range, the neutrinosphere could
be approximately modeled as a warm and low density unitary gas of neutrons.
For any unitary Fermi gas with infinitely large scattering length and zero
effective interaction range, the detail of the interaction is irrelevant due
to the absence of length scales and all systems share universal static and
dynamic properties (see e.g., [311]). Unitary gases of cold atoms have been
produced and well studied in laboratories. There are also many theoretical
calculations for unitary gases, which are, however, not available for the
neutron gas. For example, the fourth-order virial coefficients for a unitary
gas have been obtained, and can be used to extend the virial results of the
static neutrino responses up to $z\approx 1$ [312]. Based on a lattice
formulation, the static structure factors with a finite momentum transfer,
$S_{V,A}(q)$, of a unitary gas under SN conditions were also derived [313].
Similar lattice studies using the leading-order pionless effective field
theory were further performed to compute the static structure factors of
neutron matter [314].

Note that the full dynamical structure factors $S_{V,A}(\omega,q)$ based on
virial expansion or lattice calculation are not yet available. To obtain the
dynamical structure factors perturbatively, Ref. [315] considered a
pseudopotential from neutron-neutron phase shifts and computed the
contributions to the polarization functions arisen from two-loop diagrams.
Including these corrections due to neutron-neutron interaction, the dynamical
structure factors for neutron matter are computed up to the second order in
$z$. The study presented in [315] should be compared to the one employing
nuclear potentials based on chiral effective field theory in [308], where the
single-particle energies of nucleon are calculated at the Hartree-Fock mean
field level and the vertex corrections are resummed in the framework of RPA,
both incorporating an infinite number of diagrams.

#### 3.1.3 Nucleon-nucleon bremsstrahlung

Figure 3.5: Inverse mean free paths for $\nu_{e}$ in panel (a),
$\bar{\nu}_{e}$ in panel (b) and collectively for all heavy lepton flavors
$\nu_{\mu}$ in panel (c), for all four selected neutrino interaction channels
considered in SN simulations, charged current reactions ($\nu_{e}n\rightarrow
pe^{-}$, $\bar{\nu}_{e}p\rightarrow ne^{+}$), elastic neutrino nucleon/nucleus
scattering ($\nu N$), inelastic neutrino electron/positron scattering ($\nu
e^{\pm}$) and nucleon–nucleon bremsstrahlung ($\nu\bar{\nu}NN$), for which two
different treatments are employed, vacuum one-pion exchange (blue lines) [316,
261] and including approximately corrections of the $\pi NN$ vertex [262]. Not
shown are the other neutrino-pair processes, i.e. electron-positron
annihilation and neutrino-pair annihilation. The vertical solid and dash-
dotted lines mark the locations of the neutrinospheres of last inelastic and
elastic scatterings, respectively. The gray region in between marks the
presence of a scattering atmosphere, following the argumentation of Refs. [63,
151]. Credit: reprinted from Fischer et. al., A&A, 593, A103, 2016 [262],
reproduced with permission $\copyright$ ESO.

It had long been argued that the neutrino fluxes and spectra at the decoupling
are determined dominantly by the charged current rates. This turns out to be
the case during the post-bounce mass accretion phase prior to the shock
revival and onset of explosion, in which case both charged current rates for
$\nu_{e}$ and $\bar{\nu}_{e}$ are comparably large and a scattering atmosphere
develops only for the heavy lepton neutrinos, i.e. the spatial separation of
neutrinosphere of inelastic scattering at high density and elastic scattering
at lower density. The situation changes after the shock revival, when mass
accretion ceases and the PNS enters the deleptonization phase. Whereas the
neutrinospheres of last inelastic scattering dominate the neutrino decoupling
for $\nu_{e}$, an extended scattering atmosphere develops not only for the
heavy lepton neutrino but also for $\bar{\nu}_{e}$. This situation is
illustrated in Fig. 3.5 for a typical situation during the early PNS
deleptonization phase, showing the inverse mean free paths for all flavors and
the four weak reaction channels that are considered in the neutrino transport
codes employed for CCSN studies in general, charged current, elastic neutrino
nucleon/nucleus scattering, inelastic neutrino electron/positron scattering
and neutrino pair processes. The phenomenon that $\nu_{e}$ also develop a
scattering atmosphere at late times, following the argumentation of Refs. [63,
151], was realized in Ref. [62], analyzing simulation results that were based
on three-flavor Boltzmann neutrino transport [108]. This has important
consequences for the evolution of the neutrino luminosities $L_{\nu}$ and
average energies $\langle E_{\nu}\rangle$ that become increasingly similar for
all flavors during the PNS deleptonization phase, which is illustrated in Fig.
3.6 (b), as all flavors are becoming dominated by neutrino-nucleon scattering
towards late times during the PNS deleptonization, a weak process that is
flavor blind.

