# Neutrino signal dependence on gamma-ray burst emission mechanism

Tetyana Pitik  Irene Tamborra  and Maria Petropoulou

###### Abstract

Long duration gamma-ray bursts (GRBs) are among the least understood
astrophysical transients powering the high-energy universe. To date, various
mechanisms have been proposed to explain the observed electromagnetic GRB
emission. In this work, we show that, although different jet models may be
equally successful in fitting the observed electromagnetic spectral energy
distributions, the neutrino production strongly depends on the adopted
emission and dissipation model. To this purpose, we compute the neutrino
production for a benchmark high-luminosity GRB in the internal shock model,
including a dissipative photosphere as well as three emission components, in
the jet model invoking internal-collision-induced magnetic reconnection and
turbulence (ICMART), in the case of a magnetic jet with gradual dissipation,
and in a jet with dominant proton synchrotron radiation. We find that the
expected neutrino fluence can vary up to three orders of magnitude in
amplitude and peak at energies ranging from $10^{4}$ to $10^{8}$ GeV. For our
benchmark input parameters, none of the explored GRB models is excluded by the
targeted searches carried out by the IceCube and ANTARES Collaborations.
However, our work highlights the potential of high-energy neutrinos of
pinpointing the underlying GRB emission mechanism and the importance of
relying on different jet models for unbiased stacking searches.

##  1 Introduction

Gamma-ray bursts (GRBs) are irregular pulses of gamma-rays that have puzzled
astronomers for a long time [1]. Exhibiting a non-thermal spectrum, typically
peaking in $10$–$10^{4}$ keV energy band [2], bursts lasting for more than $2$
s are named long-duration GRBs and are thought to be harbored within
collapsing massive stars [3, 4, 5]. They are the brightest explosions in our
universe and can release isotropic energies as high as $10^{54}$ erg in gamma-
rays over few tens of seconds [6].

The central engine of a long-duration GRB jet can either be a hyper-accreting
black hole or a rapidly spinning magnetar. Because the central engine cannot
be directly observed, its nature can be inferred only indirectly through its
impact on the electromagnetic properties of GRBs (see, e.g., Ref. [7] and
references therein). A bipolar outflow is continuously powered for a certain
time interval, during which gravitational energy [8, 9] (for accreting
systems) or spin energy [10, 11] (for spinning-down systems) is released in
the form of thermal energy or Poynting flux energy, respectively.
Subsequently, the outflow propagates through the star and it is strongly
collimated by the stellar envelope. Once it succeeds to break out of the
stellar surface, it manifests itself as the jet responsible for the GRBs that
we observe at Earth. The dynamical evolution of the jet strongly depends on
the initial conditions of the central engine. If the magnetic field is
negligible, the evolution of the outflow can be well described by the fireball
model [12]. If instead the central engine harbours a strong magnetic field,
the jet dynamics is significantly different [13].

Gamma-ray bursts are candidate sources of ultra-high energy cosmic rays and
high energy neutrinos [14]. In the prompt phase, if the jet contains baryons,
protons and nuclei are expected to be accelerated [15]. If a photon field is
also present, photo-hadronic ($p\gamma$) interactions can lead to a
significant flux of neutrinos [16, 17, 18]. Another copious source of
neutrinos comes from hadronic collisions ($pp$ or $pn$) which, however, are
most efficient inside the progenitor star where the baryon density is large
[19, 20, 21, 15]. Given the typical GRB parameters, neutrinos produced in the
optically thin region are expected to be emitted in the TeV-PeV energy range
[16, 22, 23, 24].

The IceCube Neutrino Observatory routinely detects neutrinos of astrophysical
origin in the TeV–PeV energy range [25, 26, 27, 28]. However, despite the fact
that several sources have been proposed as possible candidates to explain the
neutrino flux that we observe [29, 22, 24, 23, 30], we are still lacking clear
evidence on the sources producing the observed neutrinos. Among the candidate
sources, high-luminosity GRBs are deemed to be responsible for less than
$10\%$ of the observed diffuse emission in the TeV energy range [27, 31]. On
the other hand, over the years, the IceCube and ANTARES Collaborations have
searched for high-energy neutrinos emitted in coincidence with GRBs observed
by the Fermi satellite [32, 33, 31], gradually placing more stringent upper
limits on somewhat optimistic GRB emission models. Recent work suggests that
current limits are still not stringent enough to rule out more realistic
estimations proposed in the literature [34, 35, 15, 36, 37, 38, 39].

Intriguingly, besides the need for increased detection sensitivity, one of the
reasons for the non-detection of GRB neutrinos could be connected to the
theoretical modeling of the neutrino emission, which is strictly linked to the
electromagnetic modeling of the jet. In fact, a comprehensive explanation of
the GRB emission and dissipation mechanism is still lacking due to the failure
of existing models in addressing all observations in the spectral and temporal
domains.

