# Some Recent Results on Neutrino Oscillations in Hypercritical Accretion

J. D. Uribe<EMAIL_ADDRESS>ICRANet, P.zza della Repubblica 10, I–65122
Pescara, Italy  Dipartimento di Fisica, Sapienza Università di Roma, P.le Aldo
Moro 5, I–00185 Rome, Italy  J. A. Rueda<EMAIL_ADDRESS>ICRANet, P.zza
della Repubblica 10, I–65122 Pescara, Italy  ICRANet-Ferrara, Dipartimento di
Fisica e Scienze della Terra, Università degli Studi di Ferrara, Via Saragat
1, I–44122 Ferrara, Italy  Dipartimento di Fisica e Scienze della Terra,
Università degli Studi di Ferrara, Via Saragat 1, I–44122 Ferrara, Italy
INAF, Istituto di Astrofisica e Planetologia Spaziali, Via Fosso del Cavaliere
100, 00133 Rome, Italy

###### Abstract

The study of neutrino flavour oscillations in astrophysical sources has been
boosted in the last two decades thanks to achievements in experimental
neutrino physics and observational astronomy. We here discuss two cases of
interest in the modelling of short and long gamma-ray bursts (GRBs):
hypercritical, i.e. highly super-Eddington spherical/disk accretion onto a
neutron star (NS)/black hole (BH). We show that in both systems the ambient
conditions of density and temperature imply the occurrence of neutrino flavour
oscillations, with a relevant role of neutrino self-interactions.

##  1 Introduction

The occurrence of neutrino flavour oscillations has been, undoubtedly,
experimentally demonstrated [1]. Of special interest here is that, it has
become clear in recent years that for the analysis of neutrino oscillations in
matter, e.g. the Mikheyev-Smirnov-Wolfenstein (MSW) effect [2, 3], refractive
effects of neutrinos on themselves, due to the neutrino self-interaction
potential, are essential.

Over the last two decades, the achievements of experimental neutrino physics
and the constant development of observational astronomy, have caused an
increasing interest in the study of the occurrence of neutrino flavour
oscillations in astrophysical sources. Although the bulk of astrophysical
analyses has been limited to supernova (SN) neutrinos, flavour oscillations
may also occur in other relativistic astrophysics sources. In particular, as
we are showing here, this phenomenon is expected to occur in known scenarios
of short- and long-duration GRBs.

The emergent picture of GRBs is that both, short-duration and long-duration
GRBs, originate in binary systems (see, e.g., [4]). Short bursts are
associated with mergers of NS-NS and/or NS-BH binaries. For this case, the
role of neutrino-antineutrino ($\nu\bar{\nu}$) annihilation leading to the
formation of an electron-positron plasma ($e^{-}e^{+}$) has been introduced
[5] (for general relativistic effects, see [6]). For long bursts, it has been
introduced binary progenitor composed of a carbon-oxygen star (COcore) and a
companion NS [7, 8]. These binaries can form in an evolutionary path including
a first SN explosion, common-envelope phases, tidal interactions and mass loss
[9]. The GRB is expected to occur when the binary experiences the second SN,
i.e. the one of the COcore. Part of the ejected matter produces a
hypercritical accretion (i.e. highly super-Eddington) process onto the NS
companion. The NS then reaches its critical mass for gravitational collapse,
hence forming a rotating BH [10, 11]. These systems have been called binary-
driven hypernovae (BdHNe), and they lead to a variety of observable emissions
from the X-rays all the way to high-energy gamma-rays (see e.g. [12, 13] for
details).

As we are showing below, a key ingredient in the above systems is a copious
emission of neutrinos during the hypercritical accretion. The high neutrino
and matter density involved suggests that a study of neutrinos oscillations
may lead to new neutrino physics in these sources. Our aim here is to compile
the main results of neutrino oscillations in the physical conditions expected
in the above scenarios of GRBs.

