# Time dependence of neutrino quantum kinetics in a core-collapse supernova

Shashank Shalgar ID and Irene Tamborra ID

###### Abstract

Our understanding of neutrino flavor conversion in the supernova core is still
preliminary, despite its likely relevance to the neutrino-driven supernova
mechanism. We present multi-angle and multi-energy numerical simulations of
neutrino quantum kinetics within a spherically symmetric shell in the
proximity of the region of neutrino decoupling. We rely on inputs from a one-
dimensional core-collapse supernova model with a mass of $18.6\ M_{\odot}$ and
find that, at early post-bounce times ($t_{\mathrm{p}b}\lesssim 0.5$ s), no
crossing is present in the angular distribution of the electron neutrino
lepton number and flavor conversion develops due to vacuum mixing. Angular
crossings appear for $t_{\textrm{pb}}\gtrsim 0.5$ s and fast flavor conversion
leads to flavor equipartition, with the spectral energy distribution of
$\nu_{e}$ ($\bar{\nu}_{e}$) and $\nu_{x}$ ($\bar{\nu}_{x}$) becoming
comparable. Notably, flavor equipartition is not a generic outcome of fast
flavor conversion, but rather a consequence of the relatively similar
properties of neutrinos of different flavors characterizing the late accretion
phase. Artificially tweaking the collision term to introduce an electron
lepton number angular crossing for $t_{\mathrm{pb}}\lesssim 0.05$ s, we
observe that flavor equipartition is not achieved. While our findings are
restricted to a specific supernova model, they suggest a rich phenomenology of
flavor conversion in the supernova core as a function of the post-bounce time
which needs to be further explored to assess its impact on the explosion
mechanism.

##  1 Introduction

In neutrino dense astrophysical sources, flavor evolution is significantly
affected by the coherent forward scattering (refraction) of neutrinos with
each other [1, 2]. Neutrino self-interaction, however, makes the neutrino
equations of motion nonlinear and the consequent flavor evolution displays a
rich phenomenology that we are still far from understanding [3, 4, 5, 6, 7].

The modification of neutrino flavor evolution due to refraction with other
neutrinos can be broadly classified into slow [8, 9, 4], fast [10, 11, 12, 13,
14, 6, 7], and collisional [15]. Slow flavor evolution is differentiated from
the fast one by the role of the vacuum frequency. In fact, fast flavor
evolution can manifest in the limit of vanishing vacuum frequency and it is
triggered by a crossing in the electron lepton number (ELN) angular
distribution of neutrinos [14, 13, 16], whereas slow flavor evolution requires
a nonzero vacuum term [9, 17]. Fast flavor evolution, as the name suggests,
occurs over very small time scales—the characteristic frequency being directly
proportional to the number density of neutrinos. On the other hand,
collisional instabilities should be triggered by neutrino collisions with the
medium in the high-density region in the proximity of neutrino decoupling [18,
19, 20, 21], but their relevance on flavor conversion physics within realistic
astrophysical environments remains to be assessed [22, 23].

Neutrinos play a crucial role in the core-collapse supernova mechanism [24,
25], as recently confirmed by multi-dimensional hydrodynamic simulations [26,
27, 28, 29]. Whether neutrino flavor conversion, triggered by any of the
flavor instabilities highlighted above, affects the core collapse mechanism is
the subject of ongoing research [30, 31, 32]. Preliminary work suggests that,
if flavor equilibration should occur as a result of such instabilities, this
would facilitate or hinder the supernova explosion according to the mass of
the collapsing star [31]. Because of the technical challenges linked to the
solution of the neutrino quantum kinetic equations (QKEs) and the vast
difference in the characteristic scales typical of the source hydrodynamics
and the ones entering the neutrino equations of motion, a self-consistent
solution of this problem is not yet available [6, 7]. Nevertheless, it is
possible to search for general trends by solving the QKEs within a simulation
shell, taking into account the interplay among flavor evolution, collisions,
and advection. In this case, a quasi-steady state configuration is found in
the aftermath of flavor conversion [33, 34, 35, 32, 22, 36, 37, 38].

An open question concerns whether flavor equipartition is a general solution
of the QKEs. While recent work seems to point in this direction [39, 36, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49], the boundary conditions as well as the
ELN distribution entering the solution of the QKEs could drastically affect
the final flavor configuration [37, 33, 34]. In this paper, we investigate the
neutrino flavor evolution in spherical symmetry, relying on input from several
post-bounce time snapshots within $1$ s of a one-dimensional hydrodynamic
simulation of a core-collapse supernova with a mass of $18.6\ M_{\odot}$ [50].
We find that, in the early accretion phase ($t_{\textrm{pb}}\lesssim 0.5$ s),
flavor conversion is driven by vacuum mixing; while, in the late accretion
phase ($t_{\textrm{pb}}\gtrsim 0.5$ s), equipartition between the neutrino
spectral energy distributions of $\nu_{e}$ and $\nu_{x}$ as well as
$\bar{\nu}_{e}$ and $\bar{\nu}_{x}$. Yet, we confirm early findings [33, 34,
37] showing that flavor equipartition is not a general flavor outcome for any
given unstable flavor configuration.

This paper is organized as follows. Section 2 introduces the neutrino QKEs and
outlines the approach adopted to model the collision term as well as the
simulation setup. In Sec. 3, we describe the main features of our benchmark
one-dimensional supernova model, compute the neutrino angular distributions of
all flavors employing a multi-energy transport scheme and illustrate as they
evolve as functions of the post-bounce time. Our main findings are presented
in Sec. 4. Finally, we discuss our results in Sec. 5 and conclude in Sec. 6.

##  2 Neutrino quantum kinetics in spherical symmetry

In this section, we introduce the neutrino equations of motion. We also
outline the numerical techniques adopted to solve the neutrino QKEs.

###  2.1 Quantum kinetic equations

Figure 1: Sketch of the simulation shell adopted to solve the QKEs. The polar
angle $\theta$ is defined with respect to the radial direction and it is a
function of radius ($r$) for any given trajectory. The radii $r_{\rm min}$ and
$r_{\rm max}$ denote the innermost and outermost radii adopted in the
simulation.

For simplicity, we consider two neutrino flavors ($\nu_{e}$ and $\nu_{x}$ with
$x=\mu,\tau$), and the corresponding antineutrinos ($\bar{\nu}_{e}$ and
$\bar{\nu}_{x}$). We model our neutrino ensemble in terms of $2\times 2$
density matrices for neutrinos and antineutrinos for each radial location
($r$), polar angle ($\theta$), and energy ($E$), as a function of time ($t$)
and under the assumption of spherical symmetry, see Fig. 1 111Note that the
polar angle $\theta$ is defined with respect to the radial direction at a
given point and it is not the emission angle used in the early papers on
neutrino-bulb model [8, 9, 4].. The diagonal elements of the density matrices
($\rho_{ii}$ with $i=e$ and $x$) represent the neutrino occupation number and
are normalized such $\int_{-1}^{1}\rho(r,\cos\theta,E,t)dE~{}d\cos\theta$
gives the local neutrino number density of flavor $i$ at a given radius
($n_{\nu_{i}}$).

