# Differentiating short gamma-ray bursts progenitors through multi-MeV
neutrinos

G. Morales,1 N. Fraija ,1
1Instituto de Astronomía, Universidad Nacional Autónoma de México, Circuito
Exterior, C.U., A. Postal 70-264, 04510, Ciudad de México, México.
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

With the most recent multi-messenger detection, a new branch in modern
astronomy has been arisen. The GW170817 event together with the short gamma-
ray burst GRB 170817A was the first-ever detection of the gravitational waves
and an electromagnetic counterpart. These detections encourage us to think
that in the following years we will detect a single event through three
different channels: including the mentioned above plus neutrinos from multiple
astrophysical sources, as those detected from SN1987A. It is believed that
short GRBs are originated in the merger of a black-hole (BH) with a neutron
star (NS) or NS-NS scenario. Particularly only in the latter case, several
simulations suggest that the magnetic field can be amplified up to $\sim
10^{15}-10^{16}$ G. Considering this effect over created thermal neutrinos
during the initial stage, we could differentiate short GRB progenitors through
the neutrino expected flavor ratio and the opacity created by the the baryon-
loaded winds ejected in each scenario. Moreover, We find that it is more
feasible to detect neutrinos from BH-NS than NS-NS systems. Finally, we also
estimate the number of neutrino events expected on ground-based detectors,
finding that it is possible to detect neutrinos from an energetic enough
source $(L\gtrsim 10^{52}$ erg s${}^{-1})$ located within a nearby vicinity,
such as, GRB170817A ($d=40$ Mpc) with Hyper-Kamiokande detector.

###### keywords:

Short Gamma-ray burst: Thermal Neutrinos: – Neutrino Oscillation

††pubyear: 2019††pagerange: Differentiating short gamma-ray bursts progenitors
through multi-MeV neutrinos–Weak $\vec{B}$ limit

##  1 Introduction

Gamma-ray bursts (GRBs) are undoubtedly one of the most exciting astrophysical
transients, coming to be the most luminous explosion in the Universe (e,g.,
see Bloom, 2011; Vedrenne G., 2010). In these events, an enormous amount of
energy (up to isotropic—equivalent $10^{55}$ erg (Atteia et al., 2017)) is
released during a short timescale (from miliseconds to minutes). Since their
discovery in the late 1960s (Klebesadel et al., 1973), astronomers have spent
many resources studying the phenomenology of GRBs, which led to the conclusion
that these events are from extragalactic origin and present a bimodal
distribution on their duration (Berger, 2014). In this way, GRBs are
classified into two subcategories: short and long ones (e.g., see Zhang &
Mészáros, 2004; Kumar & Zhang, 2015, for reviews). Short gamma-ray bursts
(sGRBs) correspond to those events with a duration typically less than two
seconds while long gamma-ray bursts (lGRBs), are those whose last over two
seconds. It is widely accepted that progenitors that give rise to both kinds
of GRBs are different. In the first case, it is believed that sGRBs are
produced due to the merger of a binary compact object system (Eichler et al.,
1989; Lee et al., 2004a, 2005; Lee & Ramirez-Ruiz, 2007; Nakar, 2007), while
long GRBs are mostly associated to the collapse of a massive star (collapsar
model) (Hjorth & et al., 2003; Woosley & Bloom, 2006; Hjorth & Bloom, 2012).
In both cases, the progenitor remains hidden because of the great photon
opacity during the initial stage. In that context, it is important to study
the progenitors of these bursts through different channels, for instance,
neutrinos.

The compact objects present during the merger are black holes (BHs) and
neutron stars (NSs), either in both BH–NS or NS–NS configuration, although by
its own nature they could exhibit some differences between them. For instance,
numerical simulations suggest that mergers with larger mass ratio as the case
of BH–NS ($E_{\rm iso}\gtrsim 10^{51}\,{\rm erg\ s}^{-1}$) produces a brighter
electromagnetic counterpart and a systematically smaller effective kick.
Similarly, these simulations have shown that during the coalescence of a
binary NS system, the magnetic field can be amplified up to several orders of
magnitude, reaching values as high as $10^{16}$ G via Kevin-Helmholtz
instabilities and turbulent amplification (Price & Rosswog, 2006; Giacomazzo
et al., 2009; Obergaulinger et al., 2010; Zrake & MacFadyen, 2013; Kiuchi et
al., 2014, 2015).

The detection of a neutrino burst in the energy range of MeV associated with
supernova SN1987A (Hirata et al., 1987) constitutes the beginning of the new
multi-messenger (photons and neutrinos) observation era, and with the first
detection of gravitational waves (GW) in 2015 by the Advanced LIGO
Collaboration (aLIGO), a second boom gives rise in the multi-messenger
scenario (Abbott et al., 2016). During the third running (O3) of aLIGO plus
Advanced Virgo, a GW signal coming from a binary NS merger was detected on
2017 August 17 at 40 Mpc in the nearby galaxy NGC 4993 (Abbott et al., 2017),
shortly after, this signal was associated with an electromagnetic counterpart
from the low-luminosity GRB 170817A and a multi-wavelength follow-up campaign
started. Nevertheless, no neutrino signal was detected in temporary or spatial
coincidence from this source in the energy range of GeV - EeV (Albert et al.,
2017). This accords with our estimates presented further down.

In this work, we study the properties of multi-MeV neutrinos produced during
the initial stage of sGRBs and how they get modified when neutrinos propagate
in media with different physical conditions. This allows us to characterize
and discriminate plausible progenitors. Furthermore, we study the effect of
magnetic field amplification in the neutrino opacity of the progenitor for the
same purpose. Finally, we present the number of expected MeV-neutrino events
with current and future ground-based detectors. It is important to mention
that hereafter, we use the convention $Q_{\rm x}\equiv Q/10^{\rm x}$ in c.g.s.
units, as well as, we adopt the natural units system where, speed of light,
reduced Planck constant, and Boltzmann constant, are all equal to the unity,
$c=\hbar=k_{B}=1$. As a summary, in Section (2), we give an overview of sGRBs
as well as their main characteristics, while in Section (3), we present the
neutrino properties splitted up into: i) neutrino effective potential, ii)
neutrino oscillation theory, and iii) neutrino processes. The mechanisms
involved during the differentiation of sGRBs progenitors are presented and
discussed in Section (4). Furthermore, in Section (5) we introduce the
neutrino detection theory, as well as, the neutrino detectors considered in
this work. Finally, we discuss and present our conclusions in Section ( 6).

##  2 Short Gamma Ray Bursts

With a typical time variability from miliseconds up to a couple of seconds and
according to the observed energy released during such events, sGRBs are the
product of the merger of two compact sources with a high-energy density black
holes and neutron stars. Because of energy losses by gravitational-wave
emission, the most popular progenitor model for sGRBs is the merger of compact
objects; NS-BH or binary NS merger. In the binary NS merger, the expected
remnant is a BH and accreting disk, although in some cases (NSs with $\sim
2\,\,M_{\odot}$; Demorest et al., 2010; Antoniadis et al., 2013), a transitory
or stable highly magnetized NS could be formed (Duncan & Thompson, 1992;
Metzger et al., 2008). In the case of NS-BH merger, a BH with a surrounding
accreting disk could appear provided that the NS is tidally disrupted out of
the BH’s horizon. High-accretion rate and rapid angular momentum play a
relevant role in the energy extraction via $\nu\bar{\nu}$ annihilation (Lee &
Ramirez-Ruiz, 2007) or MHD processes (Blandford & Znajek, 1977), and the
formation/collimation of a relativistic jet.

Based exclusively on a binary NS merger, two kind of sGRBs are generally
discussed; low-luminosity and typical sGRBs (Murguia-Berthier et al., 2017b).
Whereas the low-luminosity sGRB is produced by a mildly relativistic outflow
(Rosswog & Ramirez-Ruiz, 2003, 2002; Nagakura et al., 2014), the typical sGRB
is generated by a relativistic jet (Ramirez-Ruiz et al., 2005a; Murguia-
Berthier et al., 2014; Fraija et al., 2016b). In both cases a relativistic jet
is produced, however in the first case the jet is hampered in the advancement
by the wind expelled from the hypermassive neutron star (HMNS), thus giving
rise to a the low-luminosity sGRB with $E_{\rm iso}\simeq 10^{46}$ \-
$10^{47}$ erg. In the case of the typical sGRB, the collapse to a black hole
takes place so fast that the initial conditions of the relativistic jet are
not altered and the isotropic energy is expected in the range of $E_{\rm
iso}\simeq 10^{50}$ \- $10^{51}$ erg.

In the case of typical sGRBs, several studies have been performed in order to
describe the main accretion models in sGRBs (Popham et al., 1999; Narayan et
al., 2001). According to them, during the initial stage, a considerable amount
of the accretion energy will be converted into neutrinos within the vicinity
of the rotation axis, and subsequently they will enhance some fraction of
their acquired energy in low mass density regions, mainly by neutrino
annihilation processes, but due to neutrinos have small cross section, they
cannot transfer linear momentum to the baryons contained in the fireball,
leading to the creation of an outflow of neutrino-driven baryon-loaded winds
that expand anisotropically during the compact-object merger. The mass rate
loss during these events is of the order of $10^{-3}-10^{-4}\ M_{\odot}\rm\
s^{-1}\rm sr^{-1}$ within the first few miliseconds, being this effect non-
negligible during the evolution of the system. In a BNS merger scenario, the
presence of a strong magnetic field increase considerably the effect on the
neutrino-driven winds outflow and global HD and MHD simulations, such as
(Dessart et al., 2008; Perego et al., 2014; Siegel et al., 2014), must be
performed in order to know the main properties of these winds in both
scenarios.