##  2 Neutrino Oscillations

To study the flavour evolution of neutrinos within a particular system, a
Hamiltonian governing neutrino oscillation must be set up. The relative
strength of the potentials appearing in such Hamiltonian depends on four
elements: geometry, mass content, neutrino content and neutrino mass
hierarchy. Geometry refers to the nature of net neutrino fluxes and possible
gravitational effects. Mass and neutrino content refers to the distribution of
leptons of each flavour $(e,\mu,\tau)$ present in the medium. Finally, mass
hierarchy refers to the relative values of the masses $m_{1},m_{2},m_{3}$ for
each neutrino mass eigenstates. The equations that govern the evolution of an
ensemble of mixed neutrinos are the quantum Liouville equations

$\displaystyle i\dot{\rho}_{\mathbf{p}}=[H_{\mathbf{p}},\rho_{\mathbf{p}}]$
(1a) $\displaystyle
i\dot{\bar{\rho}}_{\mathbf{p}}=[\bar{H}_{\mathbf{p}},\bar{\rho}_{\mathbf{p}}]$
(1b)

The Hamiltonian is (see, e.g., [14, 15] and references therein)

In these equations $\rho_{\mathbf{p}}$ ($\bar{\rho}_{\mathbf{p}}$) is the
matrix of occupation numbers $(\rho_{\mathbf{p}})_{ij}=\langle
a^{\dagger}_{j}a_{i}\rangle_{\mathbf{p}}$ for neutrinos
($(\bar{\rho}_{\mathbf{p}})_{ij}=\langle\bar{a}^{\dagger}_{i}\bar{a}_{j}\rangle_{\mathbf{p}}$
for antineutrinos), for each momentum $\mathbf{p}$ and flavours $i,j$. The
diagonal elements are the distribution functions
$f_{\nu_{i}\left(\bar{\nu}_{i}\right)}\left(\mathbf{p}\right)$ such that their
integration over the momentum space gives the neutrino number density
$n_{\nu_{i}}$ of a determined flavour $i$. The off-diagonal elements provide
information about the _overlapping_ between the two neutrino flavours.
$\Omega_{\mathbf{p}}$ is the matrix of vacuum oscillation frequencies,
$l_{\mathbf{p}}$ and $\bar{l}_{\mathbf{p}}$ are matrices of occupation numbers
for charged leptons built in a similar way to the neutrino matrices, and
$\mathbf{v}_{\mathbf{p}}=\mathbf{p}/p$ is the velocity of a particle with
momentum $\mathbf{p}$ (either neutrino or charged lepton). Since the matter in
the accretion zone is composed by protons, neutrons, electrons and positrons,
$\nu_{e}$ and $\bar{\nu}_{e}$ interact with matter by both charged and neutral
currents, while $\nu_{\mu}$, $\nu_{\tau}$, $\bar{\nu}_{\mu}$ and
$\bar{\nu}_{\tau}$ interact only by neutral currents. Therefore, the behavior
of these states can be clearly divided into electronic and non-electronic
allowing us to use the two-flavour approximation. Within this approximation,
$\rho$ in Eq. (1) can be written in terms of Pauli matrices and the
polarization vector $\mathsf{P}_{\mathbf{p}}$ as:

$\small\rho_{\mathbf{p}}=\left(\begin{array}[]{cc}\rho_{ee}&\rho_{ex}\\\
\rho_{xe}&\rho_{xx}\\\
\end{array}\right)_{\mathbf{p}}=\frac{1}{2}\left(f_{\mathbf{p}}\mathbb{1}+\mathsf{P}_{\mathbf{p}}\cdot\vec{\sigma}\right),$
(3)

where $f_{\mathbf{p}}={\rm
Tr}[\rho_{\mathbf{p}}]=f_{\nu_{e}}(\mathbf{p})+f_{\nu_{x}}(\mathbf{p})$ is the
sum of the distribution functions for $\nu_{e}$ and $\nu_{x}$. Note that the
$z$ component of the polarization vector obeys

$\mathsf{P}^{z}_{\mathbf{p}}=f_{\nu_{e}}(\mathbf{p})-f_{\nu_{x}}(\mathbf{p}).$
(4)

Hence, this component tracks the fractional flavour composition of the system
and appropriately normalizing $\rho_{\mathbf{p}}$ allows to define a survival
and mixing probability

$\displaystyle
P_{\nu_{e}\leftrightarrow\nu_{e}}=\frac{1}{2}\left(1+\mathsf{P}^{z}_{\mathbf{p}}\right),$
(5a) $\displaystyle
P_{\nu_{e}\leftrightarrow\nu_{x}}=\frac{1}{2}\left(1-\mathsf{P}^{z}_{\mathbf{p}}\right).$
(5b)

On the other hand, the Hamiltonian can be written as a sum of three
interaction terms:

$\mathsf{H}=\mathsf{H}_{\mbox{\footnotesize{vac}}}+\mathsf{H}_{\mbox{\footnotesize{m}}}+\mathsf{H}_{\nu\nu}.$
(6)

where $\mathsf{H}$ is the two-flavour Hamiltonian. The first term is the
Hamiltonian in vacuum [16]:

$\mathsf{H}_{\mbox{\footnotesize{vac}}}=\frac{\omega_{\mathbf{p}}}{2}\left(\begin{array}[]{cc}-\cos
2\theta&\sin 2\theta\\\ \sin 2\theta&\cos 2\theta\\\
\end{array}\right)=\frac{\omega_{\mathbf{p}}}{2}\mathbf{B}\cdot\vec{\sigma}$
(7)

where $\omega_{\mathbf{p}}=\Delta m^{2}/2p$, $\mathbf{B}=(\sin 2\theta,0,-\cos
2\theta)$ and $\theta$ is the smallest neutrino mixing angle in vacuum.