The equations of motion that describe the evolution of the (anti)neutrino
density matrices are:

$\displaystyle i\left(\frac{\partial}{\partial
t}+\vec{v}\cdot\vec{\nabla}\right)\rho(r,\cos\theta,E,t)$ $\displaystyle=$
$\displaystyle[H(r,\cos\theta,E,t),\rho(r,\cos\theta,E,t)]+i\mathcal{C}[\rho]\
,$ (2.1) $\displaystyle i\left(\frac{\partial}{\partial
t}+\vec{v}\cdot\vec{\nabla}\right)\bar{\rho}(r,\cos\theta,E,t)$
$\displaystyle=$
$\displaystyle[\bar{H}(r,\cos\theta,E,t),\bar{\rho}(r,\cos\theta,E,t)]+i\bar{\mathcal{C}}[\bar{\rho}]\
.$ (2.2)

The total derivative on the left hand side of Eqs. 2.1 and 2.2 consists of a
partial derivative with respect to time and the advective term, which is in
spherical symmetry

$\displaystyle\vec{v}\cdot\vec{\nabla}=\cos\theta\frac{\partial}{\partial
r}+\frac{\sin^{2}\theta}{r}\frac{\partial}{\partial\cos\theta}.$ (2.3)

The advective term is such that, in the absence of any term other than the
total derivative, neutrinos travel along a straight line. General relativistic
effects are neglected in Eqs. 2.1 and 2.2, qualitatively they should not
affect the interplay between neutrino flavor conversion and transport effects
on small scales [32].

On the right hand side of Eqs. 2.1 and 2.2, the commutator of the Hamiltonian
and the density matrix takes into account neutrino flavor conversion physics.
The Hamiltonian consists of three components: the vacuum, matter, and self-
interaction terms. The matter term effectively suppresses the mixing angle in
the vacuum term; hence, we choose to adopt an effective vacuum mixing angle,
ignoring the matter term:

$H(r,\cos\theta,E,t)=H_{\textrm{vac}}(E)+H_{\nu\nu}(r,\cos\theta,t)\
\mathrm{and}\
\bar{H}(r,\cos\theta,E,t)=-H_{\textrm{vac}}(E)+H_{\nu\nu}(r,\cos\theta,t)\ .$
(2.4)

The vacuum and self-interaction terms are defined as

$H_{\textrm{vac}}(E)=\frac{\omega_{\textrm{vac}}}{2}\begin{pmatrix}-\cos
2\vartheta_{\textrm{V}}&\sin 2\vartheta_{\textrm{V}}\\\ \sin
2\vartheta_{\textrm{V}}&\cos 2\vartheta_{\textrm{V}}\end{pmatrix}\ ,\\\ $
(2.5)
$H_{\nu\nu}(r,\cos\theta,t)=\sqrt{2}G_{\textrm{F}}\xi\int_{-1}^{1}\left[\rho(r,\cos\theta^{\prime},E,t)-\bar{\rho}(r,\cos\theta^{\prime},E,t)\right]\times(1-\cos\theta\cos\theta^{\prime})dE~{}d\cos\theta^{\prime}\
.$ (2.6)

The vacuum frequency is $\omega_{\textrm{vac}}={\Delta m^{2}}/{2E}$ with
$\Delta m^{2}=2.5\times 10^{-3}$ eV2, and $\vartheta_{\textrm{V}}=10^{-3}$ rad
denotes the matter suppressed mixing angle. The self-interaction Hamiltonian,
unlike the vacuum Hamiltonian, depends on $\theta$ and is obtained integrating
over all momentum modes. The integration over the azimuthal angle results in a
constant factor $2\pi$ that has been absorbed in the definition of the density
matrices. On the other hand, the self-interaction Hamiltonian does not depend
on energy. Following the attenuation method introduced in Ref. [51], we
rescale the strength of neutrino self-interaction by a constant factor
$\zeta=10^{-3}$. The choice of $\zeta$ is such that the self-interaction
strength is always larger than $\omega$ at neutrino decoupling and the quasi-
steady state flavor state configuration is likely not affected, but the time
needed for reaching such a configuration changes with the computational time
being reduced [52, 53].

The collision term on the right hand side of Eqs. 2.1 and 2.2, represented by
$\mathcal{C}$ and $\bar{\mathcal{C}}$, includes three contributions: the
emission term ($\mathcal{C}^{\textrm{emit}}$), the absorption term
($\mathcal{C}^{\textrm{absorb}}$), and the direction changing term
($\mathcal{C}^{\textrm{dir-ch}}$) [54, 55, 28]. For all flavors of neutrinos,
pair production and Bremsstrahlung contribute to the emission and absorption
terms, whereas the beta processes only contribute to electron-type neutrinos
and antineutrinos. The direction changing term is given by the neutral current
interaction between neutrinos and nucleons, which are assumed to be elastic.
The inelastic contribution is negligible for the electron type neutrinos and
thus does not affect the presence of ELN crossings. The collision terms are
defined as follows:

$\displaystyle\mathcal{C}[\rho]$ $\displaystyle=$
$\displaystyle\mathcal{C}_{\textrm{emit}}(r,E)-\mathcal{C}_{\textrm{absorb}}(r,E)\odot\rho(r,\cos\theta,E)+\cos\theta\
\mathcal{C}_{\textrm{ani}}(r,E)\int
d\cos\theta^{\prime}\cos\theta^{\prime}\rho(r,\cos\theta^{\prime},E)$ (2.7)
$\displaystyle+$ $\displaystyle\frac{\mathcal{C}_{\textrm{dir-
ch}}(r,E)}{2}\int
d\cos\theta^{\prime}\left[-\rho(r,\cos\theta,E)+\rho(r,\cos\theta^{\prime},E)\right]\
;$

we use the symbol $\odot$ to denote the elementwise multiplication:

$\displaystyle\mathcal{C}_{\textrm{absorb}}(E)\odot\rho(\cos\theta,E)$
$\displaystyle=$
$\displaystyle\begin{pmatrix}\mathcal{C}_{\textrm{absorb}}^{ee}(E)\rho_{ee}(\cos\theta,E)&\mathcal{C}_{\textrm{absorb}}^{ex}(E)\rho_{ex}(\cos\theta,E)\cr\mathcal{C}_{\textrm{absorb}}^{xe}(E)\rho_{xe}(\cos\theta,E)&\mathcal{C}_{\textrm{absorb}}^{xx}(E)\rho_{xx}(\cos\theta,E)\end{pmatrix}\
.$ (2.8)

The emission and absorption terms are related by Kirchoff’s law for each
energy,
${\mathcal{C}_{\textrm{emit}}(E)}/{\mathcal{C}_{\textrm{absorb}}(E)}=f_{\textrm{FD}}(E)$,
where $f^{\nu_{i}}_{\textrm{FD}}(E)={dn_{\nu_{i}}}/{dE}$ is the Fermi-Dirac
energy distribution. An analogous expression holds for
$\mathcal{C}[\bar{\rho}]$. The terms $\mathcal{C}_{\textrm{ani}}$ and
$\bar{\mathcal{C}}_{\textrm{ani}}$ are included out of completeness and take
into account the anisotropic nature of direction changing neutral current
interactions. The collision terms used in this paper are the multi-energy
extension of the ones presented in Appendix A of Ref. [23], which in turn are
based on Refs. [56, 57].

###  2.2 Simulation setup

In order to solve the QKEs, we use the Julia implementation of the Adam-
Moultan multistep integrator, with adaptive order and adaptive step size for
the time variable 222The length scale characteristic of our neutrino ensemble
is not given by the inverse of the self-interaction potential, rather it is a
combination of the collision term and the self-interaction potential [33, 34].
Yet, it is important to ensure that the time step is much smaller than the
inverse of the neutrino self-interaction strength; we do this by implementing
an adaptive step size. (VCABM solver). The spatial derivatives in the
advective term are calculated using the central difference method. To
discretize the density matrix, we employ $75$ angle bins, $24$ energy bins for
$E\in[0,50]$ MeV, and $150$ radial bins for $r\in[r_{\rm{min}},r_{\rm{max}}]$
(with $r_{\rm{min}}$ ans $r_{\rm{max}}$ being the innermost and outermost
radii of our simulation shell; see Fig. 1).