It is widely accepted that GRBs empower a sharp-shaped conical relativistic
jet and based on its line of sight with respect to the observer/detector, GRBs
can also be classified into off-axis and on-axis GRBs. For instance, observers
whose field of view is within jet opening-angle ($\theta_{\rm
obs}<\theta_{j}$) see the same burst, but beyond $\theta_{j}$, the jet
emission decrease abruptly, while both, prompt emission and the afterglow are
very weak as well. Moreover because of this effect, certain energetic GRBs
viewed off-axis could be comparable with some faint llGRBs viewed on-axis,
such as GRB 980425, GRB031213, and GRB170817A, in which case it is necessary
to correct this effect using a proper relativistic transformation in the local
rest frame (Granot et al., 2005; Ramirez-Ruiz et al., 2005b). It is worth
noting that in principle, a low-luminosity sGRB viewed on-axis would seen to
be a typical sGRB viewed off-axis (Fraija et al., 2019b, c, a).

In order to incorporate GRB dynamics in our study, we rely in the most widely
accepted model called “fireball” (Cavallo & Rees, 1978), which represents the
connection between the relativistic energy outflow and the GRB central engine.
This model requires the liberation of a great concentration of radiation in a
small volume of practically baryon-free space (Piran, 1999). Moreover, during
the initial stage, thermal neutrinos are created in the energy range of MeV.
The properties of these neutrinos get modified when they propagate in a non-
vacuum medium through the MSW effect (Wolfenstein, 1978a), so these additional
aftermaths must be taken into account in the study of neutrino behavior.

In the framework of sGRBs, after the initial merger is completed and depending
on the progenitor associated, from few to dozens solar mass might be left in a
disk of debris spinning around the BH. Because of the temperature is larger,
the $e^{\pm}$ pair production, nuclei are photo-disintegrated, and the plasma
consists mainly of free $e^{\pm}$ pairs, $\gamma$-ray photons and baryons (Lee
et al., 2004b). The so-called fireball plasma connected to the progenitor is
formed as the base of the jet. According to the fireball model, the prompt
emission and afterglow are expected; the prompt emission: when inhomogeneities
in the jet lead to internal collisionless shocks (when material launched with
low velocity is cached by material with high velocity; Rees & Meszaros, 1994;
Fraija et al., 2017a) and the afterglow: when the relativistic outflow sweeps
up enough external material (Mészáros & Rees, 1997; Fraija, 2015b; Fraija et
al., 2016a; Fraija et al., 2017b). We want to emphasize that although high-
energy neutrinos are producing in the prompt emission and afterglow, in this
work we are not interested in these, only those produced during the merger and
the initial fireball.

Due to plasma ingredients are strongly coupled, fireball can be considered
spherically homogeneous. This fireball, initially opaque, will expand
adiabatically by radiation pressure until it becomes transparent to photons
and neutrinos. The initial temperature is high enough ($T\gtrsim 1$ MeV) to
surpass the binding energy in nuclei, propitiating a medium consisting
essentially of free baryons which among other parameters, such as,
temperature, electron chemical potential, and magnetic field intensity, have
an important role within the initial neutrino interactions.

Last but not least, it is important to take into account the neutrino opacity
within the fireball during the initial phase which can be described as a
function of the optical depth $\tau=r/\lambda$, where $r$ is the fireball
radius and $\lambda$ is the mean free path. The mean free path of
$\nu_{e,\mu,\tau}$ can be estimated as (Koers & Wijers, 2005)

$\displaystyle\lambda_{e}$ $\displaystyle=$ $\displaystyle 5.9\times
10^{6}\,{\rm cm}\,\,T_{7}^{-5}\,,$ (1) $\displaystyle\lambda_{\mu,\tau}$
$\displaystyle=$ $\displaystyle 2.56\times 10^{7}\,{\rm cm}\,\,T_{7}^{-5}\,.$
(2)

The difference between electron and muon/tau neutrinos corresponds to the fact
that $\nu_{e}$ interacts by charged-current (CC) and neutral current (NC),
whereas $\nu_{\mu/\tau}$ only through NC. Therefore, the optical depths for
electron and muon/tau neutrinos are

$\displaystyle\tau_{e}$ $\displaystyle=$ $\displaystyle
54\,\,E_{52}^{5/4}\,r_{0,7}^{-11/4}\,,$ (3) $\displaystyle\tau_{\mu,\tau}$
$\displaystyle=$ $\displaystyle 7.4\,\,E_{52}^{5/4}\,r_{0,7}^{-11/4}\,.$ (4)

##  3 Neutrinos

###  3.1 Neutrino effective potential in matter

As active neutrino propagates in a medium, the dynamics is affected by the
effective potentials due to the coherent interactions within it. These
interactions are given by elastic weak charged-current and neutral current
scattering (i.e., see Nötzold & Raffelt, 1988; Enqvist et al., 1991). For a
medium immersed in a magnetic field and a heat bath, the effects are
introduced through Schwinger’s proper-time method (Schwinger, 1951) and
finite-temperature field theory formalism, respectively. The neutrino
effective potential is estimated from its self-energy Feynman diagram. It is
calculated in detail in Fraija (2014b).
The dispersion relation of neutrino is

${\rm V_{eff}=k_{0}-|\vec{k}|}\,,$ (5)

where ${\rm\vec{k}}$ is calculated through the neutrino field equation in a
medium (Nötzold & Raffelt, 1988)

$[{\hbox to0.0pt{/\hss}k}-{\Sigma}(k)]\psi_{L}=0\,.$ (6)

The term $\Sigma(k)$ carries the information of medium such as the velocity,
the magnetic field and the neutrino momentum.

Following Fraija (2014b), the total neutrino self-energy is given by the
exchanges of $W$ boson (${\rm\Sigma_{W}(k)}$) and $Z$ boson ($\Sigma_{Z}$ and
$\Sigma_{t}$). Therefore, although the total neutrino self-energy can be
written as ${\rm\Sigma(k)=\Sigma_{W}(k)+\Sigma_{Z}(k)+\Sigma_{t}(k)}$, the
effective potential useful for neutrino oscillations in a medium is
${\rm\Sigma(k)=\Sigma_{W}(k)}$ (only due to the CC, $V_{\rm
eff}=V_{e}-V_{\mu,\tau}$; Babaev, 2004; Erdas et al., 1998; Sahu et al.,
2009b, a; Wolfenstein, 1978a; D’olivo et al., 1992). In other words, the
effective potential will be only dependent on electron density.

The neutrino self-energy of the W-boson exchange is (Fraija, 2014b)

$-i\Sigma(k)={\mathcal{R}}\,\biggl{[}\frac{g^{2}}{2}\int\frac{d^{4}p}{(2\pi)^{4}}\,\gamma_{\mu}S_{\ell}(p)\gamma_{\nu}\,\,W^{\mu\nu}(q)\biggr{]}\,{\mathcal{L}}\,,$
(7)

where $g^{2}=4\sqrt{2}G_{F}m_{W}^{2}$ is the weak coupling constant with
$m_{W}$ the W-boson mass and GF the Fermi coupling constant, $W^{\mu\nu}$ is
the W-boson propagator in unitary gauge (Erdas et al., 1998; Sahu et al.,
2009a), $S_{l}(p)$ is the charged lepton propagator (Fraija, 2014b). It is
worth noting although neutrino does not have charge, the magnetic field
interacts with the charged particles in a medium, and this information is
carried by the propagator of charged lepton. The terms $\gamma_{\mu}$ are the
Dirac’s matrices, and ${\mathcal{R}}$ and ${\mathcal{L}}$ are the right and
left projection operators, respectively.

Calculating the real part of neutrino self-energy (Equation 7) ${\rm
Re\Sigma(k)={\mathcal{R}}\,[a_{\perp}\hbox to0.0pt{/\hss}k_{\perp}+b\hbox
to0.0pt{/\hss}u+c\hbox to0.0pt{/\hss}b]\,{\mathcal{L}}}$ as a function of the
Lorentz scalars ($a_{\perp}$, $b$ and $c$), the dispersion relation (Equation
5) is in the form

${\rm V_{eff}=b-c\,\cos\varphi-a_{\perp}|{\vec{k}}|\sin^{2}\varphi},$ (8)

where $\varphi$ is the angle between the neutrino momentum and the direction
of magnetic field. The Lorentz scalars for the strong and weak magnetic field
limit are computing in the appendix.

####  3.1.1 Strong $\vec{B}$ limit

The neutrino effective potential (Equation 8) in the strong magnetic field
regime becomes

$\displaystyle V_{\rm
eff,s}=\frac{\sqrt{2}\,G_{F}\,m_{e}^{3}B}{\pi^{2}\,B_{c}}\biggr{[}\sum^{\infty}_{l=0}(-1)^{l}\sinh\alpha_{l}\left[F_{s}-G_{s}\cos\varphi\right]$
$\displaystyle-4\frac{m^{2}_{e}}{m^{2}_{W}}\,\frac{E_{\nu}}{m_{e}}\sum^{\infty}_{l=0}(-1)^{l}\cosh\alpha_{l}\left[J_{s}-H_{s}\cos\varphi\right]\biggr{]}\,,$
(9)

where $m_{e}$ is the electron mass, $\alpha_{l}=(l+1)\mu/T$ with $\mu$ and $T$
the chemical potential and temperature, respectively, $B_{c}=4.141\times
10^{14}\,{\rm G}$ is the critical magnetic field, $E_{\nu}$ is the neutrino
energy and the functions Fs, Gs, Js, Hs are in appendix.

####  3.1.2 Weak $\vec{B}$ limit

The neutrino effective potential (Equation 8) in the weak magnetic field limit
becomes

$\displaystyle V_{\rm
eff,w}=\frac{\sqrt{2}\,G_{F}\,m_{e}^{3}B}{\pi^{2}\,B_{c}}\biggr{[}\sum^{\infty}_{l=0}(-1)^{l}\sinh\alpha_{l}\left[F_{w}-G_{w}\cos\varphi\right]$
$\displaystyle-4\frac{m^{2}_{e}}{m^{2}_{W}}\,\frac{E_{\nu}}{m_{e}}\sum^{\infty}_{l=0}(-1)^{l}\cosh\alpha_{l}\left[J_{w}-H_{w}\cos\varphi\right]\,,$
(11)

where the functions Fw, Gw, Jw, Hw are in the appendix.