The other two terms in Eqs. (LABEL:eq:FullHam) are special since they make the
evolution equations non-linear. Even though they are very similar, we are
considering that the electrons during the accretion form an isotropic gas;
hence, the vector $\mathbf{v}_{\mathbf{q}}$ in the first integral is
distributed uniformly on the unit sphere and the factor
$\mathbf{v}_{\mathbf{q}}\cdot\mathbf{v}_{\mathbf{p}}$ averages to zero. After
integrating the matter Hamiltonian is given by:

$\mathsf{H}_{\mbox{\footnotesize{m}}}=\frac{\lambda}{2}\left(\begin{array}[]{cc}1&0\\\
0&-1\\\ \end{array}\right)=\frac{\lambda}{2}\mathbf{L}\cdot\vec{\sigma}$ (8)

where $\lambda=\sqrt{2}G_{F}\left(n_{e^{-}}-n_{e^{+}}\right)$ is the charged
current matter potential and $\mathbf{L}=(0,0,1)$.

Such simplification cannot be made with the final term. Since neutrinos are
responsible for the energy loss of the infalling material during accretion,
they must be escaping the accretion zone and the net neutrino and anti-
neutrino flux is non-zero.In this case the factor
$\mathbf{v}_{\mathbf{q}}\cdot\mathbf{v}_{\mathbf{p}}$ cannot be averaged to
zero. At any rate, we can still use Eq. (3) and obtain [17, 18, 19]:

$\mathsf{H}_{\nu\nu}=\sqrt{2}G_{F}\left[\int\\!\\!\left(1-\mathbf{v}_{\mathbf{q}}\cdot\mathbf{v}_{\mathbf{p}}\right)\left(\mathsf{P}_{\mathbf{q}}-\bar{\mathsf{P}}_{\mathbf{q}}\right)\frac{d^{3}\mathbf{q}}{\left(2\pi\right)^{3}}\right]\cdot\vec{\sigma}$
(9)

Introducing every Hamiltonian term in Eqs. (1), and using the commutation
relations of the Pauli matrices, we find the equations of oscillation for
neutrinos and anti-neutrinos for each momentum mode $\mathbf{p}$:

This set of equations is the starting point of any analysis of neutrino
oscillation in an astrophysical system. In the next sections

###  2.1 Neutrino Oscillation in Spherical Accretion

In the BdHN scenario of GRBs, the SN material first reaches the gravitational
capture region of the NS companion, namely the Bondi-Hoyle region. The
infalling material shocks as it piles up onto the NS surface forming an
accretion zone where it compresses and eventually becomes sufficiently hot to
trigger a highly efficient neutrino emission process. Neutrinos take away most
of the infalling matter’s gravitational energy gain, letting it reduce its
entropy and be incorporated into the NS. It was shown in [11] that the matter
in the accretion zone near the NS surface develops conditions of temperature
and density such that it is in a non-degenerate, relativistic, hot plasma
state. The most efficient neutrino emission channel under those conditions
becomes the electron positron pair annihilation process. The neutrino
emissivity can be approximated with a very good accuracy by [20].

where $\mathcal{F}_{k,\ell}\left(y,\eta\right)$ are the generalized Fermi
functions (see [14] for details) and
$\mathcal{F}_{k,\ell}\left(\eta\right)=\mathcal{F}_{k,\ell}\left(y=0,\eta\right)$.
For $m=0$ and $m=1$ Eq. (LABEL:eq:approximationyakovlev) gives the neutrino
and anti-neutrino number emissivity (neutrino production rate), and the
neutrino and anti-neutrino energy emissivity (energy per unit volume per unit
time) for a certain flavour $i$, respectively. Using Eq.
(LABEL:eq:approximationyakovlev) we find that the ratio of emission rates
between electronic and non-electronic neutrino flavours obey the relation