On the other hand, the scarce amount of data on high energy photons and the
related statistical challenges allow for a certain flexibility in fitting the
same set of data with different input models for GRBs–see, e.g., Refs. [40,
41, 42, 43, 44].

Different GRB models may lead to very different predictions for the neutrino
emission. The latter depends on the target photon spectrum and the properties
of the accelerated proton distribution (i.e., energy density, power-law slope,
and maximum energy), both depending on the emission and dissipation mechanisms
as well as the location of the proton acceleration region.

In this work, we compute the neutrino emission for a benchmark high-luminosity
GRB in various jet emission and dissipation scenarios. In particular, we
consider an internal shock (IS) model [45], a dissipative photosphere model in
the presence of ISs (PH-IS) [46], a three-component model (3-COMP) with
emission arising from the photosphere, the IS, and external shock [47], and
the internal-collision-induced magnetic reconnection and turbulence model
(ICMART) [48]. We also compute, for the first time, the neutrino signal
expected in two models where the jet is assumed to be magnetically dominated,
namely a magnetized jet model with gradual dissipation (MAG-DISS) [49, 50],
and a proton synchrotron emission model (p-SYNCH) [51]. Our goal is to make a
fair comparison among the proposed models for dissipation and electromagnetic
emission in GRBs for what concerns the expected neutrino signal.

This paper is organized as follows. In Sec. 2, we outline the basics of the
dynamical evolution of the GRB jets considered in this paper. The main model
ingredients as well as the proton energy distributions are reported in Sec. 3.
The neutrino production mechanism is discussed in Sec. 4. The neutrino
emission is presented in Sec. 5, first in various scenarios involving ISs,
then in the case of magnetized jets, and lastly for the proton synchrotron
mechanism. A discussion on our findings, also in the context of detection
perspectives as well as uncertainties on the input GRB parameters, and
conclusions are reported in Secs. 6 and 7, respectively. The fitting functions
adopted for the photon spectral energy distributions are listed in Appendix A.
A discussion on the dependence of the neutrino emission on the input
parameters for the magnetic model with gradual dissipation is reported in
Appendix B. A comparison of the quasi-diffuse neutrino emission with standard
input assumptions reported in the literature is provided in Appendix C.

##  2 Dynamical evolution of gamma-ray burst jets

In this section, we introduce the main physics describing the jet models
considered in this work. We present the models in the context of kinetic
dominated jets, then focus on two cases of Poynting flux dominated jets, and
the proton synchrotron model. Note that, despite the fact that the proton
synchrotron model has a Poynting luminosity larger than the kinetic one (see
Ref. [52] for a dedicated discussion), we treat it separately from the
Poynting flux dominated jets because it does not require knowledge of the jet
dynamics.

The general GRB model envisages a relativistic jet propagating with Lorenz
factor $\Gamma$, with respect to the central engine frame, and half opening
angle $\theta_{j}$. As long as $\Gamma^{-1}<\theta_{j}$, which is expected to
hold during the prompt phase [53], the radiating region can be considered
spherically symmetric. We therefore use isotropic equivalent quantities
throughout the paper.

The reference frames used in our calculation are the observer frame (on
Earth), the frame of the central engine (laboratory frame), and the jet
comoving frame. A quantity characteristic of the jet is labeled as $X$,
$\tilde{X}$, and $X^{\prime}$, in each of these frames, respectively. For
example, energy is transformed through the following relation:
$\tilde{E}=(1+z)E=(1+z)\mathcal{D}E^{\prime}$; time instead transforms as
$t=(1+z)\tilde{t}=(1+z)\mathcal{D}^{-1}t^{\prime}$, with
$\mathcal{D}=[\Gamma(1-\beta\cos\theta)]^{-1}$ being the Doppler factor,
$\beta=v/c$, $\Gamma=1/\sqrt{1-\beta^{2}}$ the Lorentz boost factor and
$\theta$ the angle of propagation of an ejecta element with respect to the
line of sight. A characteristic quantity of the jet is the isotropic-
equivalent energy, $\tilde{E}_{\rm{iso}}$, which represents the energetic
content of the outflow and it is related to the bolometric energy
$\tilde{E}_{\rm{bol}}$ through the opening angle by the following relation:
${\tilde{E}_{\rm{bol}}=(1-\cos\theta_{j})\tilde{E}_{\rm{iso}}}\approx(\theta_{j}^{2}/2)\tilde{E}_{\rm{iso}}$,
where the approximation holds for small opening angles.

The dominant source of energy in a GRB jet is related to the initial
conditions. The jet is powered by accretion onto a newly formed black hole [3]
or a rapidly spinning massive neutron star [11]. Two mechanisms are invoked to
extract energy from the central compact object and power the GRB jet: neutrino
annihilation [54, 55, 8] or tapping of the spin energy of the central object
by means of magnetic fields [10, 56].