###  3.2 Neutrino Oscillation

Neutrino oscillation is a phenomenon widely studied since second half of the
last century and even nowadays is an active field of research, being the
discovery of the neutrino massive properties one of the most important results
of the modern physics. This phenomenon, in general refers to a quantum effect
in which there exists a periodic change between the probability amplitude of
an elemental particle created with an eigenstate $\alpha$ and detected with an
eigenstate $\beta$, such as, $\alpha\neq\beta$. Thus, we are going to describe
the oscillations of thermal neutrinos propagating in a fireball medium (where
they are produced; Fraija, 2014a, 2015a). In this manner, showing up next a
summary of neutrino oscillation theory in both, vacuum and the matter is
presented .

####  3.2.1 Vacuum

Neutrinos propagating in the vacuum are not affected by external surrounding
particles and hence their amplitude probability could be easily expressed as
(Gonzalez-Garcia & Nir, 2003a)

$P(\nu_{\alpha}\rightarrow\nu_{\beta}(t))=\sum_{k>j}U_{\alpha k}^{*}U_{\beta
k}U_{\alpha j}^{*}U_{\beta k}\,\,e^{-i(E_{k}-E_{j})t}\,,$ (13)

where $E_{k}$ is the neutrino dispersion relation, which can be approximated
as ${E_{k}\approx E+({m_{k}^{2}}/{2E})}$, with $E=|{\vec{p}}|$ and
$E_{k}-E_{j}\approx\Delta m_{kj}^{2}/2E$. In the last expression $\Delta
m_{kj}^{2}$ represents the mass squared differences $\Delta m_{kj}^{2}\equiv
m_{k}^{2}-m_{j}^{2}$, such that

$P(\nu_{\alpha}\rightarrow\nu_{\beta}(t))=\sum_{k>j}U_{\alpha k}^{*}U_{\beta
k}U_{\alpha j}^{*}U_{\beta k}\,\,e^{-i(\frac{\Delta m_{kj}^{2}}{2E})t}\,.$
(14)

Likewise, we can assume that neutrino propagation time is proportional to its
distance traveled. Then, the oscillation phase is determined as
$\phi_{kj}=-{\Delta m_{kj}^{2}L}/{(2E)}\,$ and the oscillation length in the
vacuum (typical distance in which the oscillation phase is equal to a 2$\pi$
period) could be expressed as $L_{\text{osc,v}}=({4\pi E)}/{\Delta
m_{kj}^{2}}$. Thus, in order to have important oscillation effects, this
oscillation length must be greater than the distance between source and
detector, otherwise, we can only treat average oscillation effects (Jarlskog,
1985).

####  3.2.2 Matter

Wolfenstein demonstrated that neutrinos propagating in a non-vacuum medium are
affected by an effective potential which is equivalent to the refractive index
of that medium (Wolfenstein, 1978b). Later, Mikheyev and Smirnov (Mikheyev,
1986) showed that in fact, the neutrino oscillation parameters are modified
when they propagate within a material medium, currently this is known as MSW
effect. This additional potential increases the neutrino effective mass, as
well as, their mass and flavor.

###  3.3 Two-neutrino case

In this case, we consider the neutrino oscillation between eigenstates
$\alpha$ and $\beta$ with $\alpha\neq\beta$. We take into account the
equations of oscillation probabilities between electron into muon and tau
neutrino ($\nu_{e}\rightarrow\nu_{\mu}\,\,$ y
$\,\,\nu_{e}\rightarrow\nu_{\tau}$).
Using the equation of temporal evolution (Fraija, 2014b)

$i\begin{pmatrix}{\dot{\nu_{e}}}\\\
\dot{\nu_{\mu}}\end{pmatrix}=\begin{pmatrix}V_{\text{eff}}-\Delta\text{cos}2\theta\,&\,\frac{\Delta}{2}\text{sin}2\theta\\\
\frac{\Delta}{2}\text{sin}2\theta&0\end{pmatrix}\begin{pmatrix}{\nu_{e}}\\\
{\nu_{\mu}}\end{pmatrix}\,,$ (15)

with

$\Delta=\frac{\Delta
m_{\nu}^{2}}{2E_{\nu}}=\frac{m_{\nu_{e}}^{2}-m_{\nu_{\mu}}^{2}}{2E_{\nu}}\,,$
(16)

and $\theta$ the two-neutrino mixing angle, we get the oscillation probability
in a two-neutrino mixing scenario as

$P_{\nu_{e}\to\nu_{\mu}}(t)=|\psi_{\nu_{e}\to\nu_{\mu}}|^{2}=\frac{\Delta^{2}\text{sin}^{2}2\theta}{\omega^{2}}\text{sin}^{2}\left(\frac{\omega
t}{2}\right)\,,$ (17)

where
$\,\omega=\sqrt{(V_{\text{eff}}-\Delta\text{cos}2\theta)^{2}+(\Delta\text{sin}2\theta\,)^{2}},$
and the matter effects are considered using the neutrino effective potential.
In this way, the neutrino oscillation length turns out to be

$L_{\text{osc,m}}=\frac{L_{\text{osc,v}}}{\sqrt{\text{cos}^{2}2\theta(1-\frac{V_{\text{eff}}}{\Delta\text{cos}2\theta})^{2}+\text{sin}^{2}2\theta}}\,.$
(18)

In order to satisfy the resonance condition, we require the positivity of the
potential, which implies that $V_{\text{eff}}=\Delta\cos 2\theta$, and hence,
the resonance length is obtained $L_{\text{res}}=L_{\text{osc,v}}/\sin
2\theta$.

###  3.4 Three-neutrino case

In a three neutrino-mixing scenario, the evolution of neutrino flavors is
governed by the Schrödinger equation in which a neutrino state with initial
flavor $\alpha$, follows the evolution equation

$i\frac{d\vec{\nu}_{\alpha}}{dt}=H\vec{\nu}_{\alpha},$ (19)

where the effective Hamiltonian is

$H=U\cdot\mathcal{M}^{2}\cdot U^{\dagger}+\mathcal{A}\,,$ (20)

with the matrices given by

$\mathcal{M}^{2}=\frac{1}{2E_{\nu}}\begin{pmatrix}-\delta m^{2}_{21}&0&0\cr
0&0&0\cr 0&0&\delta m^{2}_{32}\cr\end{pmatrix}\,,$ (21)

and

$\mathcal{A}=\begin{pmatrix}V_{\rm eff}&0&0\cr 0&0&0\cr
0&0&0\cr\end{pmatrix}.$ (22)

The three-neutrino mixing matrix $U$ is given in (Gonzalez-Garcia & Nir,
2003a). The neutrino state is defined as

$\vec{\nu}_{\alpha}\equiv\begin{pmatrix}\nu_{e}\cr\nu_{\mu}\cr\nu_{\tau}\cr\end{pmatrix},$
(23)

The amplitude of $\nu_{\alpha}\to\nu_{\beta}$ transitions after a time $t$ is
$\phi_{\nu_{\alpha}\to\nu_{\beta}}(t)=\vec{\nu}_{\alpha}\vec{\nu}^{T}_{\beta}$
and the probability is given by
$P_{\nu_{\alpha}\to\nu_{\beta}}(t)=|\phi_{\nu_{\alpha}\to\nu_{\beta}}(t)|^{2}$,
which turns out to be (Gonzalez-Garcia & Nir, 2003b).

$\displaystyle P_{ee}$ $\displaystyle=1-4s_{13,m}^{2}c_{13,m}^{2}S_{31}\,,$
$\displaystyle P_{\mu\mu}$
$\displaystyle=1-4s_{13,m}^{2}c_{13,m}^{2}s_{23}^{4}S_{31}-4s_{13,m}^{2}s_{23}^{2}c_{23}^{2}S_{21}$
$\displaystyle\ \ \ -4c_{13,m}^{2}s_{23}^{2}c_{23}^{2}S_{32}\,,$
$\displaystyle P_{\tau\tau}$
$\displaystyle=1-4s_{13,m}^{2}c_{13,m}^{2}c_{23}^{4}S_{31}-4s_{13,m}^{2}s_{23}^{2}c_{23}^{2}S_{21}$
$\displaystyle\ \ \ -4c_{13,m}^{2}s_{23}^{2}c_{23}^{2}S_{32}\,,$
$\displaystyle P_{e\mu}$
$\displaystyle=4s_{13,m}^{2}c_{13,m}^{2}s_{23}^{2}S_{31}\,,$ (24)
$\displaystyle P_{e\tau}$
$\displaystyle=4s_{13,m}^{2}c_{13,m}^{2}c_{23}^{2}S_{31}\,,$ $\displaystyle
P_{\mu\tau}$
$\displaystyle=-4s_{13,m}^{2}c_{13,m}^{2}s_{23}^{2}c_{23}^{2}S_{31}+4s_{13,m}^{2}s_{23}^{2}c_{23}^{2}S_{21}$
$\displaystyle\ \ \ +4c_{13,m}^{2}s_{23}^{2}c_{23}^{2}S_{32}\,,$

where $\theta_{13,m}$ is the effecting mixing angle in matter given by

$\text{sin}2\theta_{13,m}=\frac{\text{sin}2\theta_{13}}{{\sqrt[]{\left(\text{cos}2\theta_{13}-\frac{2E_{\nu}V_{\text{eff}}}{\Delta
m_{32}^{2}}\right)^{2}+(\text{sin}2\theta_{13})^{2}}}}\,,$ (25)

and $S_{ij}$ corresponds to the neutrino oscillation factors defined as

$S_{ij}=\text{sin}^{2}\Bigg{(}\frac{\Delta\mu_{ij}^{2}L}{4E_{\nu}}\Bigg{)}\,.$
(26)

where the term $\Delta\mu_{ij}^{2}$, represents the squared mass diferences in
matters given by the following relations