As shown in Fig. 3.5, nucleon-nucleon ($NN$) bremsstrahlung plays a non-
negligible role for inelastic scattering of $\bar{\nu}_{e}$, $\nu_{x}$, and
$\bar{\nu}_{x}$. Neutrino pair emission from nucleon-nucleon bremsstrahlung is
also an important process for the production of $\nu_{x}$ and $\bar{\nu}_{x}$.
For the bremsstrahlung rates, two different treatments are shown, vacuum one-
pion exchange (NN) [316, 261] and taking into account approximate medium
corrections of the $\pi NN$ vertex (NN∗) [262] based on Fermi-liquid theory
[317, 318, 319]. It assumes that the nucleons are only slightly excited above
the Fermi sea, i.e. all processes occur in the vicinity of the Fermi surface
and that the vertices of the processes are dressed by $NN$ correlations. The
subsequent medium modification of the one-pion exchange process includes the
summation of all particle-hole loops and a contribution of the residual
$s$-wave $\pi NN$ interaction as well as $\pi\pi$-scattering. This involves
effectively $\rho$-meson exchange ($t$-channel) and the $\sigma$-meson as
correlated $\pi\pi$ ($s$-wave contribution). It results in the suppression of
the $NN$ bremsstrahlung opacity for all flavors (see Fig. 3.5), in the density
domain relevant for CCSNe and PNS deleptonization, up to a few times the
nuclear saturation density. At higher densities, higher order corrections
become important, due to nucleon–hole (and also $\triangle$–hole) excitations,
such that the subsequent $p$-wave pion polarization in nuclear matter leads to
the softening of the in-medium pion dispersion relation, which is not captured
by the approximate dressing of the vertex functions. Modern nuclear
interactions from chiral effective field theory have been used to study
neutrino bremsstrahlung rate based on the Landau’s theory of Fermi liquids
[320, 321, 322]. To account for the nonperturbative feature of the nuclear
force, NN bremsstrahlung rates were obtained in an independent study based on
the $T$-matrix elements extracted from measured phase shifts [323]. Using a
two-body nuclear potential from chiral effective field theory, Ref. [263]
solved the Lippmann-Schwinger equation exactly for the $T$-matrix including
non-diagonal and off-shell contributions, based on which improved
bremsstrahlung rates were computed. Compared to the case using one-pion
exchange potential, the bremsstrahlung rates from the $T$-matrix are higher at
density $\lesssim 10^{12-13}$ g/cm3 and are suppressed at higher densities
[263]. Employing the $T$-matrix rates rather than the ones based on one-pion-
exchange potential in spherically symmetric simulations has a marginal impact
on neutrino luminosities, resulting in a change of approximately $5\%$-$10\%$
[324, 325]. Additionally, it leads to a slight increase of $\sim 0.5$ MeV in
the average energies of heavy lepton flavor neutrinos.

Similarly to neutrino bremsstrahlung, the so-called modified Urca processes,
such as $\bar{\nu}_{e}+e^{-}+p+N\to n+N$ and $\bar{\nu}_{e}+p+N\to e^{+}+n+N$
with $N$ a speculator nucleon, and similarly for $\nu_{e}$, could act as
additional opacity sources for low energy neutrinos (as discussed in a recent
work [306]). The nucleon-nucleon collision leads to a broadening in the
neutrino response functions. Such a collision broadening effect was
approximately treated for the charged current neutrino-nucleon reactions (see,
e.g., [326, 306]). Note that the effects due to mean field correction and
nucleon-nucleon correlation and collision on neutrino opacities, neutrino
signals, and the related nucleosynthesis could interfere with one another
[326]. A complete and consistent treatment further taking into account the
relativistic kinematics, full hadronic weak interaction terms, and the
potential presence of light cluster and nuclear pasta is still lacking.