Our goal is to compute the “quasi-steady state” configuration achieved by the
system 333A quasi-steady state configuration does not coincide with the
solution of the boundary value problem obtained by setting the time derivative
to zero. The quasi-steady state is also not approximately close to the
solution of that boundary value problem. The reason for this counter-intuitive
assertion is that, if Eqs. 2.1 and 2.2 are evolved for a long enough time,
$\rho_{ee}$, $\rho_{xx}$ and $|\rho_{ex}|$ reach a steady state, but the phase
of $\rho_{ex}$ does not converge towards a steady state. This explains why we
need to evolve the system as a function of time instead of attempting to find
a solution to Eqs. 2.1 and 2.2 setting the time derivative to zero.. The
latter should be independent of the initial configuration, but the choice of
the initial configuration does affect the efficiency and numerical stability
of the numerical solution. As pointed out in Refs. [33, 34], the most
efficient way to tackle this issue numerically is to first solve Eqs. 2.1 and
2.2 for $H=\bar{H}=0$ to obtain the “classical steady state” solution (i.e.,
we solve the classical Boltzmann equation). We then use the classical steady
state solution as the initial configuration to solve Eqs. 2.1 and 2.2 for
$H\neq 0$ and $\bar{H}\neq 0$. This approach has two advantages. First, it is
easier to highlight the impact of flavor evolution; second, the initial
classical steady state configuration is not too different from the final
quasi-steady state. We stress, however, that the choice to use the classical
solution as the initial configuration is not unique, in principle it might be
possible to devise a better initial configuration.

For obtaining the classical steady-state solution as well as the quasi-steady
state one, the radial range has to be chosen such that the neutrino spectral
energy distributions coincide with Fermi-Dirac distributions at the minimum
radius ($r_{\textrm{min}}$); whereas the maximum radius ($r_{\textrm{max}}$)
is chosen such that the classical solution has negligible backward flux [23],
see also Fig. 1 and Table 1. These conditions serve as boundary conditions for
Eqs. 2.1 and 2.2.

Table 1: Values of $r_{\textrm{min}}$ and $r_{\textrm{max}}$ chosen for each post-bounce time snapshot in order to capture the radial region of neutrino decoupling (see also Fig. 1). $t_{\rm{pb}}$ (s) |  $r_{\textrm{min}}$ (km) |  $r_{\textrm{max}}$ (km)
---|---|---
$0.05$ | 25 | 150
$0.12$ | 25 | 100
$0.25$ | 22 | 57
$0.5$ | 20 | 35
$0.75$ | 17 | 32
$1$ | 16 | 31

The hydrodynamic and thermodynamic supernova properties employed to compute
the classical and quasi-steady state configurations are extracted from a one-
dimensional core-collapse supernova simulation whose features are illustrated
in Sec. 3.

##  3 Benchmark core-collapse supernova model

We rely on the outputs of a one-dimensional hydrodynamic simulation of a
$18.6\ M_{\odot}$ core-collapse supernova, with $1.4\ M_{\odot}$ gravitational
mass and SFHo nuclear equation of state without muons [50]. In this model, the
effects of proto-neutron star convection have been taken into account through
a mixing-length approximation [3, 58].

We focus on few selected post-bounce times ($t_{\rm pb}=0.05$, $0.12$, $0.25$,
$0.5$, $0.75$, and $1$ s) and rely on static hydrodynamic backgrounds and
thermodynamical quantities for each time snapshot to compute the classical
steady-state neutrino configuration. Figure 2 shows the radial evolution of
the main supernova properties entering the collision term in Eqs 2.1 and 2.2.

Figure 2: Radial profiles of the characteristic quantities extracted from our
core-collapse supernova model (see main text for details) at $t_{\rm
pb}=0.05$, $0.12$, $0.25$, $0.5$, $0.75$, and $1$ s. From top left to bottom
right, each panel represents the radial evolution of the baryon number
density, the electron temperature, the neutrino and electron chemical
potentials, and the neutron and electron fractions, respectively.

Since the neutrino angular distributions are not provided as an output of our
benchmark hydrodynamic simulation, we compute them following the same
procedure illustrated in Sec. III of Ref. [23], i.e. we set $r_{\rm min}$ and
$r_{\max}$ as indicated in Table 1 for each snapshot and solve Eqs. 2.1 and
2.2 for $H=\bar{H}=0$. In contrast to Ref. [23], we compute the angular
distributions of (anti)neutrinos solving Eqs. 2.1 and 2.2 for multiple energy
modes (see Sec. 2.2).

Figure 3: Left: Contour plots of the energy-integrated ELN without flavor
conversion ($\rho_{ee}-\bar{\rho}_{ee}$, classical solution) for
$t_{\textrm{pb}}=0.05$, $0.12$, and $0.25$ s, from top to bottom respectively.
At small radii, the neutrino angular distributions of all flavors are
isotropic. In the proximity of neutrino decoupling, the angular distributions
start becoming forward peaked, but no ELN crossing develops. The white shade
in the top-left corners corresponds to a region with no neutrinos. Right: Same
as the left column, but with neutrino flavor conversion included
($\rho_{ee}-\bar{\rho}_{ee}$, quantum solution); the QKEs have been evolved
for $t=1.25\times 10^{-4}$ s. For $t_{\mathrm{pb}}=0.12$ s, no flavor
evolution is seen, but for $t_{\mathrm{pb}}=0.05$ and $0.25$ s flavor
evolution occurs due to vacuum mixing, despite the absence of ELN crossings.
Figure 4: Same as Fig. 3 but for $t_{\textrm{pb}}=0.5$, $0.75$, and $1$ s,
from top to bottom respectively. In the proximity of neutrino decoupling, the
angular distributions start becoming forward peaked and ELN crossings develop.
The latter are marked by the white contour separating the red and blue
regions. In the proximity of ELN crossings, fast flavor conversion develops as
expected (the QKEs have been evolved for $t=7.5\times 10^{-5}$ s).

The left panels of Figs. 3 and 4 display the contour plot of the energy-
integrated $\rho_{ee}-\bar{\rho}_{ee}$ in the plane spanned by $\cos\theta$
and $r$ in the absence of flavor conversion (i.e., the classical steady state
configuration). One can see that, at small radii (before neutrino decoupling),
the angular distributions of neutrinos and antineutrinos are isotropic with an
overall excess of $\nu_{e}$. At larger radii, (anti)neutrinos start to
decouple from matter entering the free streaming regime. As expected, because
of the overall decrease of baryon density, neutrinos decouple at smaller radii
as the post-bounce time increases. As (anti)neutrinos approach the free-
streaming regime, an ELN crossing forms for $t_{\rm pb}\gtrsim 0.5$ s (visible
from the white contour separating the red from the blue regions in the left
panels of Fig. 4). The presence of an ELN crossing is a necessary condition
for flavor instability and becomes a sufficient condition, if periodic
boundary conditions are assumed [14, 59].

Figure 5 shows the energy-integrated ELN angular distributions, extracted at
$r_{\rm max}$, as functions of $\cos\theta$ for our selection of post-bounce
time snapshots. We can see that ELN crossings appear for $t_{\rm pb}\gtrsim
0.5$ s and are not present at early times. However, there is no monotonic
trend as a function of radius for what concerns the ELN appearance at
different $t_{\rm pb}$. It is worth noticing that Ref. [23] found ELN
crossings for all considered post-bounce time snapshots because it considered
a single-energy solution of Eqs. 2.1 and 2.2 for the sake of simplicity. We
use this set of classical steady state configurations to investigate the
differences in the flavor conversion outcome, as illustrated in the next
section.

Figure 5: Energy integrated ELN distributions as functions of $\cos\theta$ for
our selected post-bounce time snapshots. An ELN crossing appears for $t_{\rm
pb}\gtrsim 0.5$ s; the horizontal black dashed line marks ELN $=0$.

##  4 Time dependent neutrino quantum kinetics

In this section, we present our results on the solution of the QKEs for
different supernova time snapshots. First, we focus on the early accretion
phase ($t_{\rm pb}\lesssim 0.5$ s) and then we investigate the quasi-steady
state configuration during the late accretion phase ($t_{\rm pb}\gtrsim 0.5$
s). Finally, we artificially modify the collision term to investigate whether
flavor equipartition is a generic outcome of flavor conversion.