###  2.1 Kinetic dominated jets

We start with the case of a generic fireball composed of photons,
electron/positron pairs, and a small fraction of baryons (primarily protons
and neutrons), with negligible magnetic fields [57, 58]. The dynamical
evolution of the fireball is sketched in Fig. 1 and consists of three phases,
namely acceleration, coasting, and deceleration:

Figure 1: Schematic representation of a GRB jet (not in scale) where energy
dissipation takes place through relativistic shocks. The Lorentz factor
$\Gamma$ is shown as a function of the fireball radius for the case in which
the photosphere occurs in the coasting phase, so that the photospheric radius
($R_{\rm PH}$) lies above the saturation radius ($R_{\rm sat}$). The
photosphere is assumed to produce thermal $\gamma$-rays, the ISs forming at
$R_{\rm IS}$ are thought to produce non-thermal $\gamma$-rays, and the
external shock, which starts to decelerate at $R_{\rm dec}$, is responsible
for the afterglow. When energy dissipation takes place below the photosphere,
non-thermal radiation is also expected from $R_{\rm PH}$.

  1. 1.

Fireball acceleration: A hot relativistic fireball of isotropic energy
$\tilde{E}_{\rm{iso}}=\tilde{L}_{\rm{iso}}\tilde{t}_{\rm{dur}}$ is created and
launched at the radius $R_{0}$ by the central engine emitting energy with
luminosity $\tilde{L}_{\rm{iso}}$ for a time $\tilde{t}_{\rm{dur}}$. Since
after the propagation through the envelope of the progenitor star, the
fireball can be re-born [59], we adopt as size of the jet base
$R_{0}=R_{\star}\theta_{j}$, with $R_{\star}\simeq 10^{11}\,\rm{cm}$ being the
progenitor star radius. The width of the emitted shell is
${\tilde{\Delta}=c\tilde{t}_{\rm{dur}}}$. As the fireball shell undergoes
adiabatic expansion, and while the pair plasma retains relativistic
temperatures, baryons are accelerated by radiation pressure and the bulk
Lorenz factor increases linearly with radius ($\Gamma\propto R$), until it
reaches its maximum value. We assume that the latter coincides with the
dimensionless entropy per baryon $\eta=\tilde{E}_{\rm{iso}}/Mc^{2}$, where $M$
is the baryonic mass injected into the outflow. The maximum Lorenz factor is
achieved at the saturation radius $R_{\rm{sat}}=\eta R_{0}/\Gamma_{0}$, where
$\Gamma_{0}=\Gamma(R_{\star})\simeq 1$–$10$ [60] is the breakout Lorenz
factor.

  2. 2.

Fireball coasting: Beyond $R_{\rm{sat}}$, the flow coasts with
$\Gamma=\Gamma_{\rm sat}\sim\eta=\rm{const}$. As the fireball shell keeps on
expanding, the baryon density, obtained by the mass continuity equation
$\dot{M}=4\pi R^{2}\Gamma\rho^{\prime}c=\rm{const}$ [61] for a relativistic
flow with spherical symmetry, drops as

$n^{\prime}_{b}=\frac{\rho^{\prime}}{m_{p}}=\frac{\dot{M}}{4\pi
m_{p}R^{2}c\Gamma}\simeq\frac{\tilde{L}_{\mathrm{iso}}}{4\pi
R^{2}m_{p}c^{3}\eta\Gamma}\ ,$ (2.1)

where $\rho^{\prime}$ is the baryon density in the comoving frame, $R$ is the
distance from the central engine, and $\tilde{L}_{\rm iso}=\eta\dot{M}c^{2}$.
At a certain point, photons become optically thin to both pair production and
Compton scattering off free leptons associated with baryons entrained in the
fireball. Once the Thomson optical depth
($\tau_{\rm{T}}=n^{\prime}_{l}\sigma_{\mathrm{T}}R/\Gamma$) drops below $1$,
the energy that has not been converted into kinetic energy is released at the
photospheric radius $R_{\rm{PH}}$. Let $\mathcal{R}$ denote the number of
leptons per baryon ($n^{\prime}_{l}=\mathcal{R}n^{\prime}_{p}$), we can define
the critical dimensionless entropy [62]:

$\eta_{\ast}=\left(\frac{\sigma_{\mathrm{T}}\mathcal{R}\tilde{L}_{\mathrm{iso}}\Gamma_{0}}{8\pi
R_{0}m_{p}c^{3}}\right)^{1/4}\ ,$ (2.2)

where ${\sigma_{T}=6.65\times 10^{-25}\mathrm{cm}^{2}}$ is the Thomson cross
section. $\eta_{\ast}$ represents the limiting value of the Lorenz factor
which separates two scenarios: $\eta>\eta_{\ast}$ (the photosphere occurs in
the acceleration phase) and $\eta<\eta_{\ast}$ (the photosphere occurs in the
coasting phase). For our choice of parameters, we will always be in the second
case, thus we can introduce the photospheric radius as the distance such that
$\tau_{\rm{T}}=1$ [63]:

$R_{\mathrm{\rm
PH}}=\frac{\sigma_{\mathrm{T}}\tilde{L}_{\mathrm{iso}}\mathcal{R}}{4\pi\eta^{3}m_{p}c^{3}}\
.$ (2.3)

The radiation coming from the photosphere is the first electromagnetic signal
detectable from the fireball. It emerges peaking at [63]

$k_{B}\tilde{T}_{\mathrm{\rm
PH}}=k_{B}\left(\frac{\tilde{L}_{\mathrm{iso}}}{4\pi
R_{0}^{2}\sigma_{B}}\right)^{\frac{1}{4}}\left(\frac{R_{\mathrm{\rm
PH}}}{R_{\mathrm{sat}}}\right)^{-\frac{2}{3}}\ ,$ (2.4)

where $\sigma_{B}$ is the Stefan-Boltzmann constant and $k_{B}$ the Boltzmann
constant. The energy $\tilde{E}_{\rm PH}$ emerging from the photosphere is
parametrized through $\varepsilon_{\rm PH}=\tilde{E}_{\rm{PH}}/\tilde{E}_{\rm
iso}$.

Since the central engine responsible for the launch of the relativistic jet is
expected to have an erratic activity, the produced outflow is unsteady and
radially inhomogeneous. This causes internal collisions between shells of
matter emitted with time lag $t_{v}$ to occur at a distance [45]

$R_{\mathrm{IS}}\simeq\frac{2ct_{v}\Gamma^{2}}{1+z}\ ;$ (2.5)

this is the IS radius, where a fraction $\varepsilon_{\rm IS}$ of the total
outflow energy ($\tilde{E}_{\rm iso}$) is dissipated, and particles are
accelerated.

  3. 3.

Fireball deceleration: The fireball shell is eventually decelerated [58, 64,
65] by the circumburst medium that can either be the interstellar medium or
the pre-ejected stellar wind from the progenitor before the collapse. Let us
consider an external density profile [66]:

$n_{b}(R)=AR^{-s}\ ,$ (2.6)

with $s=0$ for a homogeneous medium and $s=2$ for a wind ejected at constant
speed. For a thin shell [67], the deceleration radius is defined as the
distance where the swept mass from the circumburst medium is
$m_{\rm{CMB}}=M/\eta$ [or $\Gamma(R_{\rm{dec}})=\eta/2$] [68]:

$R_{\rm{dec}}=\left(\frac{3-s}{4\pi}\frac{\tilde{E}_{\rm
K,\,iso}}{m_{p}c^{2}A\eta^{2}}\right)^{1/(3-s)}\ ;$ (2.7)

in alternative, $R_{\rm{dec}}$ can be obtained from the observed deceleration
time $t_{\rm dec}$ [69]:

$t_{\rm{dec}}\simeq 1.3(1+z)\frac{R_{\rm{dec}}}{\eta^{2}c}\ ,$ (2.8)

where $\tilde{E}_{\rm
K,\,iso}=\tilde{E}_{\rm{iso}}-\tilde{E}_{\gamma,\rm{iso}}$ is the isotropic
equivalent kinetic energy of the ouflow after $\tilde{E}_{\gamma,\rm{iso}}$
has been radiated during the prompt phase. At $R_{\rm{dec}}$, an external
shock forms and propagates into the medium, hence the deceleration radius is
essentially the initial external shock radius.

####  2.1.1 Jet model with internal shocks

For long time, the IS model [45, 70, 71] has been considered as the standard
model for the prompt emission in the literature. Among the merits of this
model there is its ability to naturally explain the variability of the
lightcurves, to provide natural sites for the dissipation of the kinetic
energy of the baryonic fireball, as well as sites for particle acceleration
and non-thermal radiation.

The erratic activity of the central engine is responsible for the creation of
an outflow that can be visualized as being composed of several shells.
Collisions of such shells with different masses and/or Lorenz factors cause
the dissipation of the kinetic energy of the jet at $R_{\rm IS}$111If there is
a large spread in the $\Gamma$ values of the shells, then $R_{\rm IS}$ can
also spread a lot [72, 73, 74]. (see Fig. 1).