$\frac{\varepsilon^{0}_{e}}{\varepsilon^{0}_{x}}\approx\frac{7}{3}.$ (12)

and because of the symmetry of the annihilation process, the neutrinos and
anti-neutrinos are produced in equal quantities. We can also find an
expression for the average neutrino energy

$\langle E_{\nu}\rangle=\langle E_{\bar{\nu}}\rangle\approx 4.1\,T$ (13)

for all neutrino flavours. The neutrino energy emissivity in Eq.
(LABEL:eq:approximationyakovlev) can be written as

$\epsilon_{e^{-}\\!e^{+}}\approx 8.69\times 10^{30}\left(\frac{T}{1\,{\rm
MeV}}\right)^{9}\,\,{\rm MeV}\,{\rm cm}^{-3}\,{\rm s}^{-1},$ (14)

which allows us to define an effective neutrino emission region [14]

$\Delta
r_{\nu}=\frac{\epsilon_{e^{-}\\!e^{+}}}{\nabla\epsilon_{e^{-}\\!e^{+}}}=\approx
0.08R_{\rm NS}.$ (15)

with $R_{\rm NS}$ the radius of the NS. Recollecting results we can make
another simplifying assumption [14]: Since the neutrino emission region is
thin, we will consider it as a spherical shell. This allows us to use the
single-angle approximation [21, 22] and simplify the last term in Eq.
(LABEL:eq:Hnu1). Precisely the _multi-angle_ term and the one responsible for
kinematic decoherence [23, 24, 25]. With the single-angle approximation and
the inverse square law of flux dilution it is possible to find the explicit
dependence in $r$ of each of the potentials in Eq. (LABEL:eq:Hnu1), namely

$\dot{M}$ $\rho$ $T$ $\eta_{e^{\mp}}$ $n_{e^{-}}\\!-n_{e^{+}}$
$T_{\nu\bar{\nu}}$ $\langle E_{\nu}\rangle$ $F^{C}_{\nu_{e},\bar{\nu}_{e}}$
$F^{C}_{\nu_{x},\bar{\nu}_{x}}$ $n^{C}_{\nu_{e}\bar{\nu}_{e}}$
$n^{C}_{\nu_{x}\bar{\nu}_{x}}$ $\sum_{i}\,n^{C}_{\nu_{i}\bar{\nu}_{i}}$
$(M_{\odot}$ s-1) (g cm${}^{-3})$ (MeV) (cm-3) (MeV) (MeV) (cm-2s-1) (cm-2s-1)
(cm${}^{-3})$ (cm${}^{-3})$ (cm${}^{-3})$ $10^{-6}$ $1.12\times 10^{7}$ 2.59
$\mp 0.193$ $3.38\times 10^{30}$ 2.93 10.61 $2.40\times 10^{38}$ $1.03\times
10^{38}$ $1.60\times 10^{28}$ $6.90\times 10^{27}$ $2.29\times 10^{28}$
$10^{-5}$ $3.10\times 10^{7}$ 3.34 $\mp 0.147$ $9.56\times 10^{30}$ 3.78 13.69
$1.84\times 10^{39}$ $7.87\times 10^{38}$ $1.23\times 10^{29}$ $5.20\times
10^{28}$ $1.75\times 10^{29}$ $10^{-4}$ $8.66\times 10^{7}$ 4.30 $\mp 0.111$
$2.61\times 10^{31}$ 4.87 17.62 $1.39\times 10^{40}$ $5.94\times 10^{39}$
$9.24\times 10^{29}$ $3.96\times 10^{29}$ $1.32\times 10^{30}$ $10^{-3}$
$2.48\times 10^{8}$ 5.54 $\mp 0.082$ $7.65\times 10^{31}$ 6.28 22.70
$1.04\times 10^{41}$ $4.51\times 10^{40}$ $7.00\times 10^{30}$ $3.00\times
10^{30}$ $1.00\times 10^{31}$ $10^{-2}$ $7.54\times 10^{8}$ 7.13 $\mp 0.057$
$2.27\times 10^{32}$ 8.08 29.22 $7.92\times 10^{41}$ $3.39\times 10^{41}$
$5.28\times 10^{31}$ $2.26\times 10^{31}$ $7.54\times 10^{31}$

Table 1: Characteristics inside the neutrino emission zone for selected values
of the accretion rate $\dot{M}$. The symbols $F^{C}$ and $n^{C}$ refer to the
neutrino flux and neutrino density at the emission region. The electron
fraction is $Y_{e}=0.5$, the pinching parameter for the neutrino spectrum is
$\eta_{\nu\bar{\nu}}=2.04$ and the.

where

$\langle v_{r}\rangle=\frac{1}{2}\left[1+\sqrt{1-\left(\frac{R_{\rm
NS}}{r}\right)^{2}}\right].$ (17)

Using Eq. (LABEL:eq:approximationyakovlev) and the hydrodynamic simulations in
[11] we can obtain the thermodynamic properties of the accreting matter at the
NS surface (see Table 1) which in turn are the initial condition to solve Eq.
(LABEL:eq:Hnu1) and obtain an approximate behaviour of oscillations.