$\displaystyle\Delta\mu_{21}^{2}$ $\displaystyle=\frac{\Delta
m_{32}^{2}}{2}\Bigg{(}\frac{\text{sin}2\theta_{13}}{\text{sin}2\theta_{13,m}}-1\Bigg{)}-E_{\nu}V_{\text{eff}}\,,$
$\displaystyle\Delta\mu_{32}^{2}$ $\displaystyle=\frac{\Delta
m_{32}^{2}}{2}\Bigg{(}\frac{\text{sin}2\theta_{13}}{\text{sin}2\theta_{13,m}}+1\Bigg{)}+E_{\nu}V_{\text{eff}}\,,$
(27) $\displaystyle\Delta\mu_{31}^{2}$ $\displaystyle=\Delta
m_{32}^{2}\Bigg{(}\frac{\text{sin}2\theta_{13}}{\text{sin}2\theta_{13,m}}\Bigg{)}\,.$

In this case, the oscillation length of the transition probability is given by

$L_{\rm osc,m}=\frac{L_{\rm osc,v}/\cos^{2}2\theta_{13}}{\sqrt{(1-\frac{V_{\rm
eff}}{V_{\rm res}})^{2}+\tan^{2}2\theta_{13}}},$ (28)

where $V_{\rm res}=\delta m^{2}_{32}\cos 2\theta_{13}/2E_{\nu}$ and $L_{\rm
osc,v}=4\pi E_{\nu}/\delta m^{2}_{32}$ is the vacuum oscillation length. The
resonance condition $V_{\rm eff}=V_{\rm Res}$ can be written as

$V_{\rm eff}=5\times 10^{-7}\frac{\delta m^{2}_{32,\rm eV}}{E_{\nu,\rm
MeV}}\,\cos 2\theta_{13}\,,$ (29)

while the resonance length is

$L_{\rm res}=\frac{4\pi E_{\nu}}{\delta m^{2}_{32}\,\sin 2\theta_{13}}\,,$
(30)

and the adiabatic condition at the resonance can be expressed as

$\kappa_{\rm res}\equiv 8\pi\,l_{\rm res}^{-2}\,\left(\frac{dV_{\rm
eff}}{dr}\right)^{-1}\geq 1\,.$ (31)

###  3.5 Best-fit oscillation parameters

####  3.5.1 Two-Neutrino mixing

Based on appearance and disappearance neutrino oscillation experiments, fluxes
of solar, atmospheric and accelerator neutrinos have provided values of the
squared-mass difference and mixing angles.

  * •

The best-fit oscillation parameters based on solar experiments are $\delta
m^{2}=(5.6^{+1.9}_{-1.4})\times 10^{-5}\,{\rm eV^{2}}$ and
$\tan^{2}\theta=0.427^{+0.033}_{-0.029}$(Aharmim & et al., 2011).

  * •

The best-fit oscillation parameters based on atmospheric experiments are
$\delta m^{2}=(2.1^{+0.9}_{-0.4})\times 10^{-3}\,{\rm eV^{2}}$ and
$\sin^{2}2\theta=1.0^{+0.00}_{-0.07}$(Abe & et al., 2011b).

  * •

The best-fit oscillation parameters based on accelerator experiments are
(Church et al., 2002) found two well defined regions of oscillation parameters
with either $\delta m^{2}\approx 7\,{\rm eV^{2}}$ or $\delta m^{2}<1\,{\rm
eV^{2}}$ compatible with both LAND and KARMEN experiments, for the
complementary confidence and the angle mixing is $\sin^{2}\theta=0.0049$. In
addition, MiniBooNE found evidence of oscillations in the 0.1 to 1.0 eV2,
which are consistent with LSND results (Athanassopoulos & et al., 1998, 1996).

  * •

Combining the solar, atmospheric and accelerator parameters, the best-fit
oscillation parameters are (Aharmim & et al., 2011; Wendell & et al., 2010)
$\delta m_{21}^{2}=(7.41^{+0.21}_{-0.19})\times 10^{-5}\,{\rm eV^{2}}$ and
$\tan^{2}\theta_{12}=0.446^{+0.030}_{-0.029}$ for $\sin^{2}\theta_{13}<0.053$
and $\delta m_{23}^{2}=(2.1^{+0.5}_{-0.2})\times 10^{-3}\,{\rm eV^{2}}$ and
$\sin^{2}\theta_{23}=0.50^{+0.083}_{-0.093}$ for $\sin^{2}\theta_{13}<0.04$.
It is worth noting that $\delta m_{21}^{2}=\delta m_{\rm sol}^{2}$ and $\delta
m_{32}^{2}=\delta m_{\rm atm}^{2}$. Here, the solar parameters correspond to
the large mixing angle solution.

####  3.5.2 Three-Neutrino Mixing

We show in Table (1) a summary of the most recent status of neutrino
oscillation parameters in a three-flavor mixing scenario performed by
global–fit analysis from (de Salas et al., 2018).

Parameter | Best-fit $\pm 1\sigma$ (NO)
---|---
$\text{sin}^{2}\theta_{12}$ | ${0.320_{-0.016}^{+0.020}}$
$\theta_{12}/\ ^{\circ}$ | $34.5_{-1.0}^{+1.2}$
$\text{sin}^{2}\theta_{23}$ | $0.547_{-0.030}^{+0.020}$
$\theta_{23}/\ ^{\circ}$ | $47.7_{-1.7}^{+1.2}$
$\text{sin}^{2}\theta_{13}$ | $0.02160_{-0.00069}^{+0.00083}$
$\theta_{13}/\ ^{\circ}$ | $8.53_{-0.15}^{+0.14}$
$\dfrac{\Delta m_{21}^{2}}{10^{-5}\text{ eV}^{2}}$ | $7.55_{-0.16}^{+0.20}$
$\dfrac{\Delta m_{31}^{2}}{10^{-3}\text{ eV}^{2}}$ | $2.50\pm 0.03$
Table 1: Summary of the neutrino oscillation parameters obtained from
global–fit analysis considering a NO (Normal Ordering) scheme ($\Delta
m_{31}^{2}>0$).

###  3.6 Neutrino processes involved

Due to the high temperatures reached during initial stage, several neutrino
emission within the plasma fireball take place, the most important processes
are (Dicus, 1972; Lattimer et al., 1991):

  * •

pairs annihilation ($e^{+}+e^{-}\to\nu_{x}+\bar{\nu}_{x}$),

  * •

plasmon decay $(\gamma\to\nu_{x}+\bar{\nu}_{x})$,

  * •

photo-neutrino emission $(\gamma+e^{-}\to e^{-}+\nu_{x}+\bar{\nu}_{x})$,

  * •

positron capture $(n+e^{+}\to p+\bar{\nu}_{e})$,

  * •

electron capture $(p+e^{-}\to n+\nu_{e})$,

with ($x=e,\mu,\tau$). Given the initial non-zero baryon density, the mean
free path of neutrinos in the fireball is constituted principally by the
interaction with nucleons. The neutrino mean free path in a non-homogeneous
medium could be expressed as a function of the neutrino cross–section and the
baryon density profiles considered.

We show in Figure (1), the neutrino resonance lengths as a function of
neutrino energy using Equations (18) and (28) in a two and three-neutrino
mixing scenario. In these plots, we have used the most recent best-fit
neutrino oscillation parameters summarized in Table (1). Excluding the solar
neutrino parameters, we find that the resonance lengths for three-flavor
neutrinos lie in a range value of $l_{\text{res}}<10^{7}$ cm, which is less
than than the typical scale of the fireball size during initial stage, i.e.,
these neutrinos will, in fact, oscillate resonantly before leaving the
fireball, implying that they will be released in each kind of flavor almost in
the same proportion.

Figure 1: Neutrino resonance lengths as a function of $E_{\nu}$ using, from
the top to bottom; solar, atmospheric, accelerator and three-flavor best-fit
neutrino parameters shown in Table (1).

These parameters also allow us to find the precise conditions by which
resonance conditions are presented. In each case, we plot in Figure (2) these
contributions for $B=10^{16}$ G and $B=10^{12}$ G, respectively. We also find
that during the magnetic field amplification, the neutrino propagation angle
also represents a considerable influence on the potential for greater angles,
but for a BH–NS merger, the contribution of $\varphi$ remains similar, even
for extreme angular values.

(a)  (b)  (c)  (d)

Figure 2:  Upper panels: Resonance conditions in the BH–NS regime with
neutrinos propagating in both, parallel (a) and perpendicular (b) direction
along the progenitor magnetic field lines for different neutrino energies
($E\nu=\left\\{1,\ 5,\ 10,\ 15\right\\}$ MeV). The comparison of these plots
shows a non-dependence behavior with the neutrino angular propagation.
Bottom panels: Just as before but for a BNS merger. The magnetic field
influence over the neutrino propagation direction modify drastically the
resonance conditions for greater $\varphi$ angles. Considering a strong
magnetic limit present in a BNS system, the neutrino effective potential
exhibits a sensitive angular dependence regarding the neutrino propagation.

##  4 Differentiating sGRBs progenitors

###  4.1 Neutrino oscillations and expected flavors

Taking the magnetic field amplification into account, we want to know how the
neutrino properties are modified when they propagate in a fireball medium. In
this context, we consider two scenarios based on the main mechanisms by which
sGRBs are produced. The first one, is the merger of two NSs, where we use an
amplified magnetic field of $10^{16}$ G, while in the second scenario, we
consider that the merger is produced during the coalescence of a BH–NS system,
with a typical magnetic field of $10^{12}$ G.

As we mentioned before, MeV-neutrinos are produced mainly by thermal processes
but only $e^{\pm}$ capture on nuclei is the responsible one for the electronic
neutrino production, then we have assumed a created neutrino proportion of
$(\nu_{e}:\nu_{\mu}:\nu_{\tau}=4:3:3)$, i.e., for every ten created neutrinos,
there will be 4 electronic, 3 muonic, and 3 tauonic neutrinos, respectively.