#### 3.1.4 Additional degrees of freedom

With slightly larger masses compared to $\mu^{\pm}$, abundant pions can also
appear in hot and dense nuclear matter. The thermal pions not only soften the
EoS but also contribute to the opacity of low energy muon neutrinos via
$\bar{\nu}_{\mu}+\mu^{-}\to\pi^{-}$ and $\nu_{\mu}+\pi^{-}\to\mu^{-}$ [242].
In addition to pions, hyperons such as $\Lambda$ and $\Sigma^{-}$ may be
present [327] and participate in weak reactions with neutrinos (see e.g.,
[328, 329, 258, 330]).

Even for purely nucleonic matter, the composition vary with density and
temperature. Besides the appearance of light clusters at low densities,
nucleons can bind into strange configurations and shapes, commonly referred to
as nuclear pasta, at low temperatures and subsaturation densities (see e.g.,
[331, 332, 333, 334, 335, 336, 337, 338, 339] and a recent review [340]). The
presence of nuclear pasta can modify the neutrino scattering rates and thus
affect the transport of neutrinos, the cooling of NS, and the related SN
neutrino signals [341, 342, 343, 344, 345, 346, 347, 348, 349, 350]. With
typical energies of a few tens MeV, neutrinos produced from SN core have
wavelengths comparable to the size of the pasta structures and can scatter
coherently on the nucleon cluster with enhanced cross sections [342, 343].
Incorporating such an enhanced cross section with the heterogeneous nuclear
pasta in SN simulations, Ref. [347] found that the neutrino luminosity
increases significantly at late times of 10 s after core collapse. However,
Ref. [348] considered the dissolution of nuclei and pasta due to finite
temperature effects and concluded that the impact of nuclear pasta on SN
neutrino opacities and signals is only modest.

### 3.2 Impact of neutrino-nucleon rates on PNS evolution and nucleosynthesis
in the $\nu$-driven wind

Accurate neutrino transport and a complete set of weak processes are both key
input physics in the simulations of the PNS deleptonization, during which the
neutrino-driven wind can be launched as described in Sec. 2.1.5. Since these
winds initially have high temperatures on the order of few MeV at the PNS
surface before being ejected [Fig. 2.5(e)], the nuclear composition are mostly
composed of free protons and neutrons in the beginning, and their ratio is
sensitively determined by the neutrino luminosities, spectra and their
evolution during the PNS deleptonization. These nucleons can combine to nuclei
once the matter reach colder regions at larger radii. The outcome of the
nucleosynthesis depends strongly on the proton-to-nucleon ratio $Y_{e}$,
especially whether $Y_{e}<0.5$ (neutron rich) or $Y_{e}>0.5$ (proton rich). As
the neutrino-nucleon reactions are fast enough, the initial value of $Y_{e}$
is simply given by [143]

$Y_{e}\approx\frac{\lambda_{\nu_{e}n}}{\lambda_{\nu_{e}n}+\lambda_{\bar{\nu}_{e}p}}~{},$
(3.2)

where $\lambda_{\nu_{e}n}$ and $\lambda_{\bar{\nu}_{e}p}$ are the rates for
$\nu_{e}$ and $\bar{\nu}_{e}$ captures on free nucleons, respectively.
Consequently, $Y_{e}$ is mainly determined by the neutrino luminosities and
their spectra, in particular the difference between $\nu_{e}$ and
$\bar{\nu}_{e}$. For example, a larger (smaller) $\langle
E_{\bar{\nu}_{e}}\rangle-\langle E_{\nu_{e}}\rangle$ leads to a lower (higher)
$Y_{e}$. Recent long-term 1D simulations found that the neutrino luminosities
and average energies of different flavors in this phase generally decrease
with time and become increasingly similar; see e.g., Fig. 3.6(b) and Refs
[148, 108]. This is mainly because the neutrino–nucleon scattering, which is
flavor blind, becomes increasingly more important than the inelastic charged
current processes, as the latter is further suppressed by the final state
blocking [62]. Thus, the resulting $Y_{e}$ in the wind are found to be around
0.5 (see e.g., Fig. 2.5(d) or Fig. 3.7). Besides $Y_{e}$, the wind dynamical
timescale and the entropy, which can be related to the high-density EoS, in
terms of the mass and radius of the nascent PNS, i.e. their ratio $M/R$ (see
e.g., Ref. [143]), also play important roles in determining the
nucleosynthesis outcome. Typically, a more compact PNS that results from a
softer high-density EoS features higher entropy in the neutrino-driven wind.