###  4.1 Early accretion phase: no ELN crossings, minimal flavor conversion

Although we do not find ELN crossings at early times ($t_{\mathrm{pb}}\lesssim
0.5$ s), this does not imply the absence of flavor evolution as displayed in
the right panels of Fig. 3 for $t_{\textrm{pb}}=0.05$ and $0.25$ s. Such
flavor evolution is triggered by the vacuum mass term. Earlier work concluded
that slow flavor evolution should be suppressed at early post-bounce times due
to the large ratio of the self-interaction strength and the vacuum term [60,
61, 62, 63, 64, 65]. However, as recently shown in Ref. [66], if the ELN
distribution is on the verge of developing a crossing, the vacuum term can
induce a flavor instability even when the self-interaction strength is large
(inducing a crossing in the effective ELN distribution). This is exactly what
happens at $t_{\rm pb}=0.05$ and $0.25$ s. We verify this conjecture by
performing the linear stability analysis without rescaling the self-
interaction strength. As shown in Fig. 6, a non-zero growth rate $k$ of the
off-diagonal term of the density matrix is found.

Figure 6: Left: Growth rate of the off-diagonal term of the density matrix for
$t_{\mathrm{pb}}=0.05$ s without rescaling of the self-interaction strength
and adopting the average energy of the spectral energy distributions of
$\nu_{e}$ and $\bar{\nu}_{e}$. The nonzero growth rate highlights the presence
of a flavor instability, despite the absence of an ELN crossing (cf. left
panels of Fig. 3). Right: Same as the left, but for $t_{\mathrm{pb}}=0.25$ s.
The growth rate is much smaller than that for $t_{\mathrm{pb}}=0.05$ s, but it
is non-zero.

The right panels of Fig. 3 show that, unlike for the flavor instability
induced by an ELN crossing (see Sec. 4.2), flavor equipartition does not take
place. This can be attributed to a growth rate that does not dominate over the
advective length scale as in the case of fast flavor instability and the
different shape of the ELN angular distribution (which exhibits a larger
difference between the properties of $\nu_{e}$ and $\bar{\nu}_{e}$ with
respect to later post-bounce times).

###  4.2 Late accretion phase: appearance of ELN crossings and fast flavor
conversion

The right panels of Fig. 4 show the contour plots of the energy-integrated
$\rho_{ee}-\bar{\rho}_{ee}$ at the simulation time $t=7.5\times 10^{-5}$ s in
the plane spanned by $\cos\theta$ and $r$ and in the presence of flavor
conversion for $t_{\textrm{pb}}=0.5$, $0.75$ and $1$ s (from top to bottom
respectively). We find that flavor conversion develops in the proximity of the
ELN crossings (the latter being denoted by the white region in between the red
and blue ones) for $t_{\textrm{pb}}=0.5$, $0.75$ and $1$ s. Flavor conversion
is therefore driven by fast flavor instabilities, as expected from the
preliminary findings of Ref. [23] in the single-energy approximation. In fact,
Ref. [23] found that the growth rate of the fast flavor instability is four
orders of magnitude larger than the collisional instability one for these time
snapshots (cf. their Fig. 6). Reference [23] also found that collisional
instabilities are present within small spatial regions at smaller radii, yet
we do not see any significant flavor conversion linked to them.

Figure 7: Left: Contour plots of the energy-integrated quasi-steady state
configuration of $\rho_{ee}-\rho_{xx}$ (quantum solution) for
$t_{\textrm{pb}}=0.5$, $0.75$, and $1$ s, from top to bottom respectively (the
simulation has been evolved for $t=7.5\times 10^{-5}$ s). Right: Same as the
left column but for antineutrinos. As a result of flavor conversion, flavor
equipartition is achieved at large radii, after neutrino decoupling, for
neutrinos and antineutrinos ($\rho_{ee}\simeq\rho_{xx}$ and
$\bar{\rho}_{ee}\simeq\bar{\rho}_{xx}$; see also Fig. 9).

In order to investigate whether flavor equipartition is achieved as a result
of flavor conversion, Fig. 7 displays the energy-integrated
$\rho_{ee}-\rho_{xx}$ for the quasi-steady-state solution obtained after
flavor conversion for neutrinos (antineutrinos) on the left (right) panels.
Note that, in the region of neutrino trapping, there is an excess of $\nu_{e}$
over $\nu_{x}$ due to the positive chemical potential of $\nu_{e}$; while the
opposite is true for antineutrinos. For all analyzed configurations for
$t_{\rm pb}\gtrsim 0.5$ s, we find flavor equipartition is achieved for
neutrinos and antineutrinos. This is also visible from the left panels of Fig.
9, where the radial evolution of the neutrino number density of the different
flavors after flavor conversion (quantum solution) is compared to the one
obtained neglecting flavor conversion (classical solution). In particular, as
shown in the right panels of Fig. 9 for $t_{\rm pb}=0.5$ (top two panels) and
$1$ s (bottom two panels), the angle-integrated energy distribution of
$\nu_{e}$ ($\bar{\nu}_{e}$) is identical to the one of $\nu_{x}$
($\bar{\nu}_{x}$) in the quasi-steady state configuration and at large radii
($r_{\textrm{max}}$ has been chosen to extract the spectral distributions
plotted in Fig. 9, although any radius after neutrino decoupling shows a
similar trend).

Figure 8: Left:  Angle and energy integrated number densities for neutrinos
(first and third row) and antineutrinos (second and fourth row) as functions
of radius for $t_{\rm pb}=0.5$ (top panels) and $1$ s (bottom panels). The
dashed curves correspond to the classical steady state solution obtained in
the absence of flavor conversion, while the solid curves represent the quasi-
steady state configuration obtained including flavor conversion and extracted
after $1.25\times 10^{-4}$ s of simulation time. After neutrino decoupling
flavor equipartition is achieved and the number densities of the different
flavors become identical to each other. The radius of neutrino decoupling is
marked through vertical lines in each plot and is minimally affected by flavor
conversion. Right: Angle-integrated spectral energy distributions as functions
of the neutrino energy, extracted at $r_{\textrm{max}}$ for the classical
steady state solution (dashed curves) and the quasi-steady state one (solid
curves). Because of flavor conversion, the spectral energy distribution of
$\nu_{e}$ ($\bar{\nu}_{e}$) almost coincides with the one of $\nu_{x}$
($\bar{\nu}_{x}$) for both post-bounce times.  Figure 9: Left:  Angle and
energy integrated number densities for neutrinos (first and third row) and
antineutrinos (second and fourth row) as functions of radius for $t_{\rm
pb}=0.5$ (top panels) and $1$ s (bottom panels). The dashed curves correspond
to the classical steady state solution obtained in the absence of flavor
conversion, while the solid curves represent the quasi-steady state
configuration obtained including flavor conversion and extracted after
$7.5\times 10^{-5}$ s of simulation time. After neutrino decoupling flavor
equipartition is achieved and the number densities of the different flavors
become identical to each other. The radius of neutrino decoupling is marked
through vertical lines in each plot and is minimally affected by flavor
conversion. Right: Angle-integrated spectral energy distributions as functions
of the neutrino energy, extracted at $r_{\textrm{max}}$ for the classical
steady state solution (dashed curves) and the quasi-steady state one (solid
curves). Because of flavor conversion, the spectral energy distribution of
$\nu_{e}$ ($\bar{\nu}_{e}$) almost coincides with the one of $\nu_{x}$
($\bar{\nu}_{x}$) for both post-bounce times.