These neutrinos will cross a fireball medium before being released and
therefore the effective potential must be used in order to perform the
propagation neutrino properties in both regimes. It is important to notice
that since neutrinos are already polarized into incoherent mass eigenstates
after leaving the high–density source, then the neutrino oscillation
phenomenon in vacuum are suppressed (Lunardini & Smirnov, 2001; Romao et al.,
2015; Smirnov, 2005; Kneller et al., 2008). For that reason, this effect will
no longer be considered in this work. In this manner, the neutrino expected
ratio on Earth will only be determined by the outgoing flavor ratio after
neutrino oscillations take place within the source. Thus, considering both
magnetic field scenarios, we first compute the transition probabilities from
Equation (3.4) using in the first case, the effective potential in the weak
limit with $B=10^{12}$ G and in the second case, the potential in the strong
limit with $B=10^{16}$ G. In both scenarios, we consider a medium with
$r=10^{7}$ cm, $T=1$ MeV, $\mu=1$ keV and $\varphi=0^{\circ}$. These results
allow us to obtain, in each case, the neutrino flavor ratio on Earth as a
function of neutrino energies within MeV–range which we show in Figure (3).

(a)

(b)

Figure 3: Neutrino flavor ratio expected on Earth as function of energy from a
sGRB considering in the first case, a magnetic field of $B=10^{12}$ G (left
panel) and in the second case a magnetic field of $B=10^{16}$ G (right panel).

We found again the same strong dependence of the magnetic field. For instance,
we can see that the ratio remains constant during the propagation when we
considered a magnetic field of $B=10^{12}$ G despite of the energies regarded
in our calculations, while during a BNS merger, the flavor ratio is
fluctuating for several neutrino energies. In other words, our estimates show
that if we detect MeV–neutrinos for the same source with a flavor variation
for different energies, we can assure that the merger took place within an
environment without any amplification of the magnetic field, but in the
opposite case, if we always detect the same proportion even for different
energies, we can determine that the magnetic field did not have any influence
during the coalescence and, therefore, the central engine turns out to be a
black hole and a neutron star.

The reason we found this behavior in Figure (3) is attributed to the direct
influence that magnetic field has over the neutrino effective potential. Thus,
to notice these differences that arise from this magnetic field amplification,
we have computed Equations (40) and (44) in both regimes. These results are
shown in Figure (4), where we have plotted each contribution of
$V_{\text{eff}}$ with the complementary values fixed to typical values of each
regime due to the potential is a multi-variable function. In each case, we
have considered typical values lying within the range of $1\leq E_{\nu}\leq
20$ MeV, $1\leq T\leq 10$ MeV and $1\leq\mu\leq 1000$ eV. In this manner, we
show in panel (4a) the contribution of $V_{\text{eff}}$ as a function of
neutrino energy, while the chemical potential contribution is shown in panel
(4b). Furthermore, we exhibit the thermal contribution of the potential for:
i) a parallel propagation angle, (4c) and ii) for several $\varphi$ angles,
(4d). In these panels, we can notice that the potential increases up to
several orders of magnitude because of magnetic field and propagation angle
influence. The strong dependence on these variables suggests that in order to
understand the internal nature of a GRB as well as the internal magnetic field
topology in both scenarios, these effects must be taken into account.

(a)

(b)

(c)

(d)

Figure 4: Neutrino effective potential as a function of neutrino energy (a),
chemical potential (b), temperature (c) and neutrino propagation angle (d) ,
in both progenitor scenarios for multi-MeV neutrinos. In subplot (d), we
represent the contribution to the potential using a magnetic field value of
$B=10^{12}$ and $B=10^{16}$ G. The magnetic field amplification present during
a BNS merger evinces a substantial influence over the neutrino effective
potential behavior.

###  4.2 Neutrino opacity: on/off–axis mechanism

(a)

(b)

Figure 5:  Left panel: Density profiles and neutrino opacity to the neutrino-
driven and magnetically-driven winds that surround the progenitor in both
scenarios according with the analysis made by (Murguia-Berthier et al., 2017a)
for a radius of $10^{7}$, $E_{\nu}=20$ MeV and considering both neutrino
processes during propagation: neutrino capture on neutron (solid line) and
antineutrino capture on protons (dashed line).
Right panel: Neutrino opacity in neutrino-driven (dashed lines) and
magnetically-driven winds (solid lines) for multiple half-opening angles and
considering only neutrino capture processes.

Neutrino opacity product of the baryon loaded-winds can be expressed as
$\tau_{\nu(\bar{\nu})}=r/\lambda=\sigma_{\nu(\bar{\nu})}rn$, with $r$ the
fireball radius and $n$ the number density, which is displayed in term of
baryonic density as $n=m_{p}\rho_{\rm w}(r,\theta_{j})^{-1}$ with $m_{p}$ the
proton mass. Then, opacity turns out to be

$\tau_{\nu(\bar{\nu})}=\sigma_{\nu(\bar{\nu})}rm_{p}\rho_{\rm
w}(r,\theta_{j})^{-1}\,,$ (32)

with $\sigma_{\nu(\bar{\nu}})$ obtained theoretically for MeV–neutrinos
(antineutrinos) from (Tubbs & Schramm, 1975):

$\sigma_{\nu(\bar{\nu})}=\dfrac{G_{F}^{2}}{\pi}(3\alpha^{2}+1)E_{\nu}^{2}g(E_{\nu})\,,$
(33)

with

$g(E_{\nu})=\left(1\pm\dfrac{Q}{E_{\nu}}\right)\left[1\pm
2\dfrac{Q}{E_{\nu}}+\dfrac{Q^{2}-(\pm
m_{e}^{2})}{E_{\nu}^{2}}\right]^{1/2}\,,$ (34)

where the sign $\pm$ corresponds to the neutrino (+) and antineutrino (-)
treatment, $Q=1.3$ MeV the proton-neutron mass difference, $\alpha=-1.26$ the
nuclear axial coupling coefficient. The neutrino cross–section lies in the
range between $10^{-43}$ and $10^{-40}$ cm2, for our studied multi-MeV
neutrinos.

In order to estimate the number of neutrinos released in an extreme event,
such a GRB, mostly we need to compute the baryon density profiles
$\rho(\theta_{j})$ of the wind-like circum–media. Recent hydrodynamical (HD)
and magneto-hydrodynamical (MHD) simulations have been performed to achieve
this purpose (Perego et al., 2014; Siegel et al., 2014). Murguia-Berthier et
al. (2017a) compiled this results in both regimes, finding the density
profiles showed in Figure (5), where we can observe a considerable increment
in the density for angles close to the equatorial plane due to the magnetic
field amplification. Once we know the density profiles and neutrino cross-
section, from Equation (32) we also have plotted the neutrino opacity around
the progenitor originated by circum–media winds as a function of the half-
opening angle $\theta_{j}$ and neutrino energy, respectively. In these plots,
the most noteworthy result is the strong dependence on the magnetic field in a
BNS merger, in this case we observe that, for example, using a typical
$E_{\nu}=20$, MeV the neutrinos are opaque for $\theta_{j}>62.1^{\circ}$,
while for greater neutrino energies, the angle of transparency decrease as low
as $\theta_{j}=54.1^{\circ}$ for $E_{\nu}=30$ MeV and indeed, considering
extreme energies like $E_{\nu}=100$ MeV, we find that neutrinos propagating at
$\theta_{j}>38.2^{\circ}$ will not be released at all, while on the other
hand, during a BH–NS coalescence, neutrinos are isotropically released. The
above directly implies that where a magnetic field amplification effect took
place, neutrinos are confined in a preferential direction along the jet
propagation path and using this result, we can be able to distinguish between
central engines if, for example, we identify the electromagnetic counterpart
of an off-axis GRB and instead; we do not detect neutrinos from the same
source.

##  5 Detection

###  5.1 Detectors

(a)

(b)

Figure 6:
Left panel: Number of neutrino expected events on several ground-based
neutrino telescopes as a function of typical thermal neutrino energies
$E_{\nu}$ using as a case study the GW170817/GRB170817A event. In this plot,
we have considered the values of $L_{\nu}=5\times 10^{48}$ erg s-1, $d=40$ Mpc
and $T_{\text{burst}}=2$ s.
Right panel: Number of neutrino expected events from sGRBs in Hyper-Kamiokande
detector as a function of redshift considering an average neutrino energy of
$E_{\nu}=20$ MeV. Additionally we compare the expected neutrino events for
several sGRB sources with known redshift and isotropic-equivalent
luminosities.

####  5.1.1 Super-Kamiokande

Super-Kamiokande (SK) is an underground neutrino observatory built 1000 m
below the surface in a Japanese mine. At present, SK is the biggest water
Čerenkov detector with a base of 39 m in diameter and a height of 42 m, having
the capacity to contain 50 kton of ultra-pure water. SK has an array of more
than 13,000 photomultipliers (PMTs) placed in two sections; 11,129 within the
inner region and 1,885 PMTs in the outer region. The outreach of SK is to
perform studies in solar, accelerator, and atmospheric neutrinos as well as
proton decays in the MeV energy range (Fukuda et al., 2003).

####  5.1.2 Hyper-Kamiokande

Hyper-Kamiokande (HK) will be a third-generation Čerenkov detector (replacing
the current SK detector) located also in a Japanese mine near to its
predecessor SK. It is planned to start building in April 2020 and it is
expected to start operating in the middle of the following decade. The
original design (Abe & et al., 2011a; Hyper-Kamiokande Working Group et al.,
2014) shows a design of two almost-cylindrical tanks containing (0.56) million
metric tons of ultra-pure water. This detector will be equipped with 99,000
PMTs uniformly placed within the tanks. Complementing the tasks of SK, HK will
also perform detection for neutrinos produced in both, terrestrial and extra-
terrestrial sources, as well as studies in Particle Physics, such as, CP
violation in the leptonic sector, proton decay and neutrino oscillation
phenomena (Abe et al., 2018).

####  5.1.3 Deep Underground Neutrino Experiment

The DUNE (Deep Underground Neutrino Experiment) experiment will consist of two
neutrino experiment, the first one placed near to Fermi National Laboratory
Acceleration Facility in Illinois, USA (short-baseline program) while the
second one will be built within the SURF facilities in South Dakota (long-
baseline program). Altogether DUNE will have a fiducial mass of 40 kton of
liquid argon, within four cryostats adapted with Liquid Argon Time Projection
Chambers (LArTPCs) and analogously to its counterpart (HK) it is expected to
start operating as of the second half of the next decade. Their main focus
will be the studies of accelerator neutrinos and the measure of its mixing
parameters, but also it will perform studies in the detection of astrophysical
neutrinos and the search for proton decay (Acciarri et al., 2016).