Inverted Hierarchy

Figure 1: Neutrino flavour evolution for inverted hierarchy. Electron neutrino
survival probability is shown as a function of the radial distance from the NS
surface. The curves for the electron anti-neutrino match the ones for electron
neutrinos.

Normal Hierarchy

Figure 2: Electron neutrino and anti-neutrino flavour evolution for normal
hierarchy. The survival probability is shown as a function of the radial
distance from the NS surface.

In Figs. 1 and 2 we show the solution of Eqs. (LABEL:eq:Hnu1) for both normal
and inverted hierarchies using a monochromatic spectrum dominated by the
average neutrino energy for $\dot{M}=10^{-2},10^{-3},10^{-4}$ and $5\times
10^{-5}M_{\odot}$ s-1. For the inverted hierarchy, there is no difference
between the neutrino and anti-neutrino survival probabilities. This should be
expected since for these values of $r$ the matter and self-interaction
potentials are much larger than the vacuum potential, and there is virtually
no difference between Eqs. (LABEL:eq:Hnu1). Also, note that the anti-neutrino
flavour proportions in Tab. 1 remain virtually unchanged for normal hierarchy
while the neutrino flavour proportions change drastically around the point
$\lambda_{r}\sim\omega_{r}$. From these solutions we can calculate the
oscillation length to be

$t_{\rm osc}\approx(0.05-1)\,\,{\rm km}$ (18)

which agree with the algebraic estimations in [23, 24]. Clearly, the full
equations of oscillations are highly non-linear so the solution may not
reflect the real neutrino flavour evolution. Concerning the single-angle
approximation, it is discussed in [23, 24, 25] that in the more realistic
multi-angle approach, kinematic decoherence happens for both mass hierarchies.
And in [26] the conditions for decoherence as a function of the neutrino
flavour asymmetry have been discussed. It is concluded that if the symmetry of
neutrinos and anti-neutrinos is broken beyond the limit of $O(25\%)$, i.e., if
the difference between emitted neutrinos and anti-neutrinos is roughly larger
than 25% of the total number of neutrinos in the medium, decoherence becomes a
sub-dominant effect. As a direct consequence of the peculiar symmetric
situation we are dealing with, in which neutrinos and anti-neutrinos are
produced in similar numbers, bipolar oscillations happen and, as we have
already discussed, they present very small oscillation length as shown in Eq.
(18). Note also that the bipolar oscillation length depends on the neutrino
energy. Therefore, the resulting process is equivalent to an averaging over
the neutrino energy spectrum and an equipartition among different neutrino
flavours is expected [24]. Although, for simplicity, we are dealing with the
two neutrino hypothesis, this behavior is easily extended to the more
realistic three neutrino situation. We assume, therefore, that at few
kilometers from the emission region neutrino flavour equipartition is a
reality:

$\nu_{e}:\nu_{\mu}:\nu_{\tau}=1:1:1.$ (19)

After leaving the emission region, beyond $r\approx R_{\rm NS}+\Delta
r_{\nu}$, where $\Delta r_{\nu}$ is the width defined in Eq. (15), the
effective neutrino density quickly falls in a asymptotic behavior
$\mu_{r}\approx 1/r^{4}$. The decay of $\lambda_{r}$ is slower. Hence, very
soon the neutrino flavour evolution is determined by the matter potential.
Matter suppresses neutrino oscillations and we do not expect significant
changes in the neutrino flavour content along a large region. Nevertheless,
the matter potential can be so small that there will be a region along the
neutrino trajectory in which it can be compared with the neutrino vacuum
frequencies and the higher and lower resonant density conditions will be
satisfied. Using the results in [27, 14] we can include the matter effects an
compare in Tab. 2 the flavour content at the emission region and after
decoherence and the MSW resonance.