Part of the dissipated energy, $\varepsilon_{\rm IS}\tilde{E}_{\rm iso}$, is
used for particle acceleration. Non-thermal electrons (protons) receive a
fraction $\varepsilon_{e}$ ($\varepsilon_{p}$), while a fraction
$\varepsilon_{B}$ goes into the amplification of magnetic fields. In this
scenario, electrons emit synchrotron radiation in the fast cooling regime. The
radiated energy can thus be expressed as
$E^{\prime}_{\gamma,\mathrm{iso}}=\varepsilon_{e}\varepsilon_{\rm
IS}E^{\prime}_{\rm iso}$ and the magnetic field as

$B^{\prime}=\sqrt{8\pi\frac{\varepsilon_{B}}{\varepsilon_{e}}\frac{E^{\prime}_{\gamma,\mathrm{iso}}}{V^{\prime}_{\mathrm{iso}}}}\
.$ (2.9)

The protons co-accelerated with electrons interact with the prompt photons
through photo-hadronic interactions and produce neutrinos, as discussed in
Sec. 4. Here $V^{\prime}_{\rm iso}=4\pi R^{2}_{\gamma}\Gamma c\tilde{t}_{\rm
dur}$ represents the isotropic volume of the jet in the comoving frame.

Within a more realistic setup, various collisions between plasma shells occur
along the jet. Scenarios involving collisions of multiple shells have been
considered [72, 73, 74] and can lead to lower neutrino fluxes. Yet, in this
work, since we aim to compare different jet models, we adopt one
representative shell with average parameters and spectral properties for
simplicity.

####  2.1.2 Jet model with a dissipative photosphere and internal shocks

In the class of photospheric models, it is assumed that the dominant radiation
observed in the prompt phase is produced in the optically thick region below
the photosphere [75]. Depending on the presence of dissipative processes
acting in the optically thick parts of the outflow, photospheric models can be
classified in non-dissipative or dissipative ones.

In the presence of a non-dissipative photosphere, according to the standard
fireball model, the thermal radiation advected with the flow and unaffected by
the propagation is released at $R_{\mathrm{PH}}$, see Fig. 1. Depending on the
dimensionless entropy of the outflow (see Eq. 2.2), this component can be very
bright or highly inefficient and is characterized by the fraction
$\varepsilon_{\rm{PH}}=(\eta/\eta_{\ast})^{8/3}$ [63].

For a dissipative photosphere, strong subphotospheric dissipation is required
in the optically thick inner parts of relativistic outflows in order to
account for the detected non-thermal spectra [76, 77, 78, 79, 80, 81, 82]. In
this scenario, the spectral peak and the low-energy spectrum below the peak
are formed by quasi-thermal Comptonization of seed photons by mildly
relativistic electrons when the Thomson optical depth of the flow is
$1\lesssim\tau_{T}\lesssim 100$ [83, 84]. In the literature, several sub-
photopsheric dissipative mechanisms have been proposed, including ISs at
small-radii [79], collisional nuclear processes [85], or dissipation of
magnetic energy [83]. One of the most attractive features of these models is
their ability to naturally explain the observed small dispersion of the sub-
MeV peak and the high prompt emission efficiency [86, 59, 87], that the
standard version of the IS model cannot easily explain.

The scenario explored in this work considers the main prompt emission as being
released at $R_{\rm PH}$ with a non-thermal spectrum. These photons cross the
IS region and interact with energetic protons accelerated at the IS to produce
neutrinos. We do not consider neutrino production below and at the
photosphere, as this would result in neutrino energies well below the PeV
range that we are interested in (i.e., GeV neutrinos produced in proton-
neutron collisions in the ejecta [88, 89] or TeV neutrinos produced via $pp$
interactions of protons accelerated at sub-photospheric ISs [90, 91, 92, 20,
93]). Indeed, photo-hadronic interactions in the opaque region do not lead to
efficient production of high-energy neutrinos because of inefficient Fermi
acceleration that limits the maximum proton energy to low values [94]. We
stress that the wording “dissipative photosphere” in this work is meant to
highlight the non-thermal nature of the photospheric spectrum and it should
not be associated with the neutrino production region below the photosphere,
as usually done in the literature (see, e.g. [90]).

####  2.1.3 Jet model with three emission components

The three-component GRB model was introduced in Ref. [47], where the authors
found that a thermal component described by a black body (BB) spectrum, a Band
spectrum (sometimes statistically equivalent to a cut-off power-law, CPL) and
a non-thermal power-law (PL) spectrum at high energies (with or without cut-
off) represent a globally better description of the data than the Band
spectral fit for a number of bursts. We refer the reader to Appendix A for
details on the spectral energy distributions of photons.

As argued in Ref. [47], the physical interpretation proposed for the three
components is the following. The BB component, given its weakness, is
interpreted as thermal photospheric emission of a magnetized jet not strongly
affected by sub-photospheric dissipation. The non-thermal emission fitted by
the Band (or CPL) component, given the observed variability, is assumed to be
produced in the optically thin region of the jet from relativistic electrons.
The third PL (or CPL) component, which extends over at least $5$ decades in
energy and sometimes emerges with a slight temporal delay with respect to the
trigger of the burst, is the one with the least clear origin. Because of its
initial temporal variability, it is assumed to be of internal origin; e.g., it
might be due to inverse Compton processes, even if this scenario is not able
to explain the extension of such a component to lower energies or the temporal
delay. Finally, the fact that in some cases the PL component becomes dominant
at the end of the bursts and lasts longer than the prompt emission led to
identify it with the emergence of an early afterglow, which corresponds to the
start of deceleration of the outflow.