In the left panels of Fig. 9, the decoupling radii for each neutrino flavor
are marked by vertical lines. The decoupling radius has been computed as the
radius at which the flux factor is equal to $1/3$ [33, 67, 68]. We can see
that the decoupling radius is minimally affected by flavor conversion (cf.
dashed vs. solid lines); in fact, as shown in Ref. [33], the impact of flavor
conversion on the decoupling physics strongly depends on the initial ELN
distribution and how similar the electron and non-electron flavor emission
properties are in the absence of flavor conversion.

###  4.3 Flavor equipartition is not a generic flavor outcome of fast flavor
conversion

The neutrino properties obtained in the late accretion phase tend to be
similar to each other for the different post-bounce times that we have
considered (cf. Figs. 3, 4 and 9). In order to assess whether flavor
equipartition should be expected to be the general flavor outcome of any
quasi-steady state configuration, we increase the emission term of
$\bar{\nu}_{e}$ at $t_{\textrm{pb}}=0.05$ s by $20\%$ to induce an ELN
crossing. Solving the QKEs numerically, we find that flavor equipartition is
not reached in the quasi-steady state solution, as shown in Fig. 10.

This flavor outcome can be explained by looking at the number density of
$\nu_{e}$ and $\bar{\nu}_{e}$ in the region of flavor instability: the two
differ substantially compared to the configurations obtained at later post-
bounce times for which flavor equipartition was found. The observed quasi-
steady state configuration is due to the fact that the collision term tends to
enhance the $\nu_{e}$ number density and suppress the $\nu_{x}$ one. On the
other hand, flavor conversion tends to push $\nu_{e}$ and $\nu_{x}$ towards
each other. As a result, the $\nu_{x}$ number density tends to change more
(moving upwards) in the quasi-steady state configuration than the $\nu_{e}$
one with respect to the classical solution, being the impact of the collision
term smaller for the non-electron flavors. As a consequence, the decoupling
radii are also more largely affected by flavor conversion than observed for
the other flavor configurations (cf. Fig. 4).

Figure 10: Left: Angle and energy integrated neutrino number densities for
each flavor as functions of radius for the modified configuration obtained by
changing the emission term of $\bar{\nu}_{e}$ by $20\%$ at
$t_{\textrm{pb}}=0.05$ s. The quasi-steady state is extracted at $t=2\times
10^{-4}$ s (simulation time), starting from the classical steady state
solution. It can be seen that $\nu_{e}$ does not reach equipartition with
$\nu_{x}$. Right: The same as the left panel, but for antineutrinos.

Figure 11: Left: Angle-integrated spectral energy distributions plotted for
$200$ selected simulation time steps between $1\times 10^{-4}$ and $2\times
10^{-4}$ s, and extracted at $r=125$ km for the modified configuration
obtained by increasing by $20\%$ the emission term of $\bar{\nu}_{e}$ at
$t_{\textrm{pb}}=0.05$ s. The energy spectra fluctuate in time by almost
$10\%$, with the $\nu_{e}$ spectrum going approximately in equipartition with
$\nu_{x}$. However, equipartition in the antineutrino sector is not achieved.
Right: Same as the left panel, but for $r=150$ km. A comparable variation of
$10\%$ in the energy distributions is observed in time once the quasi-steady
state configuration is reached. In this case, the antineutrinos approach
equipartition, but this is not the case for neutrinos.

Figure 11 shows the spectral energy distributions extracted for $200$ selected
time steps at $r=125$ km and $150$ km, in order to investigate the temporal
evolution of the quasi-steady flavor configuration. The neutrino spectra
continue to oscillate in time, changing by as much as $10\%$. However, the
system is approximately in partial equipartition for the neutrino sector and
not for the antineutrino sector at $r=125$ km (cf. left panel). This trend
changes as a function of radius, e.g. the opposite is true for $r=150$ km (cf.
right panel), and in general clearly shows that flavor equipartition is not
achieved for this flavor configuration.

We note that the time over which flavor evolution occurs is much shorter than
the time scale over which the collision term changes. The continued temporal
evolution of the spectral energy distributions suggests that the physically
relevant quantities with potential feedback on the source physics should be
the average of the bands in Fig. 11.

##  5 Discussion

In order to assess the possible impact of our findings on neutrino flavor
conversion physics on the supernova mechanism, Fig. 12 shows the ratio of the
neutrino heating rates obtained relying on the quasi-steady state
configuration when the QKEs are solved with and without the commutator term
encapsulating the flavor conversion physics. The heating rate is defined as

$\displaystyle\dot{\epsilon}$ $\displaystyle=$
$\displaystyle\dot{\epsilon}_{\nu_{e}}+\dot{\epsilon}_{\bar{\nu}_{e}}\quad\textrm{where,}$
(5.1) $\displaystyle\dot{\epsilon}_{\nu_{e}}$ $\displaystyle=$
$\displaystyle\sigma_{0}\left(\frac{1+3g_{\textrm{A}}}{4}\right)\int_{0}^{\infty}dE\left(\frac{E+Q}{{m_{e}c^{2}}}\right)^{2}\sqrt{(E+Q)^{2}-m_{e}^{2}}$
(5.2) $\displaystyle\times$
$\displaystyle\left[1-\left(\frac{m_{e}c^{2}}{E+Q}\right)\right]^{\frac{1}{2}}\left(1-1.01\frac{E}{m_{n}}\right)\left(1-f_{e^{-}}\right)\frac{dn_{\nu_{e}}}{dE}$
$\displaystyle\dot{\epsilon}_{\bar{\nu}_{e}}$ $\displaystyle=$
$\displaystyle\sigma_{0}\left(\frac{1+3g_{\textrm{A}}}{4}\right)\int_{m_{e}+Q}^{\infty}dE\left(\frac{E-Q}{{m_{e}c^{2}}}\right)^{2}\sqrt{(E-Q)^{2}-m_{e}^{2}}$
(5.3) $\displaystyle\times$
$\displaystyle\left[1-\left(\frac{m_{e}c^{2}}{E-Q}\right)\right]^{\frac{1}{2}}\left(1-7.1\frac{E}{m_{p}}\right)\frac{dn_{\bar{\nu}_{e}}}{dE}\
,$

where $E$ denotes the energy or $\nu_{e}$ or $\bar{\nu}_{e}$, $Q=1.2933$ MeV
denotes the $Q$-value of the $\beta$-reaction, $m_{e}=0.511$ MeV is the mass
of the electron, $\sigma_{0}$ is the characteristic neutrino interaction cross
section ($4G_{\textrm{F}}^{2}m_{e}^{2}/\pi\approx 1.7\times 10^{-44}$ cm2),
$(1-f_{e^{-}})$ is the electron Pauli-blocking factor, and $g_{A}=1.27$ is the
axial coupling.

Figure 12 shows that, for all cases except $t_{\textrm{pb}}=0.12$ s (for which
we do not see any flavor conversion), an increase in the heating rate by
$15$–$25\%$ occurs. This is not surprising since the flavor evolution
increases the number density of electron type neutrinos in the high energy
tail of the spectral distribution, and a larger heating rate is expected for
larger neutrino energy. While we do not compute the neutrino cooling rate and
do not take into account the feedback that the neutrino flavor conversion
physics could have on the supernova hydrodynamic background, our results
suggest a non-trivial impact of neutrino flavor conversion physics on the
supernova mechanism, supporting earlier parametric studies [31, 30, 32].

Figure 12: Left: Ratio between the neutrino heating rates obtained adopting
the quasi-steady state angle-integrated neutrino distributions after solving
the QKEs and the ones computed from the classical solution (i.e., neglecting
flavor conversion) for early times, $t_{\rm{pb}}=0.05$, $0.12$ and $0.25$ s.
At radii larger than the one of neutrino decoupling, the heating rate is
increased by $15$–$25\%$, except for the case of $t_{\rm{pb}}=0.12$ s. Right:
Same as the left panel but for late accretion times, $t_{\rm{pb}}=0.5$, $0.75$
and $1$ s.