###  5.2 Number of neutrino expected events

It is possible to estimate the numbers of events to be expected on Earth as

$N_{\rm
ev}=VN_{A}\,\rho_{N}\int_{t^{\prime}}\int_{E^{\prime}}\sigma^{\bar{\nu}_{e}p}_{cc}\frac{dN}{dE}\,dEdt\,,$
(35)

where $V$ is the effective volume of water, $N_{A}=6.022\times 10^{23}$ g-1 is
the Avogadro’s number, $\rho_{N}=(M_{\rm fiducial}/V)=2/18\,{\rm g\,cm^{-3}}$
is the nucleons density in water or $\sigma^{\bar{\nu}_{e}p}_{cc}\simeq
9\times 10^{-44}\,E^{2}_{\bar{\nu}_{e}}/{\rm MeV}^{2}$ is the cross-section
(Bahcall, 1989), $dt$ is the neutrino emission time and $dN/dE$ is the
neutrino spectrum. Taking into account the relation between the neutrino
luminosity $L_{\bar{\nu}_{e}}$ and flux $F_{\bar{\nu}_{e}}$,
$L_{\bar{\nu}_{e}}=4\pi
D^{2}_{z}F_{\bar{\nu}_{e}}\braket{E_{\bar{\nu}_{e}}}=4\pi
D^{2}_{z}E^{2}_{\bar{\nu}_{e}}dN/dE_{\bar{\nu}_{e}}$, then the number of
events is

$\displaystyle N_{\text{ev}}$ $\displaystyle\simeq$
$\displaystyle\frac{N_{A}\,\rho_{N}\sigma^{\bar{\nu}_{e}p}_{cc}}{4\pi
D^{2}_{z}\braket{E_{\bar{\nu}_{e}}}}V_{\rm
det}\,\braket{E_{\bar{\nu}_{e}}}\,,$ (36)

where $D_{z}$ is the distance from neutrino production to Earth,
$\braket{E_{\bar{\nu}_{e}}}$ is the average energy of electron antineutrino
and $E_{T,\bar{\nu}_{e}}=\int L_{\bar{\nu}_{e}}dt$ is the total energy emitted
(Mohapatra & Pal, 2004; Fraija et al., 2014).

The number of neutrino expected events can be obtained through the Equation
(36). As a case of study we compute the number of plausible neutrino events
coming from GRB170817, for this purpose we take into account the reported
initial parameters for this event with $L_{\nu}=5\times 10^{48}$ erg s-1,
$d=40$ Mpc and $T_{\text{burst}}=2$ s (Abbott et al., 2017). We represent in
the left panel of Figure (6) the expected events in HK, DUNE and SK detectors
as a function of neutrino energy. We found that even though this GRB was in a
nearby location (40 Mpc), it had an atypical low isotropic luminosity that
diminishes the initial neutrino flux. Multiple observations performed for
several collaborations in different energy scales (Albert et al., 2017) have
verified that, in fact, no neutrino signal has been detected on Earth from
this source.

Furthermore, we realized that HK detection capacity is more than a order of
magnitude greater than DUNE experiment an its predecessor SK, being HK the
best candidate so far to perform extragalactic neutrino detections in the near
future, with this in mind, we present in the right panel of Figure (6) the
number of neutrino expected events in HK detector collecting data information
for the most significant sGRBs with a measured redshift thus far (Berger,
2014; D’Avanzo et al., 2014; Jin et al., 2018). In our calculations, we have
considered an Einstein–de Sitter (EdS) Universe with $h=0.673$,
$\Omega_{m}=0.315$, $\Omega_{\Lambda}=0.0.685$ parameters, being $\Omega_{m}$
and $\Omega_{\Lambda}$ the pressureless matter density and the dark energy
density of the $\rm{\Lambda CDM}$ Universe (Patrignani et al., 2016),
respectively. We also notice that most of the sGRBs have their origin in a
host galaxy with $z>1$ and have a typical equivalent isotropic luminosity
between $10^{51}-10^{53}$ erg s-1, being GRB090524 the brightest one, almost
reaching the value of $10^{54}$ erg s -1. In the same plot we can see the
atypical GRB associated with the GW detection in 2017, coming to be the least
luminous and the closest GRB in this sample.

The most remarkable feature on this plot is the fact to know the most
plausible sensitive detection region in HK for multi–MeV neutrinos produced in
sGRBs, we can see that, for instance, a close event like GRB170817A but even
more energetic could lead to the production of neutrinos that HK can actually
observe.

##  6 Conclusion

We can conclude that multi-MeV neutrinos play an important role in the
understanding of transient events where radiation cannot break free during
early phases. In that context, we study the effect on the neutrino effective
potential product of a magnetic field amplification during the merger of both
kind of progenitors that give rise to sGRBs. We found that even when the
potential is dependant of multiple physical parameters, only the magnetic
field intensity as well as the neutrino propagation angle represent the
greatest contribution to the potential, coming to be up to four orders of
magnitude bigger in the NS–NS scenario. This become of crucial importance
because through this outcome we can be able to infer the intrinsic topology
composition of the involved magnetic field during the merger. On the other
hand, in order to figure out how the neutrinos properties are altered, we have
computed the neutrino oscillations and resonance lengths in both magnetic
field regimes for these multi-MeV neutrinos, finding that, in fact, they will
oscillate resonantly before leaving the fireball with a predominant $\nu_{e}$
survival rate affecting the final expected neutrino ratio on Earth. After
computing the neutrino probabilities we realized that through this effect we
can discriminate the progenitors based on the flavor ratio expected. For
instance, taking two arbitrary energy values during a BNS merger, we expect a
flavor ratio of $(\nu_{e}:\nu_{\mu}:\nu_{\tau}=1.1871:0.9071:0.9059)$ for
$E_{\nu}=10$ MeV and $(\nu_{e}:\nu_{\mu}:\nu_{\tau}=1.1871:0.9071:0.9059)$ for
$E_{\nu}=30$ MeV, while for a BH–NS merger, we will expect the same ratio of
$(\nu_{e}:\nu_{\mu}:\nu_{\tau}=1.2:0.9:0.9)$ for both chosen energies.

With respect to the released neutrinos, we identify two different behaviors in
both configurations: i) in a NS–NS merger; neutrinos are collimated along the
rotation axis, mainly because of the gradual angular decrease of the opacity
within the surrounding medium during the magnetic field amplification and ii)
during a BH–NS merger; the medium remains transparent towards neutrinos,
regardless the angle by which they leave the source, meaning that neutrinos
are released almost isotropically. This result provides us of a unique
trustworthy method to discriminate the participating progenitors during the
merger through this on-axis/off-axis method, since if somehow we are able to
detect an off-axis GRB with a line of sight greater than the critical angle
(for instance, through their afterglow), and jointly we receive neutrinos
within temporal and spatial dependency associated with this source, without
any doubt, we can determine that the GRB progenitors are the result of NS-NS
merger where the magnetic field amplification took place. In similar
conditions but with an on-axis event configuration, we cannot infer the
progenitors through this method and instead, we need to make use of the
neutrino oscillation parameters and effective potential variations in order to
identify the initial central engine configuration.

Our estimates also predict that for an energetic enough sGRB $(L\gtrsim
10^{52}$ erg s${}^{-1})$ located within a nearby vicinity, such as, GRB170817A
($d=40$ Mpc), we can be able to detect neutrinos from this source with the HK
detector. So far, no MeV-EeV neutrino has been found from sources with the
characteristics presented above, which actually agrees with our results.

## Acknowledgements

We thank J. Beacom, W. H. Lee, P. Veres, E. Moreno and A. Marinelli for useful
discussions. This work was supported by PAPIIT-UNAM IG100414.

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## Appendix

### Strong $\vec{B}$ limit

In the strong magnetic field approximation ($B/B_{c}\gg 1$), the Lorentz
scalars are

$\displaystyle
a_{\rm\perp,s}=-\frac{\sqrt{2}G_{F}}{m_{W}^{2}}\biggl{[}\biggl{(}(E_{\nu_{e}}+k_{3})(N^{0}_{e}-\bar{N}^{0}_{e})\biggr{)}+\frac{eB}{\pi^{2}}\int^{\infty}_{0}dp_{3}\,\frac{m_{e}^{2}}{E_{0}}\,(f_{e,0}+\bar{f}_{e,0})\biggr{]},$
$\displaystyle b_{\rm s}=$
$\displaystyle\sqrt{2}G_{F}\biggl{[}\biggl{(}1+\frac{3}{2}\frac{m_{e}^{2}}{M_{W}^{2}}+\frac{eB}{M_{W}^{2}}+\frac{E_{\nu_{e}}k_{3}}{M_{W}^{2}}+\frac{E^{2}_{\nu_{e}}}{M_{W}^{2}}\biggr{)}(N^{0}_{e}-\bar{N}^{0}_{e})-\frac{eB}{2\pi^{2}M_{W}^{2}}\int^{\infty}_{0}\,dp_{3}\biggl{\\{}2\,k_{3}E_{e,0}+2E_{\nu_{e}}\biggl{(}E_{e,0}-\frac{m_{e}^{2}}{2E_{e,0}}\biggr{)}\biggr{\\}}$
(37) $\displaystyle\hskip
395.49312pt\times(f_{e,0}+\bar{f}_{e,0})\biggr{]}\,,$

and

$\displaystyle c_{\rm s}=$
$\displaystyle\sqrt{2}G_{F}\biggl{[}\biggl{(}1+\frac{1}{2}\frac{m_{e}^{2}}{M_{W}^{2}}+\frac{eB}{M_{W}^{2}}-\frac{E_{\nu_{e}}k_{3}}{M_{W}^{2}}-\frac{k^{2}_{3}}{M_{W}^{2}}\biggr{)}(N^{0}_{e}-\bar{N}^{0}_{e})-\frac{eB}{2\pi^{2}M_{W}^{2}}\int^{\infty}_{0}\,dp_{3}\biggl{\\{}2E_{\nu_{e}}\biggl{(}E_{e,0}-\frac{m_{e}^{2}}{2E_{e,0}}\biggr{)}\biggr{\\}}$
(39) $\displaystyle\hskip
281.6821pt+2k_{3}\biggl{(}E_{e,0}-\frac{3m_{e}^{2}}{2E_{e,0}}\biggr{)}\biggr{\\}}(f_{e,0}+\bar{f}_{e,0})\biggr{]}\,,$

where the number density of electrons can be written as
$n_{e}^{0}(\mu,T,B)=\frac{eB}{2\pi^{2}}\int^{\infty}_{0}dp_{3}f_{e,0}$ with
$f_{e,0}=f(E_{e,0})=\frac{1}{e^{\beta(E_{e,0}-\mu)}+1}$ and
$E^{2}_{e,0}=(p^{2}_{3}+m_{e}^{2})$. The quantity $E_{0}$ corresponds to the
electron energy in the lowest Landau level.
Assuming that the chemical potentials ($\mu$) of the electrons and positrons
are much smaller than their energies ($\mu\leq$Ee,0), the fermion distribution
function can be written as a sum given by
$f_{e,0}\approx\sum^{\infty}_{l=0}(-1)^{l}e^{-\beta(E_{e,0}-\mu)(l+1)}$.
Replacing Equations (39) in (8) and solving these integral-terms, the neutrino
effective potential in the strong magnetic field regime becomes