###  2.2 Poynting flux dominated jets

When the central compact object is a rapidly rotating black hole threaded by
open magnetic field lines, it is possible to tap the black hole spin energy to
produce Poynting-flux dominated jets [10]. The electromagnetic luminosity of
this jet is much larger than the kinetic luminosity associated to matter.

A characteristic parameter is the magnetization $\sigma$, defined as the ratio
of the Poynting luminosity and the kinetic luminosity:

$\sigma(R)\equiv\frac{L_{B}}{L_{K}}=\frac{B^{2}(R)}{4\pi\Gamma\rho(R)c^{2}}=\frac{B^{\prime
2}(R)}{4\pi\rho^{\prime}(R)c^{2}}\ ,$ (2.10)

where $B^{\prime}(R)$ and $\rho^{\prime}(R)$ are the magnetic field strength
and matter density in the comoving frame at a certain distance $R$ from the
central engine. Hence, the total jet luminosity at any radius is
$L(R)=[1+\sigma(R)]L_{K}(R)$. In this work we consider two models for
magnetized jets: the ICMART model and the gradual magnetic dissipation model,
which we briefly introduce below.

####  2.2.1 ICMART model

Figure 2: Schematic representation of a Poynting flux dominated jet (not in
scale) in the ICMART model. The Lorentz factor $\Gamma$ is shown as a function
of jet radius. The radiation from the photosphere ($R_{\rm PH}$) and ISs
($R_{\rm IS}$) is strongly suppressed and can be at most $1/(\sigma+1)$ of the
total jet energy (see Eq. 2.10); typical values for the magnetization
parameter $\sigma$ are shown. The emitting region is located at $R_{\rm
ICMART}$, where magnetic reconnection causes a strong discharge of magnetic
energy and the emission of gamma-rays. The magnetization at $R_{\rm ICMART}$
is $\sigma_{\rm in}$ and $\sigma_{\rm end}$ in the beginning and at the end of
an ICMART event, respectively.

The ICMART model [48] considers Poynting flux dominated jets, whose energy is
dissipated and radiated away at very large radii from the central engine, as
shown in Fig. 2. The main motivation behind this model relies on the non-
detection (or detection of a very weak) photospheric component in the spectra
of some GRBs, hinting that the jet composition cannot be largely Poynting flux
dominated at the photosphere.

The GRB central engine intermittently ejects an unsteady jet with variable
Lorentz factor and with a nearly constant degree of magnetization
$\sigma_{0}\equiv\sigma(R_{0})$. Such a jet is composed by many discrete
magnetized shells which collide at $R_{\mathrm{IS}}$ (see Eq. 2.5 and Fig. 2).
Yet, the kinetic energy dissipated at the ISs is smaller by a factor
[$1+\sigma(R_{\mathrm{IS}})$] with respect to the energy available in the
traditional IS model. Hence, the total energy emitted in radiation could be
completely negligible at this stage.

In the optically thin region, the early internal collisions have the role of
altering, and eventually destroying, the ordered magnetic field configuration,
triggering the first reconnection event. The ejection of plasma from the
reconnection layer would disturb the nearby ambient plasma and produce
turbulence, facilitating more reconnection events which would lead to a
runaway catastrophic release of the stored magnetic field energy at the radius
defined as $R_{\mathrm{ICMART}}$. This would correspond to one ICMART event,
which would compose one GRB pulse. Other collisions that trigger other
reconnection-turbulence avalanches would give rise to other pulses.

This model successfully reproduces the observed GRB lightcurves with both fast
and slow components [95]. The slow component, related to the central engine
activity, would be caused by the superposition of emission from all the mini-
jets due to multiple reconnection sites, while the erratic fast component
would be related to the mini-jets pointing towards the observer.

####  2.2.2 Magnetized jet model with gradual dissipation

In this scenario, the energy dissipation through reconnection starts below the
jet photosphere and occurs gradually over a wide range of radii [96, 13], as
schematically shown in Fig. 3. Following Refs. [49, 50], we consider
magnetized outflows with a striped-wind magnetic field structure, where energy
is gradually dissipated through magnetic reconnection until the saturation
radius. This model can naturally explain the double-hump electromagnetic
spectra sometimes observed [50].

Figure 3: Schematic representation of a Poynting flux dominated jet (not in
scale) in the gradual energy dissipation model. The Lorentz factor $\Gamma$ is
shown as a function of jet radius. The radiation from the photosphere ($R_{\rm
PH}$) can be very bright, depending on the initial magnetization $\sigma_{0}$
of the outflow. Typical values for the magnetization parameter $\sigma$ are
shown. The emitting region is located between $R_{\rm PH}$ and $R_{\rm sat}$,
where magnetic reconnection causes the dissipation of magnetic field energy,
the emission of thermal gamma-rays at $R_{\rm PH}$ and synchrotron radiation
from accelerated electrons in the optically thin region up to $R_{\rm sat}$.