Since the neutron-richness of the neutrino-driven wind is directly affected by
the energy spectra of $\nu_{e}$ and $\bar{\nu}_{e}$, which can depend on the
evolution history of the PNS and the decoupling of neutrinos of all flavors,
in recent years, a number of studies investigated the impact of both the
charged-current and neutral-current neutrino nucleon interactions on the PNS
evolution and the nucleosynthesis conditions [288, 326, 262, 324, 306]. In
particular, Refs. [288, 326] found that including the medium modification to
the charged-current rates can significantly lower the $Y_{e}$ to slightly
neutron-rich during the early neutrino-driven wind phase. More recently, Ref.
[260] investigated in detail the improved description including consistent
treatment of full kinematics, weak magnetism contributions, as well as the
(inverse) neutron decay reactions (see Sec. 3.1.1), in accordance with the
underlying EoS. Figure 3.6 compares radial profiles of selected quantities at
three post-bounce times during the PNS deleptonization in the panel (a) as
well as the corresponding neutrino luminosities and average energies evolution
in the panel (b), for four different treatments of the charged-current weak
processes involving the unbound baryon: i) the elastic rates without inelastic
contributions and corrections due to weak magnetism; ii) elastic approximation
but taking into account nucleon recoil and weak magnetism correction
formulated in Ref. [286]; iii) and iv) the fully inelastic rates of Ref. [260]
without and with neutron decay (see Sec. 3.1). In particular, it was found
that the improved treatment results in higher $Y_{e}$ values inside the PNS as
shown in Fig. 3.6(a). This is due to the modified condition of weak
equilibrium established as well as the reduction of the $\bar{\nu}_{e}$
average energies during the initial deleptonization phase before the impact of
the charged-current reactions diminishes, which reduces the deleptonization
timescale indicated by the lower central temperatures at smaller radii at
intermediate and later times of about $>5~{}{\rm s}$ post bounce, as shown in
Fig. 3.6(b).

(a) Radial profiles (b) Evolution of neutrino luminosities and average
energies

Figure 3.6: PNS evolution, comparing four different treatments of the
electronic charged-current reactions (see Sec. 3.1). Panel (a): Radial
profiles of selected quantities (from top to bottom), temperature $T$, rest
mass density $\rho$, electron fraction $Y_{e}$ and mean field potential
difference between neutrons and protons, $U_{n}-U_{p}$, at three selected
post-bounce times. Panel (b): Evolution of the neutirno luminosities $L_{\nu}$
and average energies $\langle E_{\nu}\rangle$, for all flavors ($\nu_{x}$ and
$\bar{\nu}_{x}$ denote collectively $\nu_{\mu/\tau}$ and
$\bar{\nu}_{\mu/\tau}$). (Figures reprinted from Ref. [260])

Moreover, as shown in Fig. 3.6(b), the inclusion of weak magnetism and recoil
mainly suppresses the opacity for $\bar{\nu}_{e}$ and thus shifts the
neutrinosphere of last inelastic scattering to higher densities and
temperatures. Therefore, the average energies of $\bar{\nu}_{e}$ will be
enhanced. Naively, one would expect a reduced $Y_{e}$, or an enhanced neutron
fraction, due to a larger spectral difference between $\nu_{e}$ and
$\bar{\nu}_{e}$. However, this is not the case since weak magnetism also
reduces the cross section for $\bar{\nu}_{e}$ capture on protons and
$\lambda_{\bar{\nu}_{e}p}$ in Eq. (3.2) as well. The reduction in
$\lambda_{\bar{\nu}_{e}p}$ is primary and the net effect is to enhance $Y_{e}$
to be closer to 0.5 within a few seconds post core bounce (see e.g., the red
and black lines in the upper panel of Fig. 3.7). Compared to the elastic
approach with approximate inclusion of weak magnetism and recoil [286], the
full kinematics treatment with weak magnetism gives rise to larger charged
current opacities for $\nu_{e}$, and $\bar{\nu}_{e}$ to a lesser extent, which
reduces the $\nu_{e}$ luminosity more evidently. Consequently, the ejected
winds turn more neutron rich (see the black line in the upper panel of Fig.
3.7). However, when inverse neutron decay is considered as an additional
source for $\bar{\nu}_{e}$, the spectral difference between $\nu_{e}$ and
$\bar{\nu}_{e}$ decreases significantly, turning material slightly neutron
rich at the early deleptonization stage within $\sim 3$ s post bounce, and
proton rich later.