$\displaystyle
V_{eff,s}=\frac{\sqrt{2}\,G_{F}\,m_{e}^{3}B}{\pi^{2}\,B_{c}}\biggr{[}\sum^{\infty}_{l=0}(-1)^{l}\sinh\alpha_{l}\left[F_{s}-G_{s}\cos\varphi\right]-4\frac{m^{2}_{e}}{m^{2}_{W}}\,\frac{E_{\nu}}{m_{e}}\sum^{\infty}_{l=0}(-1)^{l}\cosh\alpha_{l}\left[J_{s}-H_{s}\cos\varphi\right]\biggr{]}\,,$
(40)

where $m_{e}$ is the electron mass, $\alpha_{l}=(l+1)\mu/T$ with $\mu$ and $T$
the chemical potential and temperature, respectively, $B_{c}=4.141\times
10^{14}\,{\rm G}$ is the critical magnetic field, $E_{\nu}$ is the neutrino
energy and the functions Fs, Gs, Js, Hs are

$\displaystyle F_{s}$ $\displaystyle=$
$\displaystyle\left[1+\frac{m_{e}^{2}}{m^{2}_{W}}\left(\frac{3}{2}+2\frac{E^{2}_{\nu}}{m^{2}_{e}}+\frac{B}{B_{c}}\right)\right]K_{1}(\sigma_{l})\,,\hskip
28.45274ptG_{s}=\left[1+\frac{m_{e}^{2}}{m^{2}_{W}}\left(\frac{1}{2}-2\frac{E^{2}_{\nu}}{m^{2}_{e}}+\frac{B}{B_{c}}\right)\right]K_{1}(\sigma_{l})\,,$
(41) $\displaystyle J_{s}$ $\displaystyle=$
$\displaystyle\frac{3}{4}K_{0}(\sigma_{l})+\frac{K_{1}(\sigma_{l})}{\sigma_{l}}\,,\hskip
108.12054ptH_{s}=\frac{K_{1}(\sigma_{l})}{\sigma_{l}}\,,$ (42)

with $\sigma_{l}=(l+1)m_{e}/T$.

### Weak $\vec{B}$ limit

In the weak field approximation ($B/B_{c}\ll 1$), the potential in this
regimen can be written as

$\displaystyle
a_{\rm\perp,w}=-\frac{\sqrt{2}G_{F}}{m_{W}^{2}}\biggl{[}\biggl{\\{}E_{\nu_{e}}(n_{e}-\bar{n}_{e})+k_{3}(n_{e}^{0}-\bar{n}_{e}^{0})\biggr{\\}}+\frac{eB}{2\pi^{2}}\int^{\infty}_{0}dp_{3}\int^{\infty}_{0}(2-\delta_{n,0})\biggl{(}\frac{m_{e}^{2}}{E_{n}}-\frac{2neB}{E_{n}}\biggr{)}(f_{e,n}+\bar{f}_{e,n})\,dn\biggr{]},$
$\displaystyle b_{\rm w}$ $\displaystyle=$
$\displaystyle\sqrt{2}G_{F}\biggl{[}\biggl{(}1+\frac{E_{\nu_{e}}^{2}}{m_{W}^{2}}\biggr{)}(n_{e}-\bar{n}_{e})+\frac{E_{\nu_{e}}k_{3}}{m_{W}^{2}}(n_{e}^{0}-\bar{n}_{e}^{0})-\frac{eB}{\pi^{2}m_{W}^{2}}\int^{\infty}_{0}dp_{3}\int^{\infty}_{0}(2-\delta_{n,0})E_{\nu_{e}}\biggl{\\{}E_{n}\delta_{n,0}+\biggl{(}E_{n}-\frac{m_{e}^{2}}{2E_{n}}\biggr{)}\biggr{\\}}$
$\displaystyle\hskip 367.04037pt\times(f_{e,n}+\bar{f}_{e,n})\,dn\biggr{]}\,,$

and

$\displaystyle c_{\rm
w}=\sqrt{2}G_{F}\biggl{[}\biggl{(}1-\frac{k_{3}^{2}}{m_{W}^{2}}\biggr{)}(n_{e}^{0}-\bar{n}_{e}^{0})-\frac{E^{2}_{\nu_{e}}}{m_{W}^{2}}(n_{e}-\bar{n}_{e})-\frac{eB}{\pi^{2}m_{W}^{2}}\int^{\infty}_{0}dp_{3}\int^{\infty}_{0}(2-\delta_{n,0})E_{\nu_{e}}\biggl{\\{}\biggl{(}E_{n}-\frac{m_{e}^{2}}{E_{n}}\biggr{)}\delta_{n,0}$
$\displaystyle+\biggl{(}E_{n}-\frac{3}{2}\frac{m_{e}^{2}}{E_{n}}-\frac{2neB}{E_{n}}\biggr{)}\biggr{\\}}(f_{e,n}+\bar{f}_{e,n})\,dn\biggr{]}.$
(43)

where the electron number density is
$n_{e}(\mu,T,B)=\frac{eB}{2\pi^{2}}\int_{0}^{\infty}dp_{3}\int^{\infty}_{0}\frac{(2-\delta_{n,0})\,dn}{e^{\beta(E_{e,n}-\mu)}+1}$
and electron distribution function is
$f_{e,n}=f(E_{e,n},\mu)=\frac{1}{e^{\beta(E_{e,n}-\mu)}+1}$, with
$\bar{f}_{e,n}(\mu,T)=f_{e,n}(-\mu,T)$ and
$E_{e,n}={p_{3}^{2}+m_{e}^{2}+2neB}^{1/2}$.
Replacing Equations (43) in 8 and solving these integral-terms, the neutrino
effective potential in the strong magnetic field regime becomes

$\displaystyle
V_{eff,is(w)}=\frac{\sqrt{2}\,G_{F}\,m_{e}^{3}B}{\pi^{2}\,B_{c}}\biggr{[}\sum^{\infty}_{l=0}(-1)^{l}\sinh\alpha_{l}\left[F_{w}-G_{w}\cos\varphi\right]-4\frac{m^{2}_{e}}{m^{2}_{W}}\,\frac{E_{\nu}}{m_{e}}\sum^{\infty}_{l=0}(-1)^{l}\cosh\alpha_{l}\left[J_{w}-H_{w}\cos\varphi\right]$
(44)

where the functions Fw, Gw, Jw, Hw are

$\displaystyle F_{w}$ $\displaystyle=$
$\displaystyle\biggl{(}2+2\frac{E^{2}_{\nu}}{m^{2}_{W}}\biggr{)}\biggl{(}\frac{K_{0}(\sigma_{l})}{\sigma_{l}}+2\frac{K_{1}(\sigma_{l})}{\sigma_{l}^{2}}\biggr{)}\frac{B_{c}}{B}-K_{1}(\sigma_{l})\,,\hskip
28.45274ptG_{w}=K_{1}(\sigma_{l})-\frac{2B_{c}}{B}\frac{E^{2}_{\nu}}{m^{2}_{W}}\biggl{(}\frac{K_{0}(\sigma_{l})}{\sigma_{l}}+2\frac{K_{1}(\sigma_{l})}{\sigma_{l}^{2}}\biggr{)}\,,$
$\displaystyle J_{w}$ $\displaystyle=$
$\displaystyle\biggl{(}\frac{1}{2}+\frac{3B_{c}}{B\,\sigma_{l}^{2}}\biggr{)}K_{0}(\sigma_{l})+\frac{B_{c}}{B}\biggl{(}1+\frac{6}{\sigma_{l}^{2}}\biggr{)}\frac{K_{1}(\sigma_{l})}{\sigma_{l}}\,,\hskip
45.52458ptH_{w}=\biggl{(}\frac{1}{2}+\frac{B_{c}}{B\,\sigma_{l}^{2}}\biggr{)}K_{0}(\sigma_{l})+\frac{B}{B_{c}}\biggl{(}\frac{2}{\sigma_{l}^{2}}-\frac{1}{2}\biggr{)}\frac{K_{1}(\sigma_{l})}{\sigma_{l}}\,.$


Let $\mathcal{F}$ be a right exact functor from category $\mathcal{C}$ to category $\mathcal{D}$. Then higher derived functor of $\mathcal{F}$ is a collection of functors $\mathcal{F} = \mathcal{F}_0$, $\mathcal{F}_1$, $\mathcal{F}_2$, $\dots$ such that if

\[ \begin{tikzcd}
0 \arrow{r} & A \arrow{r} & B \arrow{r} & C \arrow{r} & 0
\end{tikzcd}\]

is a short exact sequence, where $A, B, C \in \mathcal{C}$, then

\[ \begin{tikzcd}
\dots \arrow{r} & \mathcal{F}_1(A) \arrow{r} & \mathcal{F}_1(B) \arrow{r} & \mathcal{F}_1(C) \arrow{r} & \mathcal{F}_0(A)\\
\arrow{r} & \mathcal{F}_0(B) \arrow{r} & \mathcal{F}_0(C) \arrow{r} & 0
\end{tikzcd}\]

is a long exact sequence.