Our findings should be considered with caution as they have been obtained
relying on a spherically symmetric core-collapse supernova model. While we
observe a non-trivial appearance of ELN crossings as a function of the post-
bounce time in the classical solution of the Boltzmann equations, multi-
dimensional core-collapse supernova simulations show evidence of large scale
asymmetries in the neutrino emission (e.g. due to the LESA instability [69])
and the location of flavor instabilities may be impacted [70].

In order to facilitate the numerical solution of the QKEs, we choose to
attenuate the self-interaction strength by a constant factor $10^{-3}$ (see
Eq. 2.6), nevertheless for all pot-bounce times we have that $\mu\gg\omega$ at
neutrino decoupling. When flavor equipartition is reached, it is unlikely that
the rescaling of the self-interaction strength can affect the results.
However, for the earlier accretion phase ($t_{\textrm{pb}}\lesssim 0.5$ s),
flavor equipartition is not reached, and it is possible that this scaling
might slightly affect the exact final flavor outcome and the time needed to
reach a quasi-steady state solution.

The energy-dependent solution of the QKEs has consequences on the development
of ELN crossings (cf. the corresponding single-energy solution presented in
Ref. [23]) and therefore the presence of flavor instabilities; yet, for all
post-bounce times, we find that a quasi-steady state configuration is
achieved—cf. also Refs. [71, 51]. These findings support recent attempts
aiming at forecasting the quasi-steady state flavor configuration due to
flavor conversion or designing methods to incorporate such physics into
neutrino-radiation-hydrodynamic simulations [41, 42, 36, 72, 35, 73, 74]; on
the other hand, caution should be adopted in the forecast of the quasi-steady
state neutrino configuration because of the dependence of the latter on the
boundary conditions of the simulation box and features of the ELN distribution
[37, 41, 34].

We find that flavor equipartition is obtained for $t_{\rm{pb}}\gtrsim 0.5$ s.
Such results may seem to confirm recent work in this direction [39, 36, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49]. However, as shown through the flavor
configuration obtained by tweaking the collision term in Sec. 4.3, we stress
that flavor equipartition depends on the initial conditions and should not be
considered a general outcome of global flavor simulations [37, 33, 34].

Similar to Ref. [22], we explore the flavor conversion physics for a range of
neutrino flavor configurations linked to different post-bounce times. While
our results are in general agreement with the ones of Ref. [22], we scan a
broader range of post-bounce times highlighting time-dependent features of the
flavor outcome. Moreover, differently from Ref. [22], we do not artificially
tweak the electron abundance to favor the appearance of ELN crossings; this
has implications on the relative relevance between collisional and fast flavor
instabilities. We also confirm the preliminary findings of Refs. [23, 70, 22],
where it was concluded that collisional instabilities do not affect the flavor
composition significantly.

##  6 Conclusions

Understanding the neutrino flavor distribution resulting from flavor
conversion occurring in the dense core of a supernova is crucial in order to
investigate the impact of this physics on the supernova explosion mechanism.
We numerically solve the neutrino quantum kinetic equations (QKEs) in a multi-
energy and multi-angle framework and within a spherically symmetric shell,
tracking neutrino decoupling.

We solve the QKEs adopting as inputs static hydrodynamic backgrounds extracted
from a one-dimensional core-collapse supernova simulation with a mass of
$18.6\ M_{\odot}$ at post-bounce times $t_{\rm{pb}}=0.05$, $0.12$, $0.25$,
$0.5$, $0.75$, and $1$ s. First, neglecting flavor conversion, we compute the
angular distributions of neutrinos and antineutrinos, following Ref. [23].
Differently from Ref. [23] and because of the multi-energy numerical solution
of the QKEs, we find that crossings in the electron neutrino lepton number
(ELN) angular distributions appear for $t_{\rm{pb}}\gtrsim 0.5$ s, but not at
earlier post-bounce times.

We adopt the classical steady-state solution to solve the QKEs, this time
taking into account neutrino flavor conversion. Fast flavor conversion is
responsible for making the spectral energy distribution of $\nu_{e}$
($\bar{\nu}_{e}$) equal to the one of $\nu_{x}$ ($\bar{\nu}_{x}$) for
$t_{\rm{pb}}\gtrsim 0.5$ s, i.e. flavor equipartition is reached. Despite the
changes in the neutrino energy and angular distributions of all flavors, we
find that the flavor-dependent neutrino decoupling radii are negligibly
affected for the investigated flavor configurations; this is because of the
relatively similar properties between the electron and the non-electron
flavors characterizing the late accretion phase that facilitate flavor
equipartition, but overall negligibly affect the energy and angle integrated
flavor-dependent number density at decoupling. On the other hand, ELN
crossings are not found for $t_{\rm{pb}}\lesssim 0.5$ s, but flavor conversion
develops because of vacuum mixing, which effectively induces a crossing in the
ELN angular distributions.

Remarkably, flavor equipartition is not a generic outcome of fast flavor
conversion, but rather the result of the interplay between collisions, flavor
conversion and advection, when the neutrino emission properties of the
electron and non-electron flavors are reasonably comparable to each other.
Modifying the collision term artificially for a time snapshot extracted at
$t_{\rm{pb}}\lesssim 0.5$ s to induce a crossing in the ELN angular
distribution, we find that flavor equipartition is not achieved. This is
because the differences in the emission properties of different flavors are
larger in the early accretion phase and the interplay between flavor
conversion and collisions cannot equilibrate the electron and non-electron
flavors.

Our findings should be expanded to draw conclusions on the flavor conversion
phenomenology expected in a broader range of core-collapse supernova models.
Moreover, a self-consistent implementation of flavor conversion in supernova
hydrodynamic simulations may dynamically change the neutrino emission
properties as a function of the post-bounce time, in turn possibly affecting
the flavor conversion phenomenology. Despite its intrinsic limitations, our
work highlights a very interesting, non-trivial flavor phenomenology in core-
collapse supernovae, calling for further investigation in order to assess its
impact on the neutrino-driven explosion mechanism and nucleosynthesis.

## Acknowledgments

This project has received support from the Danmarks Frie Forskningsfond
(Project No. 8049-00038B), the European Union (ERC, ANET, Project No.
101087058), and the Deutsche Forschungsgemeinschaft through
Sonderforschungbereich SFB 1258 “Neutrinos and Dark Matter in Astro- and
Particle Physics” (NDM). Views and opinions expressed are those of the authors
only and do not necessarily reflect those of the European Union or the
European Research Council. Neither the European Union nor the granting
authority can be held responsible for them.