Finally, we note that for accretion rates $\dot{M}<5\times\\!10^{-5}M_{\odot}$
s-1, either the matter potential is close enough to the vacuum potential and
the MSW condition is satisfied, or both the self-interaction and matter
potentials are so low that the flavor oscillations are only due to the vacuum
potential. In both cases, bipolar oscillations are not present [14]. Without
bipolar oscillations, it is not possible to guarantee that decoherence will be
complete and Eq. (19) is no longer valid.

$n^{0}_{\nu_{e}}/n$ $n^{0}_{\bar{\nu}_{e}}/n$ $n^{0}_{\nu_{x}}/n$
$n^{0}_{\bar{\nu}_{x}}/n$ $n_{\nu_{e}}/n$ $n_{\bar{\nu}_{e}}/n$
$n_{\nu_{x}}/n$ $n_{\bar{\nu}_{x}}/n$ Normal Hierarchy $\frac{1}{6}$
$\frac{1}{6}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$
$\frac{1}{6}+\frac{1}{6}\sin^{2}\theta_{12}$ $\frac{1}{6}$
$\frac{1}{3}-\frac{1}{6}\sin^{2}\theta_{12}$ Inverted Hierarchy $\frac{1}{6}$
$\frac{1}{6}$ $\frac{1}{3}$ $\frac{1}{3}$
$\frac{1}{6}+\frac{1}{6}\cos^{2}\theta_{12}$ $\frac{1}{3}$
$\frac{1}{3}-\frac{1}{6}\cos^{2}\theta_{12}$ $\frac{1}{6}$

Table 2: Fraction of neutrinos and anti-neutrinos for each flavour after
decoherence and matter effects. $n=2\sum_{i}n_{\nu_{i}}$.

###  2.2 Neutrino Oscillations in Accretion Disks

In the same BdHN scenario of Sec. 2, part of the SN ejecta keeps bound to the
newborn Kerr BH, forming an accretion disk [28]. In order to study
analytically the properties of accretion disks, different models make
approximations that allow casting the physics of an accretion disk as a two-
or even one-dimensional problem. Here, we will consider neutrino-cooled
accretion disks (NCADs) which are steady-state [15, 29], axisymmetric, thin,
alpha-disks models with the following parameters: $\dot{M}$ the accretion
rate, $\alpha$ the alpha-viscosity and $a$ the spin of the BH [15, 30, 31, 29,
32, 33, 34, 35]. The procedure to analyse the dynamics of oscillations is
similar to the one in 2.1. The first step is find the neutrino flavour
distributions to establish the initial conditions for Eq. (LABEL:eq:Hnu1),
then we have to find each of the potentials and finally solve the equation. To
do this we first solve the hydrodynamic model in the absence of oscillations.

Figure 3: Properties of accretion disks in the absence of oscillations with
$M=3M_{\odot}$, $\alpha=0.01$, $a=0.95$ for accretion rates
$\dot{M}=1M_{\odot}$ s-1, $\dot{M}=0.1M_{\odot}$ s-1 and
$\dot{M}=0.01M_{\odot}$ s-1, respectively.

In Fig. 3 we find the neutrino number densities and energies inside the disk.
Note that, as in Sec. 2.1, the energies of neutrinos are comparable to the
ones in spherical accretion and the number of neutrinos and anti-neutrinos are
equal. There is also a significant excess of electron neutrinos over non-
electron neutrinos. However, there are several key differences that make the
analysis in accretion disks more complex. First, an accretion disk has an
effective thickness $H$ and neutrinos can be produced at any point inside the
disk. This means that it is not possible to set a _surface of emission_ as
before and due to the lack of spherical symmetry does not allow to use the
single-angle approximation. second, close to the BH the effects of curvature
may not be negligible, implying that in Eq. (1), when applying the Liouville
operator, a term proportional rate of change of the energy of neutrinos
$\dot{\mathbf{p}}$ may be present. To simplify the equations of oscillation we
consider the local rest frame of the disk (see [36, 37] for details) and make
a set of assumptions:

Figure 4: Survival provability for electron neutrinos and anti-neutrinos for
the accretion disk with $\dot{M}=0.1M_{\odot}$ s-1 at
$r=9r_{s},10r_{s},11r_{s},12r_{s}$.

Figure 5: Comparison of density and temperature between thin disks with and
without neutrino flavour equipartition for selected accretion rates.

  1. i

Due to axial symmetry, the neutrino density is constant along the $\mathbf{z}$
direction. Moreover, since neutrinos follow null geodesics, we can set
$\dot{p}_{z}\approx\dot{p}_{\phi}=0$. Also, within the thin disk
approximation, the neutrino and matter densities are constant along the
$\mathbf{y}$ direction and the momentum change due to curvature along this
direction can be neglected, that is, $\dot{p}_{y}\approx 0$.