Now we will show how to compute higher derived functors. Given $A \in \mathcal{C}$, our aim is to compute $\mathcal{F}_i(A)$. We find a projective resolution $(P_., d_.)$ of $A$.

\[ \dots \rightarrow P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow A \rightarrow 0   \]

Applying functor $\mathcal{F}$ on the above resolution gives us a chain complex $( \mathcal{F}(P_.), \mathcal{F}(d_.))$. By definition,

\[ \mathcal{F}_i(A) = H_i(\mathcal{F}(P_.), \mathcal{F}(d_.))  \]

\[  \mathcal{F}_0(A) = \mathcal{F}(A). \]

The functor $\mathcal{F}_i(A)$ doesn't depend on the resolution $(P_.,d_.)$.

For proof, we refer to [6].

Short exact sequence gives rise to long exact sequence of homologies.

This is stated in theorem 6.1.1. Hence $\mathcal{F}_i$ are derived functor of $\mathcal{F}$.

Let $\mathcal{C}$ be a category of right $R$ modules where $R$ is an algebra over a field $\mathbb{K}$ and let $\mathcal{D}$ be category of vector spaces over $\mathbb{K}$. Let $N$ is a given left $R$-module. Then $\mathcal{F}_N : M \rightarrow M \otimes_R N$ is a right exact functor from $\mathcal{C}$ to $\mathcal{D}$. We compute derived functors of $\mathcal{F}_N$. They are called

\[ \mathcal{F}_{N,i}(M) = \Tor_i^R(M,N)  \]

Now we will show how to compute some derived functors from our Anick resolution. Let $A$ be an augmented algebra over $\mathbb{K}$. This makes $\mathbb{K}$ a left and right module. Our aim is to construct the structure of

\[ \Tor_i^A(\mathbb{K},\mathbb{K}) \]

We have our Anick resolution of $\mathbb{K}$

\begin{equation} \begin{tikzcd}
\dots \arrow{r}{d_2} & C_1 \otimes_{\mathbb{K}} A \arrow{r}{d_1} & C_0 \otimes_{\mathbb{K}} A \arrow{r}{d_0} & \mathbb{K} \arrow{r} & 0
\end{tikzcd} \end{equation}

We tensor each element and differentials of the above resolution with $\mathbb{K}$ over $A$. Hence we get

\[ ( C_i \otimes_{\mathbb{K}} A ) \otimes_A \mathbb{K} = C_i \otimes_{\mathbb{K}} \mathbb{K} \simeq C_i. \]

Hence we obtain the following sequence

\[ \begin{tikzcd}
\dots \arrow{r}{d_3 \otimes_A \mathbb{K}} & C_2 \arrow{r}{d_2 \otimes_A \mathbb{K}} & C_1 \arrow{r}{d_1 \otimes_A \mathbb{K}} & C_0.
\end{tikzcd}\]

Let $c \in C_i \otimes 1 \simeq C_i$. Then $d_i(c)$ is of the form

\[ d_i(c) = \sum_l c^l \otimes \sigma ^l  \]

for some $c^l$ and $\sigma^l$. Then

\[ d_i \otimes_A \mathbb{K} (c) = \sum_l v^l \cdot \epsilon ( r^l)  \]

for some $v^l, r^l$ where $\epsilon: A \rightarrow \mathbb{K}$ is the augmentation map. In particular if $A$ is graded i.e. $A = \oplus_i A_i$ with $A_0 = \mathbb{K}$ then

\[ \epsilon|_{A_i} = 0 \hspace{2mm} \text{for} \hspace{2mm} i \neq 0.  \]

A resolution is said to be minimal if

\[ d_i \otimes_A \mathbb{K} = 0 \hspace{2mm}\text{for all}\hspace{1mm}i.  \]

If we have a resolution of the form given in equation (87) and if it is minimal, then

\[   \Tor_i^A(\mathbb{K},\mathbb{K}) \simeq C_i. \]

For algebra $B_1$, its Anick resolution is minimal. But for algebra $B_n$ for $n >1$, its Anick resolution is not minimal as $d_3 \otimes_A \mathbb{K} \neq 0$.

For more details about derived functors and minimal resolution we refer to [5], [6] and [17].

§.§ Computation of $A_\infty$ algebra from Anick resolution

In 1960's Jim Stasheff introduced the concept of $A_\infty$ spaces and $A_\infty$ algebras to study 'group like' topological spaces. In this section we give a brief introduction about $A_\infty$ algebras and computation of $A_\infty$ structures on $\Ext$-algebras. For more details we refer to [7], [8] and [22].

An $A_\infty$-algebra over a field $\mathbb{K}$ (also called a 'strongly homotopy associative algebra') is a $\mathbb{Z}$-graded vector space

\[ A = \bigoplus_{n \in \mathbb{Z}} A^n  \]

endowed with graded maps

\[ m_n : A^{\otimes n} \rightarrow A, \hspace{2mm} n \geq 1,  \]

of degree $2-n$ satisfying the following relations.

* $m_1m_1 = 0$, i.e. $(A,m_1)$ is a differential complex.

* We have

\[ m_1m_2 = m_2( m_1 \otimes \id + \id \otimes m_1) \]

as maps $A^{\otimes2} \rightarrow A$. Here $\id$ denotes the identity map of the space $A$. So $m_1$ is a graded derivation with respect to multiplication $m_2$.

* We have

\[ m_2(\id \otimes m_2 - m_2 \otimes \id) = m_1m_3 +   \]

\[ m_3( m_1 \otimes \id \otimes \id + \id \otimes m_1 \otimes \id + \id \otimes \id \otimes m_1)  \]

as maps $A^{\otimes3} \rightarrow A$. This implies that $m_2$ is associative up to homotopy.

* More generally, for $n \geq 1$, we have

\[ \sum (-1)^{r+st} m_u ( \id^{\otimes r} \otimes m_s \otimes \id^{\otimes t}) = 0  \]

where the sum runs over all decompositions $n = r+s+t$ and $u = r+t+1$.

Following is an immediate consequence of the definition 7.2.1:

If $m_n$ vanishes for all $n \geq 3$, then $A$ is an associative differential graded algebra and conversely each differential graded algebra yields an $A_\infty$-algebra structure with $m_n = 0$ for all $n \geq 3$.

Let $C$ be an $A_\infty$-algebra and let $E = HC$ be the cohomology ring of $C$. There exists an $A_\infty$ structure on $E$ with $m_1=0$ and $m_2$ induced by multiplication on $C$.

For proof and more details about the theorem we refer to [23].

Furthermore the $A_\infty$ structure on $E$, mentioned in theorem 7.2.1 is unique up to quasi-isomorphism[22] of $A_\infty$ algebras.

§.§.§ Merkulov's construction

Let $A$ be a $\mathbb{K}$-algebra and let $(Q_.,d_.)$ be a minimal graded projective resolution of $\mathbb{K}$ by $A$-modules with $Q_0=A$. Here the resolution can be taken as Anick resolution and it is possible to construct a minimal resolution from Anick resolution. Let $U = \Hom_A(Q,Q)$ be the differential graded endomorphism algebra of $(Q_.,d_.)$ with multiplication given by composition. The minimality of the resolution implies that $\Hom_A(Q_.,\mathbb{K})$ is equal to its cohomology.

We compute the structure of an $A_\infty$  algebra on $\Hom_A(Q_.,\mathbb{K})$ by specifying the data of a strong deformation retraction from $U$ to $\Hom_A(Q_.,\mathbb{K})$. We choose maps $i : \Hom_A(Q_.,\mathbb{K}) \rightarrow U$, $p : U \rightarrow \Hom_A(Q_.,\mathbb{K})$ and a chain homotopy map $G: U \rightarrow U$ such that $i$ and $p$ are degree $0$ morphisms of complexes and $G$ is a homogeneous $\mathbb{K}$-linear map of degree $-1$ such that $pi = \id_{\Hom(Q,d)}$ and $\id_U - ip = \delta G - G\delta$, where $\delta$ is the differential on $U$ given by $\delta(f) = df - (-1)^{|f|}fd$ ($|f|$ is the degree of $f$).

We define a family of homogeneous $\mathbb{K}$-linear maps $\{ \lambda_n : U^{\otimes n} \rightarrow U \}_{n \geq 2}$ with $\deg(\lambda_n) = 2-n$ as follows:

* $\lambda_2$ is multiplication on $U$.

* $G\lambda_1 = -\id_U$.

* $\lambda_n = \sum_{\substack{s+t=n \\ s,t \geq 1}} (-1)^{s+1} \lambda_2(G\lambda_s \otimes G\lambda_t)$.

Let $m_1 =0$ and for $n \geq 2$, let $m_n = p\lambda_n i^{\otimes n}$. Then $(\Hom(Q_.,\mathbb{K}), m_1, m_2, m_3, \dots )$ is an $A_\infty$-algebra satisfying the conditions of theorem 7.2.1.

Merkulov's theorem applies to the subcomplex $i(\Hom_A(Q_.,\mathbb{K})$ of $U$ and gives structure maps $m_n = ip\lambda_n$ which together with the subcomplex forms an $A_\infty$ structure.

$ E = \Ext_A( \mathbb{K}, \mathbb{K})$ is isomorphic to $HU$.

So we have seen how to get an $A_\infty$ algebra structure on $\Ext$-algebras.

If one use the Merkulov's construction to compute $A_\infty$-algebra structure associated to algebra $\mathbb{K}\langle x \rangle /(x^2)$ by constructing its Anick resolution then one can find that all higher maps in $A_\infty$ structure vanish for $n \geq 3$. A Koszul algebra canonically carries an $A_\infty$ structure will all higher maps vanish for $n \geq 3$ [7]. Algebra $\mathbb{K}\langle x \rangle /(x^2)$ is Koszul and hence the above computation through Merkulov's construction verifies this fact.

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