## References

  * [1] J. T. Pantaleone, _Neutrino flavor evolution near a supernova’s core_ , _Phys. Lett. B_ 342 (1995) 250–256, [astro-ph/9405008].
  * [2] P. Langacker and J. Liu, _Standard Model contributions to the neutrino index of refraction in the early universe_ , _Phys. Rev. D_ 46 (1992) 4140–4160, [hep-ph/9206209].
  * [3] A. Mirizzi, I. Tamborra, H.-T. Janka, N. Saviano, K. Scholberg, R. Bollig et al., _Supernova Neutrinos: Production, Oscillations and Detection_ , _Riv. Nuovo Cim._ 39 (2016) 1–112, [1508.00785].
  * [4] H. Duan, G. M. Fuller and Y.-Z. Qian, _Collective Neutrino Oscillations_ , _Ann. Rev. Nucl. Part. Sci._ 60 (2010) 569–594, [1001.2799].
  * [5] S. Chakraborty, R. Hansen, I. Izaguirre and G. G. Raffelt, _Collective neutrino flavor conversion: Recent developments_ , _Nucl. Phys. B_ 908 (2016) 366–381, [1602.02766].
  * [6] I. Tamborra and S. Shalgar, _New Developments in Flavor Evolution of a Dense Neutrino Gas_ , _Ann. Rev. Nucl. Part. Sci._ 71 (2021) 165–188, [2011.01948].
  * [7] S. Richers and M. Sen, _Fast Flavor Transformations_ , pp. 1–17.  Springer Nature Singapore, Singapore, 2022.  [2207.03561].  10.1007/978-981-15-8818-1_125-1.
  * [8] H. Duan, G. M. Fuller and Y.-Z. Qian, _Collective neutrino flavor transformation in supernovae_ , _Phys. Rev. D_ 74 (2006) 123004, [astro-ph/0511275].
  * [9] H. Duan, G. M. Fuller, J. Carlson and Y.-Z. Qian, _Simulation of Coherent Non-Linear Neutrino Flavor Transformation in the Supernova Environment. 1. Correlated Neutrino Trajectories_ , _Phys. Rev. D_ 74 (2006) 105014, [astro-ph/0606616].
  * [10] R. F. Sawyer, _Speed-up of neutrino transformations in a supernova environment_ , _Phys. Rev. D_ 72 (2005) 045003, [hep-ph/0503013].
  * [11] R. F. Sawyer, _The multi-angle instability in dense neutrino systems_ , _Phys. Rev. D_ 79 (2009) 105003, [0803.4319].
  * [12] R. F. Sawyer, _Neutrino cloud instabilities just above the neutrino sphere of a supernova_ , _Phys. Rev. Lett._ 116 (2016) 081101, [1509.03323].
  * [13] S. Chakraborty, R. S. Hansen, I. Izaguirre and G. G. Raffelt, _Self-induced neutrino flavor conversion without flavor mixing_ , _JCAP_ 03 (2016) 042, [1602.00698].
  * [14] I. Izaguirre, G. G. Raffelt and I. Tamborra, _Fast Pairwise Conversion of Supernova Neutrinos: A Dispersion-Relation Approach_ , _Phys. Rev. Lett._ 118 (2017) 021101, [1610.01612].
  * [15] L. Johns, _Collisional Flavor Instabilities of Supernova Neutrinos_ , _Phys. Rev. Lett._ 130 (2023) 191001, [2104.11369].
  * [16] T. Morinaga, _Fast neutrino flavor instability and neutrino flavor lepton number crossings_ , _Phys. Rev. D_ 105 (2022) L101301, [2103.15267].
  * [17] S. Hannestad, G. G. Raffelt, G. Sigl and Y. Y. Y. Wong, _Self-induced conversion in dense neutrino gases: Pendulum in flavour space_ , _Phys. Rev. D_ 74 (2006) 105010, [astro-ph/0608695].
  * [18] L. Johns and Z. Xiong, _Collisional instabilities of neutrinos and their interplay with fast flavor conversion in compact objects_ , _Phys. Rev. D_ 106 (2022) 103029, [2208.11059].
  * [19] I. Padilla-Gay, I. Tamborra and G. G. Raffelt, _Neutrino fast flavor pendulum. II. Collisional damping_ , _Phys. Rev. D_ 106 (2022) 103031, [2209.11235].
  * [20] Y.-C. Lin and H. Duan, _Collision-induced flavor instability in dense neutrino gases with energy-dependent scattering_ , _Phys. Rev. D_ 107 (2023) 083034, [2210.09218].
  * [21] Z. Xiong, M.-R. Wu, G. Martínez-Pinedo, T. Fischer, M. George, C.-Y. Lin et al., _Evolution of collisional neutrino flavor instabilities in spherically symmetric supernova models_ , _Phys. Rev. D_ 107 (2023) 083016, [2210.08254].
  * [22] H. Nagakura and M. Zaizen, _Basic characteristics of neutrino flavor conversions in the postshock regions of core-collapse supernova_ , _Phys. Rev. D_ 108 (2023) 123003, [2308.14800].
  * [23] S. Shalgar and I. Tamborra, _Do neutrinos become flavor unstable due to collisions with matter in the supernova decoupling region?_ , _Phys. Rev. D_ 109 (2024) 103011, [2307.10366].
  * [24] S. A. Colgate and R. H. White, _The Hydrodynamic Behavior of Supernovae Explosions_ , _Astrophys. J._ 143 (1966) 626.
  * [25] H. A. Bethe and J. R. Wilson, _Revival of a stalled supernova shock by neutrino heating_ , _Astrophys. J._ 295 (1985) 14–23.
  * [26] H. T. Janka, T. Melson and A. Summa, _Physics of Core-Collapse Supernovae in Three Dimensions: a Sneak Preview_ , _Ann. Rev. Nucl. Part. Sci._ 66 (2016) 341–375, [1602.05576].
  * [27] B. Müller, _Hydrodynamics of core-collapse supernovae and their progenitors_ , _Astrophysics_ 6 (2020) 3, [2006.05083].
  * [28] A. Mezzacappa, E. Endeve, O. E. Bronson Messer and S. W. Bruenn, _Physical, numerical, and computational challenges of modeling neutrino transport in core-collapse supernovae_ , _Liv. Rev. Comput. Astrophys._ 6 (2020) 4, [2010.09013].
  * [29] A. Burrows and D. Vartanyan, _Core-Collapse Supernova Explosion Theory_ , _Nature_ 589 (2021) 29–39, [2009.14157].
  * [30] J. Ehring, S. Abbar, H.-T. Janka, G. G. Raffelt and I. Tamborra, _Fast neutrino flavor conversion in core-collapse supernovae: A parametric study in 1D models_ , _Phys. Rev. D_ 107 (2023) 103034, [2301.11938].
  * [31] J. Ehring, S. Abbar, H.-T. Janka, G. G. Raffelt and I. Tamborra, _Fast Neutrino Flavor Conversions Can Help and Hinder Neutrino-Driven Explosions_ , _Phys. Rev. Lett._ 131 (2023) 061401, [2305.11207].
  * [32] H. Nagakura, _Roles of Fast Neutrino-Flavor Conversion on the Neutrino-Heating Mechanism of Core-Collapse Supernova_ , _Phys. Rev. Lett._ 130 (2023) 211401, [2301.10785].
  * [33] S. Shalgar and I. Tamborra, _Neutrino decoupling is altered by flavor conversion_ , _Phys. Rev. D_ 108 (2023) 043006, [2206.00676].
  * [34] S. Shalgar and I. Tamborra, _Neutrino flavor conversion, advection, and collisions: Toward the full solution_ , _Phys. Rev. D_ 107 (2023) 063025, [2207.04058].
  * [35] H. Nagakura and M. Zaizen, _Connecting small-scale to large-scale structures of fast neutrino-flavor conversion_ , _Phys. Rev. D_ 107 (2023) 063033, [2211.01398].
  * [36] Z. Xiong, M.-R. Wu, M. George and C.-Y. Lin, _Robust integration of fast flavor conversions in classical neutrino transport_ , 2403.17269.
  * [37] M. Cornelius, S. Shalgar and I. Tamborra, _Perturbing fast neutrino flavor conversion_ , _JCAP_ 02 (2024) 038, [2312.03839].
  * [38] S. Shalgar, I. Padilla-Gay and I. Tamborra, _Neutrino propagation hinders fast pairwise flavor conversions_ , _JCAP_ 06 (2020) 048, [1911.09110].
  * [39] Z. Xiong, M.-R. Wu, M. George, C.-Y. Lin, N. K. Largani, T. Fischer et al., _Fast neutrino flavor conversions in a supernova: emergence, evolution, and effects_ , 2402.19252.