  2. ii

In the local rest frame of the disk, the normalized radial momentum of a
neutrino can be written as $p_{x}=\pm\frac{r}{\sqrt{r^{2}-2Mr+M^{2}a^{2}}}$
(see [15] for details). Hence, the typical scale of the change of momentum
with radius is $\Delta
r_{p_{x},\text{eff}}=\left|\frac{d\ln{p_{x}}}{dr}\right|^{-1}=\frac{r\left(r^{2}-2Mr+M^{2}a^{2}\right)}{M\left(Ma^{2}-r\right)}$,
which obeys $\Delta r_{p_{x},\text{eff}}>r_{s}$ for $r>2r_{\text{in}}$. This
means that we can assume $\dot{p}_{x}\approx 0$ up to regions very close to
the inner edge of the disk.

  3. iii

We define an effective distance $\Delta
r_{\rho,\text{eff}}=\left|\frac{d\ln\left(n_{e^{-}}-n_{e^{+}}\right)}{dr}\right|^{-1}$.
For all the systems we evaluated we found that is comparable to the height of
the disk $(\Delta r_{\rho,\text{eff}}\sim 2-5$ $r_{s}$). This means that at
any point of the disk we can calculate neutrino oscillations in a small
regions assuming that both the electron density and neutrino densities are
constant.

  4. iv

We neglect energy and momentum transport between different regions of the disk
by neutrinos that are recaptured by the disk due to curvature. This assumption
is reasonable except for regions very close to the BH but is consistent with
the thin disk model (see, e.g.,[29]). We also assume initially that the
neutrino content of neighbouring regions of the disk (different values of $r$)
do not affect each other. As a consequence of the results discussed above, we
assume that at any point inside the disk and at any instant of time an
observer in the local rest frame can describe both the charged leptons and
neutrinos as isotropic gases around small enough regions of the disk.

Figure 6: Total optical depth (left scale) and mean free path (right scale)
for neutrinos and anti-neutrinos of both flavours for accretion disks with
$\dot{M}=1M_{\odot}$ s-1 and $0.01M_{\odot}$ s-1, between the inner radius and
the ignition radius.

All assumptions are sensible except iv, which is considerably restrictive.
However, we can build our analysis on top of it and use the same results of
Sec. 2.1 to generalize the model. Note that with our assumptions, the last
term in Eq. (LABEL:eq:Hnu1) is again simplified. When we calculate the
oscillation in different point of the disk (see Fig. 4) we obtain fast flavour
transformations with oscillation lengths of the order

$t_{\rm osc}\approx 10^{-6}\,{\rm s}$ (20)

Keeping this in mind and given the symmetry between neutrinos and anti-
neutrinos in Fig. 3 we note that in [26] it was shown that if the symmetry
between neutrinos and anti-neutrinos is not broken beyond the limit of 25%,
kinematic decoherence is still the main effect of neutrino oscillations.
Additionally, in [24] it is shown that for asymmetric $\nu\bar{\nu}$ gas, even
an infinitesimal anisotropy triggers an exponential evolution towards
equipartition. Decoherence happens within a few oscillation cycles of
oscillation so we can expect an steady-state, thin disk model to achieve
flavour equipartition and is the result of a non-vanishing flux term (which is
present in accretion disks due to the increasing density towards the BH) such
that at any point, (anti)-neutrinos travelling in different directions, do not
experience the same self-interaction potential due to the multi-angle term in
the integral of Eq. (LABEL:eq:FullHam). This effect is of the neutrino mass
hierarchy and neutrino flavour equipartition is achieved for both hierarchies.
Within the disk dynamic, this is equivalent to imposing the condition

$\langle P_{\nu_{e}\to\nu_{e}}\rangle=\langle
P_{\bar{\nu}_{e}\to\bar{\nu}_{e}}\rangle=0.5.$ (21)