The bottom panel of Fig. 3.7 shows the corresponding nucleosynthesis yields in
the neutrino-driven wind with these four different treatments of the charged
current neutrino-nucleon rates for $\nu_{e}$ and $\bar{\nu}_{e}$ as described
above. As already found in previous studies, the neutrino-driven wind at early
times are not neutron rich enough to support an $r$-process, but the light
neutron capture nuclei with atomic numbers of $38<Z<42$ could be produced
under the slightly neutron-rich conditions encountered in simulations without
the inclusion of inverse neutron decay. Taking into account the inverse
neutron decay, however, the neutron fraction is reduced and the
nucleosynthesis path terminates even before reaching the light neutron capture
nuclei.

Figure 3.7: Evolution of the neutrino-driven wind $Y_{e}$ (top panel) and
elemental yields from integrated nucleosynthesis analysis (bottom panel),
comparing the different treatments of the charged-current weak rates
introduced above. The data are obtained from simulations of the PNS
deleptonization within general relativistic neutrino radiation hydrodynamics
based on Boltzmann neutrino transport. (Figure reprinted from Ref. [260])

In fact, most of the ejected matter in the SN neutrino wind is turned slightly
proton-rich by neutrino interactions, which may allow a nucleosynthesis
process called $\nu p$-process [351, 10, 11, 352] to occur. Upon reaching
colder regions, the ejected proton-rich matter freezes-out mainly into
$\alpha$ nuclei, with the excess protons left as free particles. In principle,
heavier nuclei can be synthesized from the freeze-out abundances by subsequent
proton captures, interrupted by $\beta^{+}$ decays. This situation is realized
in explosive hydrogen burning in X-ray bursters. However, in this scenario the
mass flow is strongly hampered by the increasing Coulomb barrier of the
produced nuclei and by the so-called waiting points. These are $\alpha$ nuclei
(56Ni, 64Ge…) which have relatively long $\beta$ half lives and for which
proton captures are strongly hindered by the small or negative proton binding
energies of the final nuclei. In contrast to X-ray bursts, the free protons in
the SN wind are immersed in a strong flux of neutrinos emitted by the PNS. By
absorption of $\bar{\nu}_{e}$ on ree protons, these can be turned into a fresh
supply of free neutrons which can interact with the already produced nuclei.
In this way the matter flux can overcome the waiting point nuclei by $(n,p)$
reactions and can continue to produce intermediate mass nuclei up to the mass
range $A\sim 80-100$ or above. The final abundance distribution and the
efficiency of $\nu p$ process nucleosynthesis can depend sensitively on
several factors, including the $Y_{e}$ value of the wind, the $\bar{\nu}_{e}$
flux, as well as the formation of the wind termination shock [353, 354].
Although recent 1D SN simulations tend to suggest that the $\nu p$ process
does not occur very efficiently in producing $p$-nuclei heavier than e.g.,
94Mo, see e.g., Ref. [355, 260] and Fig. 3.7, other unaccounted effects such
as nuclear correlation on the neutrino-nucleon rates [306, 308], the flavor
oscillations of neutrinos, which will be discussed in Sec. 4.2.1, or the
multidimensional nature (see below) may still alter this conclusion.

In principle, the waiting points can also be overcome by absorbing
antineutrinos which would change the proton to a neutron inside the nucleus.
But the rates are noticeably disfavored compared to the two-step process by
$p(\bar{\nu_{e}},e^{+})n$ and $(n,p)$ reactions. Ref. [356] has explored the
influence of the various charged- and neutral-current neutrino-nucleus
reactions on the $\nu p$ process abundances and finds negligible effect. Also,
during the long-term PNS evolution muons should be considered as well. In
particular, as the deleptonizing and cooling PNS is evolving towards the cold
remnant neutron star, the muon abundance increases continuously as the
electron and muon chemical potentials become increasingly similar. However, at
present, no impact on the neutrino fluxes and the spectra nor their evolution
could be identified from the inclusion of muonic weak processes (c.f. the
appendix of Ref. [357]).