  * [40] J. D. Martin, J. Carlson, V. Cirigliano and H. Duan, _Fast flavor oscillations in dense neutrino media with collisions_ , _Phys. Rev. D_ 103 (2021) 063001, [2101.01278].
  * [41] M. Zaizen and H. Nagakura, _Characterizing quasisteady states of fast neutrino-flavor conversion by stability and conservation laws_ , _Phys. Rev. D_ 107 (2023) 123021, [2304.05044].
  * [42] M. Zaizen and H. Nagakura, _Simple method for determining asymptotic states of fast neutrino-flavor conversion_ , _Phys. Rev. D_ 107 (2023) 103022, [2211.09343].
  * [43] Z. Xiong, M.-R. Wu, S. Abbar, S. Bhattacharyya, M. George and C.-Y. Lin, _Evaluating approximate asymptotic distributions for fast neutrino flavor conversions in a periodic 1D box_ , _Phys. Rev. D_ 108 (2023) 063003, [2307.11129].
  * [44] E. Grohs, S. Richers, S. M. Couch, F. Foucart, J. P. Kneller and G. C. McLaughlin, _Neutrino fast flavor instability in three dimensions for a neutron star merger_ , _Phys. Lett. B_ 846 (2023) 138210, [2207.02214].
  * [45] S. Richers, D. Willcox and N. Ford, _Neutrino fast flavor instability in three dimensions_ , _Phys. Rev. D_ 104 (2021) 103023, [2109.08631].
  * [46] S. Richers, H. Duan, M.-R. Wu, S. Bhattacharyya, M. Zaizen, M. George et al., _Code comparison for fast flavor instability simulations_ , _Phys. Rev. D_ 106 (2022) 043011, [2205.06282].
  * [47] S. Bhattacharyya and B. Dasgupta, _Fast Flavor Depolarization of Supernova Neutrinos_ , _Phys. Rev. Lett._ 126 (2021) 061302, [2009.03337].
  * [48] S. Bhattacharyya and B. Dasgupta, _Elaborating the ultimate fate of fast collective neutrino flavor oscillations_ , _Phys. Rev. D_ 106 (2022) 103039, [2205.05129].
  * [49] M.-R. Wu, M. George, C.-Y. Lin and Z. Xiong, _Collective fast neutrino flavor conversions in a 1D box: Initial conditions and long-term evolution_ , _Phys. Rev. D_ 104 (2021) 103003, [2108.09886].
  * [50] “Garching core-collapse supernova data archive.” https://wwwmpa.mpa-garching.mpg.de/ccsnarchive/data/Bollig2016/.
  * [51] H. Nagakura and M. Zaizen, _Time-Dependent and Quasisteady Features of Fast Neutrino-Flavor Conversion_ , _Phys. Rev. Lett._ 129 (2022) 261101, [2206.04097].
  * [52] S. Shalgar, I. Tamborra and M. Bustamante, _Core-collapse supernovae stymie secret neutrino interactions_ , _Phys. Rev. D_ 103 (2021) 123008, [1912.09115].
  * [53] I. Padilla-Gay, S. Shalgar and I. Tamborra, _Multi-Dimensional Solution of Fast Neutrino Conversions in Binary Neutron Star Merger Remnants_ , _JCAP_ 01 (2021) 017, [2009.01843].
  * [54] A. Burrows, S. Reddy and T. A. Thompson, _Neutrino opacities in nuclear matter_ , _Nucl. Phys. A_ 777 (2006) 356–394, [astro-ph/0404432].
  * [55] H.-T. Janka, _Explosion Mechanisms of Core-Collapse Supernovae_ , _Ann. Rev. Nucl. Part. Sci._ 62 (2012) 407–451, [1206.2503].
  * [56] T. A. Thompson, _Topics in the theory of core-collapse supernovae_ , Ph.D. thesis, The University of Arizona., 2002.
  * [57] E. O’Connor, _An Open-Source Neutrino Radiation Hydrodynamics Code for Core-Collapse Supernovae_ , _Astrophys. J. Suppl._ 219 (2015) 24, [1411.7058].
  * [58] A. Weiss, W. Hillebrandt, H. C. Thomas and H. Ritter, _Cox and Giuli’s Principles of Stellar Structure_.  Cambridge Scientific Publishers LTD, Cambridge, UK, 2006.
  * [59] T. Morinaga and S. Yamada, _Linear stability analysis of collective neutrino oscillations without spurious modes_ , _Phys. Rev. D_ 97 (2018) 023024, [1803.05913].
  * [60] A. Esteban-Pretel, A. Mirizzi, S. Pastor, R. Tomàs, G. G. Raffelt, P. D. Serpico et al., _Role of dense matter in collective supernova neutrino transformations_ , _Phys. Rev. D_ 78 (2008) 085012, [0807.0659].
  * [61] S. Sarikas, I. Tamborra, G. G. Raffelt, L. Huedepohl and H.-T. Janka, _Supernova neutrino halo and the suppression of self-induced flavor conversion_ , _Phys. Rev. D_ 85 (2012) 113007, [1204.0971].
  * [62] N. Saviano, S. Chakraborty, T. Fischer and A. Mirizzi, _Stability analysis of collective neutrino oscillations in the supernova accretion phase with realistic energy and angle distributions_ , _Phys. Rev. D_ 85 (2012) 113002, [1203.1484].
  * [63] S. Chakraborty, T. Fischer, A. Mirizzi, N. Saviano and R. Tomàs, _No collective neutrino flavor conversions during the supernova accretion phase_ , _Phys. Rev. Lett._ 107 (2011) 151101, [1104.4031].
  * [64] S. Chakraborty, T. Fischer, A. Mirizzi, N. Saviano and R. Tomàs, _Analysis of matter suppression in collective neutrino oscillations during the supernova accretion phase_ , _Phys. Rev. D_ 84 (2011) 025002, [1105.1130].
  * [65] S. Chakraborty, G. Raffelt, H.-T. Janka and B. Müller, _Supernova deleptonization asymmetry: Impact on self-induced flavor conversion_ , _Phys. Rev. D_ 92 (2015) 105002, [1412.0670].
  * [66] P. Dedin Neto, I. Tamborra and S. Shalgar, _Fast Conversion of Neutrinos: Energy Dependence of Flavor Instabilities_ , 2312.06556.
  * [67] I. Tamborra, L. Huedepohl, G. Raffelt and H.-T. Janka, _Flavor-dependent neutrino angular distribution in core-collapse supernovae_ , _Astrophys. J._ 839 (2017) 132, [1702.00060].
  * [68] M.-R. Wu, I. Tamborra, O. Just and H.-T. Janka, _Imprints of neutrino-pair flavor conversions on nucleosynthesis in ejecta from neutron-star merger remnants_ , _Phys. Rev. D_ 96 (2017) 123015, [1711.00477].
  * [69] I. Tamborra, F. Hanke, H.-T. Janka, B. Müller, G. G. Raffelt and A. Marek, _Self-sustained asymmetry of lepton-number emission: A new phenomenon during the supernova shock-accretion phase in three dimensions_ , _Astrophys. J._ 792 (2014) 96, [1402.5418].
  * [70] R. Akaho, J. Liu, H. Nagakura, M. Zaizen and S. Yamada, _Collisional and fast neutrino flavor instabilities in two-dimensional core-collapse supernova simulation with Boltzmann neutrino transport_ , _Phys. Rev. D_ 109 (2024) 023012, [2311.11272].
  * [71] C. Kato, H. Nagakura and M. Zaizen, _Flavor conversions with energy-dependent neutrino emission and absorption_ , _Phys. Rev. D_ 108 (2023) 023006, [2303.16453].
  * [72] S. Abbar, M.-R. Wu and Z. Xiong, _Application of neural networks for the reconstruction of supernova neutrino energy spectra following fast neutrino flavor conversions_ , _Phys. Rev. D_ 109 (2024) 083019, [2401.17424].
  * [73] H. Nagakura, L. Johns and M. Zaizen, _Bhatnagar-Gross-Krook subgrid model for neutrino quantum kinetics_ , _Phys. Rev. D_ 109 (2024) 083013, [2312.16285].
  * [74] S. Abbar, M.-R. Wu and Z. Xiong, _Physics-informed neural networks for predicting the asymptotic outcome of fast neutrino flavor conversions_ , _Phys. Rev. D_ 109 (2024) 043024, [2311.15656].