The jet is injected at $R_{0}\sim 10^{7}$ cm with magnetization $\sigma_{0}\gg
1$ and Lorentz factor $\Gamma_{0}=\sqrt{\sigma_{0}+1}\approx\sigma_{0}$. As it
propagates, the magnetic field lines of opposite polarity reconnect, causing
the magnetic energy to be dissipated at a rate [96]:

$\dot{E}_{\mathrm{diss}}=-\frac{dL_{B}}{dR}=-\frac{d}{dR}\left(\frac{\sigma}{\sigma+1}L\right)\propto
R^{1/3}\ ,$ (2.11)

where $\sigma(R)$ is obtained from the conservation of the total specific
energy ${\Gamma(R)\sigma(R)=\Gamma_{0}\sigma_{0}}$.

The Lorentz factor of the flow evolves as [96]

$\Gamma(R)=\Gamma_{\mathrm{sat}}\left(\frac{R}{R_{\mathrm{sat}}}\right)^{1/3}\
,$ (2.12)

until the saturation radius $R_{\mathrm{sat}}=\lambda\Gamma^{2}_{\rm sat}$
(see Fig. 3), where $\lambda$ is connected to the characteristic length scale
over which the magnetic field lines reverse polarity. This length scale can be
related to the angular frequency of the central engine (e.g., of millisecond
magnetars) or with the size of the magnetic loops threading the accretion disk
[97].

Motivated by results of particle-in-cell (PIC) simulations of magnetic
reconnection in magnetically dominated electron-proton plasmas [98, 99, 100],
we assume that half of the dissipated energy in Eq. 2.11 is converted in
kinetic energy of the jet, while the other half goes into particle
acceleration and is redistributed among electrons and protons222Rough energy
equipartition between magnetic field, protons and electron-positron pairs is
also found in kinetic simulations of reconnection in pair-proton plasmas
[101].. In particular, the fraction of energy which goes into electrons is
[100]

$\varepsilon_{e}\approx\frac{1}{4}\left(1+\sqrt{\frac{\sigma}{10+\sigma}}\right)\
,$ (2.13)

while the one that goes into protons has been extracted from Fig. 20 of Ref.
[100] and is $\varepsilon_{p}\sim 1-\varepsilon_{e}$. A fraction $\xi$ of
electrons injected into the dissipation region are accelerated into a power-
law distribution $n^{\prime}_{e}(\gamma^{\prime}_{e})\propto\gamma^{\prime-
k_{e}}_{e}$ in the interval
$[\gamma^{\prime}_{e,\mathrm{min}},\gamma^{\prime}_{e,\mathrm{max}}]$ with the
minimum electron Lorentz factor being

$\gamma^{\prime}_{e,\mathrm{min}}(R)=\frac{k_{e}-2}{k_{e}-1}\frac{\varepsilon_{e}}{2\xi}\sigma(R)\frac{m_{p}}{m_{e}}\
,$ (2.14)

and $\gamma^{\prime}_{e,\mathrm{max}}$ is the maximum electron energy obtained
by equating the acceleration time and the total cooling time. The power-law
slope of the accelerated particles in relativistic reconnection depends on the
plasma magnetization in a way that harder spectra ($k_{e}<2$) are obtained for
$\sigma\gg 1$ [102, 103, 104]. Here, we adopt the following parameterization
for the electron power-law slope [100]:

$k_{e}(\sigma)\approx 1.9+0.7/\sqrt{\sigma}\ .$ (2.15)

The proton spectrum will be discussed in detail in Sec. 5.2.2.

###  2.3 Proton synchrotron model

Recently, Refs. [105, 40, 106] have analyzed the spectra of a sample of GRBs
for which data down to the soft X-ray band and, in some cases, in the optical
are available. This extensive work has established the common presence of a
spectral break in the low energy tail of the prompt spectra and led to realize
that the spectra could be fitted by three power-laws. The spectral indices
below and above the break are found to be $\alpha_{1}\simeq-2/3$ and
$\alpha_{2}\simeq-3/2$ respectively, while the photon index of the third PL is
$\beta<2$. The values of all photon indexes are consistent with the predicted
values for the synchrotron emission in a marginally fast cooling regime [107].
However, if electrons are responsible for the prompt emission, then the
parameters of the jet have to change drastically with respect to the standard
scenario, in which the emission takes place at relatively small radii and with
strong magnetic fields in situ. One possible way out to this has been
discussed in Ref. [51], where protons are considered to be the particles which
radiate synchrotron emission in the marginally fast cooling regime; in this
way, it is possible to recover the typical emitting region size at
$R_{\gamma}\simeq 10^{13}$ cm.