We note that most of the above discussions are based on 1D simulations, which
cannot capture several multidimensional effects that may be important in
affecting the neutrino properties and the wind outflow conditions. For
instance, recent multi-dimensional long term simulations carried over several
seconds post bounce generally found that the neutrino-driven wind can be
rather aspherical due to the sustained fallback materials, depending on the
mass of the progenitor and the asphericity of the explosion [149, 95, 150,
72]. In particular, Ref. [72] examined the detailed hydrodynamical and
thermodynamical conditions, as well as $Y_{e}$ in the neutrino-driven wind
from 3D simulations with spectral transport of neutrinos. They found that the
wind properties can resemble those obtained with 1D models for lighter
progenitors, but differ more significantly from 1D cases for heavier
progenitors. However, the resulting nucleosynthesis patterns still seem to be
in broad agreement with what is obtained in 1D simulations, as the $Y_{e}$,
entropy, and expansion velocities allow only the production of heavy elements
up to $Z\sim 40$ and $A\sim 90$. Moreover, preliminary studies suggested that
the convection inside the PNS can possibly alter the decoupling of neutrinos
of different flavors and therefore changing the resulting $Y_{e}$ in the wind
[358], or even generate the gravito-acoustic waves that may strongly affect
the wind dynamics and the nucleosynthesis [359].

Finally, we note that there are exceptions, in which the neutrinos are not
primarily responsible for the determination of the nucleosynthesis conditions
if the matter is ejected relativistically. Among them are electron-capture SN
[66] and the alternative scenarios briefly mentioned at the end of sec. 2.1.4
– the magneto-rotational SNe in which a jet forms and quark-hadron phase
transition driven SN explosions. Both these latter cases result in the initial
ejection of very neutron rich material, while slightly neutron rich pockets
are ejected in electron-capture supernovae, followed by the standard neutrino-
driven wind on a longer timescale on the order of several seconds, as
discussed above.

### 3.3 Neutrino-nucleus interaction

The neutrinos produced in various stages of the SN can interact with nuclei in
different ways with potential relevance for the SN dynamics, nucleosynthesis
and detection. In this section we will briefly discuss how the interaction
between SN neutrinos and nuclei can be described and which consequences they
have on these various aspects.

Neutrino reactions on nuclei can be described in perturbation theory. The
formalism for the calculation of the various neutrino-nucleus reaction cross
sections has been developed in Ref. [360]. It is based on a state-by-state
evaluation connecting initial and final nuclear states by multipole operators
weighted by an appropriate phase space factor. For the typical low energies
and momentum transfers involved, the description reduces to a nuclear
structure problem where mainly allowed and first-forbidden transitions are of
importance. Furthermore, the energies are too low to allow for charged-current
reactions induced by muon and tau neutrinos. These neutrino families
contribute only to inelastic neutrino-nucleus scattering which is neutrino
flavor blind.

Fermi transitions, which only contribute to elastic neutrino scattering and to
$(\nu_{e},e^{-})$ reactions, can only connect isobaric analog states, where
the transition strength is given by the Fermi sum rule. There has been
significant experimental and theoretical progress in determining the Gamow-
Teller (GT) strength functions in nuclei. Here one distinguishes between GT-
and GT+ strengths, depending on whether a neutron is changed into a proton
(GT-) or a proton into a neutron (GT+). The former contributes to
($\nu_{e},e^{-})$ reactions, while the latter to ($\bar{\nu}_{e},e^{+}$). For
inelastic scattering the relevant strength is GT0 where the identity of the
proton or neutron struck by the neutrino is conserved. The GT- and GT+
strengths can be studied by intermediate energy charge-exchange reactions like
$(p,n)$, $(^{3}$He,$t)$ or $(n,p)$, $(d,^{2}$He), respectively. Recent reviews
on this subject can be found in [361, 362]. As important results these studies
revealed that the GT distributions are strongly fragmented in the final
nucleus and the total GT strengths are noticeably reduced compared to the
expectations from a model of independent particles [363]. Both observations
are related to strong correlations among the nucleons. Such correlations are
accounted for in the nuclear shell model [364] and indeed the experimentally
determined GT strengths are well reproduced by large-scale shell model
calculations [365, 364, 366], if the latter are renormalized by a constant
factor [367, 368, 369]. The GT- distribution shows a noticeable concentration
of the strength in the so-called GT resonance whose centroid is usually also
well reproduced within the quasiparticle random phase approximation (QRPA)
which, however, does not describe the fragmentation as well as the shell model
(see for example [370]).