Within this condition, we can compare the behaviour of disks with and without
flavour equipartition. Figure 5 shows that equipartition increases the disk
density and reduces the temperature where the neutrino emission is important.
The effect is mild for low accretion rates while very pronounced for high
ones. Thus result can be explained as follows: for low accretion rates the
neutrino optical depth for all flavors is $\tau_{\nu\bar{\nu}}\lesssim 1$ (see
Fig. 6), hence neutrinos, regardless of their flavour, are free to leave the
disk. When the initial (mainly electron flavour) is redistributed among both
flavours, the total neutrino cooling remains virtually unchanged and the disk
evolves as if equipartition had never occurred, save the new emission flavour
content. On the other hand, when accretion rates are high, the optical depth
obeys
$\tau_{\nu_{x}}\approx\tau_{\bar{\nu}_{x}}\lesssim\tau_{\bar{\nu}_{e}}<\tau_{\nu}\sim
10^{3}$. The $\nu_{e}$ cooling is more heavily suppressed than the others.
When flavours are redistributed, the new $\nu_{x}$ particles a free to escape,
enhancing the total cooling with a consequent reduction of the temperature. As
the temperature decreases, a lower internal energy allows for a higher matter
density. The net impact of flavour equipartition is to make the disk evolution
less sensitive to $\nu_{e}$ opacity. It can be shown (see [15] for details)
that it increases the total cooling efficiency by the precise factor

$\frac{1}{2}\left(1+\frac{\langle E_{\nu_{x}}\rangle}{\langle
E_{\nu_{e}}\rangle}\frac{1+\tau_{\nu_{e}}}{1+\tau_{\nu_{x}}}\right).$ (22)

The main difference with the previous system is that, for similar accretion
rates, the density of the accretion disk can be high dense to impede, or even
trap neutrinos within it. However, since electron and non-electron neutrinos
have different cross sections, the flavour transformations affect not only the
dynamics of the disk, but also the neutrino flavour content emerging from the
disk. This, in turn, affects the energy deposition rate of the process
$\nu+\bar{\nu}\mapsto e^{-}+e^{+}$. In particular, it leads to a deficit of
electron neutrinos and a smaller energy deposition rate with respect to
previous estimates not accounting by flavour oscillations inside the disk. The
exact value of the reduction factor depends on the $\nu_{e}$ and $\nu_{x}$
optical depths but it can be as high as $\sim 5$. We refer the reader to [15]
for further details on this subject.

##  3 Concluding Remarks

We have outlined the implications of neutrino oscillations in two different
accreting systems within the BdHN scenario of GRBs. In both, spherical
accretion and disk accretion, the emission of neutrinos is a crucial
ingredient since they act as the main cooling process that allows the
accretion onto the NS (or onto the BH) to proceed at very high rates of up to
$1~{}M_{\odot}$ s-1. Also, the ambient conditions of density and temperature
imply the occurrence of neutrino flavour oscillations, with a relevant role of
neutrino self-interactions.

We have seen that in spherical accretion the density of neutrinos on top the
NS implies that neutrino self-interactions dominate the flavour evolution,
leading to collective effects. The latter induce quick flavour conversions
with an oscillation length as small as $(0.05$–$1)$ km. Far from the NS
surface, the neutrino density decreases and so the matter potential and MSW
resonances dominate the flavour oscillations. Owing to the above, the neutrino
flavour content emerging from the system is completely different with respect
to the one created at the bottom of it, namely on top the NS accreting
surface.

Concerning disk accretion onto a BH, we saw that the number densities of
electron neutrinos and anti-neutrinos are very similar. As a consequence of
this particular environment, very fast pair conversions,
$\nu_{e}\bar{\nu}_{e}\rightleftharpoons\nu_{x}\bar{\nu}_{x}$, induced by
bipolar oscillations, are obtained for the inverted mass hierarchy case with
high oscillation frequencies. However, due to the interaction between
neighbouring regions of the disk, the onset of kinematic decoherence with a
timescale comparable to the oscillation length induces flavour equipartition
among electronic and non-electronic neutrinos throughout the disk. Therefore,
the neutrino content emerging from the disk is very different from the one
that is usually assumed (see e.g. [38, 39]).

Flavour equipartition, while leaving anti-neutrino cooling practically
unchanged, it enhances neutrino cooling by allowing the energy contained (and
partially trapped inside the disk due to high opacity) within the $\nu_{e}$
gas to escape in the form of $\nu_{x}$, rendering the disk insensible to the
electron neutrino opacity. The variation of the flavour content in the
emission flux implies a loss in the electron neutrino luminosity and an
increase in non-electron neutrino luminosity and $L_{\bar{\nu}_{e}}$. As a
consequence, the total energy deposition rate of the process $\nu+\bar{\nu}\to
e^{-}+e^{+}$ is reduced.

These results are only a first step toward the analysis of neutrino
oscillations in a novel relativistic astrophysics context that can have an
impact on a wide range of astrophysical phenomena: from $e^{-}e^{+}$ plasma
production above BHs in GRB models, to r-process nucleosynthesis in disk winds
and possible MeV neutrino detectability.

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