The shell model has been used to calculate the GT distributions for the about
100 mid-mass nuclei in the iron-nickel-region which dominate the core
composition in the early phase of the collapse [365, 371]. These GT
distributions have then been used to calculate appropriate electron capture
rates, from which the $(\nu_{e},e^{-}$) rates are derived by detailed balance.
The relation of electron capture and $(\nu_{e},e^{-}$) rates have also been
exploited in SN simulations at higher densities. However, as the electron and
neutrino energies are larger at this stage of the collapse, the rates were
derived by QRPA calculations [49, 372], including contributions from forbidden
transitions and most importantly using partial occupation numbers as derived
from shell model Monte Carlo calculations at finite temperatures accounting
for the relevant nuclear correlations [373, 374]. All rates have been
evaluated taking the finite temperature of the environment into account which
is quite important for electron capture at nuclear shell gaps [375, 362, 376,
377, 378]. Electron antineutrinos are much less abundant during collapse than
electron neutrinos. The ($\bar{\nu}_{e},e^{+}$) rates are derived from
positron capture rates by detailed balance.

The only neutral-current cross section measured for nuclei heavier than the
deuteron is the transition from the ground state to the $T=1$ state at 15.11
MeV in 12C [379, 380]. Hence one has to use indirect experimental information
to validate $(\nu,\nu^{\prime}$) cross sections. Due to the relatively low
energies of SN neutrinos, allowed GT0 transitions dominate the cross sections.
As the GT0 strength is closely related to M1 excitations of nuclei, precision
M1 data obtained by inelastic electron scattering are a useful constraint to
the extent that the isoscalar and orbital pieces present in the M1 data can be
neglected. Here three experimental observations help: 1) the isovector
component dominates over the isoscalar piece; 2) the orbital and spin M1
responses are usually experimentally well separated with the major orbital
strength, for example the collective scissors mode, residing at significantly
lower energies (at about 2-4 MeV) than the major spin excitations (at about
7-9 MeV) [381]; 3) the orbital part is strongly suppressed in spherical nuclei
[382]. Detailed studies of spherical $pf$ shell nuclei performed within the
shell model reproduce the M1 data obtained by inelastic electron scattering
very well [383], including the strong dominance of isovector over isoscalar
excitations. We note that results calculated from the shell model GT0 strength
or the experimental or shell model M1 data agreed quite well [383], justifying
the use of the shell model for the calculation of the allowed contributions to
the inelastic neutrino-nucleus cross sections.

In Refs. [384, 385] a hybrid model, in which the allowed transitions were
derived within the shell model and the forbidden within the QRPA approach, has
been proposed to calculate inelastic neutrino-nucleus scattering. This hybrid
model was then extensively used to calculate such cross sections for many mid-
mass (pf shell) nuclei [386]. Calculations of inelastic neutrino-nucleus
scattering for selected nuclei, in which all multipole contributions are
derived within the QRPA, were reported in Refs. [387, 388, 389, 390].

During the collapse the neutrino-nucleus reactions occur in a finite-
temperature environment in which nuclei exist as a thermal ensemble rather
than in the ground state. For small neutrino energies this has a significant
effect on the cross sections which are strongly enhanced compared to those at
$T=0$ [383, 386]. Finite temperature effects are relatively unimportant once
the neutrino energy is large enough to excite the main GT0 strength at 7–9
MeV.

Figure 3.8: Normalized $\nu_{\mathrm{e}}$ number spectra radiated during the
shock breakout burst as seen by a distant observer at rest. Results are shown
for simulations with three different nuclear EoSs. For better comparison of
the strongly time-dependent spectra during this evolution phase, integration
in a window of 8 ms around the peak luminosity was performed. Inelastic
neutrino scattering off nuclei (dashed lines) leads mostly to energy losses of
high-energy neutrinos and thus reduces the high-energy tails of the spectra.
(Figure adapted from [391]).

It has been pointed out [392] that inelastic neutrino-nucleus scattering can
contribute to the shock revival and the thermalization of neutrinos with
matter. A first study performed with relatively simplified neutrino-nucleus