##  3 Main model ingredients

In this section, we outline some of the quantities characterizing the
energetics and geometry of the jet for all models. We also introduce the
target particle distributions.

###  3.1 Reference model parameters

The gamma-ray emission is assumed to originate from an isotropic volume
$V^{\prime}_{\rm{iso,s}}=4\pi R_{\gamma}^{2}\Delta^{\prime}_{s}$, where
${\Delta^{\prime}_{s}=R_{\gamma}/2\Gamma}$ is the comoving thickness of the
emitting shell and $R_{\gamma}$ is the distance from the central engine where
the electromagnetic radiation is produced. Dissipation–whether it occurs in
the photosphere, in the optically thin region (e.g., ISs) or external
shocks–causes the conversion of a fraction $\varepsilon_{d}$ of the total jet
energy $\tilde{E}_{\rm{iso}}$ into thermal energy, bulk kinetic energy, non-
thermal particle energy, and magnetic energy. The energy stored in
relativistic electrons, protons, and magnetic fields in the emitting region
can be parameterized through the fractions $\varepsilon_{p}$,
$\varepsilon_{e}$ and $\varepsilon_{B}$, respectively.

These parameters ignore the detailed microphysics at the plasma level, but
allow to establish a direct connection with the observables. The dissipation
efficiency $\varepsilon_{\rm IS}=0.2$ has been chosen by following Refs. [70,
72]. In addition, for the IS models, we rely on PIC simulations of mildly
relativistic shocks in electron-ion plasma. Recently, a relatively long 2D PIC
simulation has been performed [108] and it has been shown that quasi-parallel
shocks can be efficient particle accelerators. The energy fractions going into
non-thermal protons, electrons and the turbulent magnetic field are found to
be $\varepsilon_{p}\simeq 0.1$, $\varepsilon_{e}\simeq 5\times 10^{-4}$ and
$\varepsilon_{B}\gtrsim 0.1$, respectively. These results are valid for a
shock with Lorentz factor $\Gamma_{\rm sh}=1.5$. We know, on the other hand,
that efficient energy dissipation through ISs can take place only if a large
spread in Lorentz gamma factors is present within a kinetic dominated jet
[72], which would lead to $1\lesssim\Gamma_{\rm sh}\lesssim 5$. Unfortunately,
as of today, the midly relativistic regime is poorly explored, and a study of
the transition regime from non relativistic to ultrarelativistic is still
necessary. Nevertheless, the results from [108] can be considered as
indicative for our case. Given that for the relativistic regime it has been
found $\varepsilon_{e}\lesssim 0.1$ [109], we conservatively adopt
$\varepsilon_{e}=0.01$, while $\varepsilon_{p}=0.1$ and $\varepsilon_{B}=0.1$.
For the magnetized jet models, instead, these parameters are found to depend
on the magnetization of the jet, as we will see later. Finally, at the
relativistic external shock, in the deceleration phase, we adopt
$\varepsilon_{e}=4\times 10^{-2}$, $\varepsilon_{B}=10^{-4}$ and
$\varepsilon_{p}=1-\varepsilon_{e}-\varepsilon_{B}$, which turn out to be
constrained by observations for a number of cases [110, 111, 112]. In
addition, we use $k_{e}=k_{p}=2.5$ for the power slope, motivated by PIC
simulations for ultra-relativistic shocks [113].

For what concerns the energetics of our reference jet, motivated by recent
observations of GRB afterglows [114], we choose
${\tilde{E}_{\rm{iso}}=3.4\times 10^{54}}$ ergs, where a typical opening angle
of $\theta_{j}=3$ degrees is adopted. Our benchmark Lorenz factor is
$\Gamma=\Gamma_{\rm sat}=300$ [115, 116]. The duration of the burst is taken
to be $\tilde{t}_{\rm dur}=100\,{\rm s}/(1+z)$, where $z=2$ is the redshift we
adopt for our reference GRB. Finally we use $t_{v}=0.5$ s as the variability
timescale of the GRB lightcurve [31]. The parameters adopted for each model
for our benchmark GRB are summarized in Table 1.

Table 1: Characteristic parameters assumed for our benchmark GRB jet for the
scenarios considered in this paper: internal shock (IS) model, dissipative
photosphere model with internal shocks (PH-IS), three components model
(3-COMP), ICMART model, magnetized jet model with gradual dissipation (MAG-
DISS), and proton synchrotron model (p-SYNCH). In the case of quantities
varying along the jet, the variability range is reported. For the magnetic
model with gradual dissipation, the electron fraction, the electron power-law
index, and the proton power-law index are defined in Eqs. 2.13, and 2.15,
respectively.

Parameter Symbol Model IS PH-IS 3-COMP ICMART MAG-DISS p-SYNCH Total jet
energy $\tilde{E}_{\rm iso}$ $3.4\times 10^{54}$ erg n/a Jet opening angle
$\theta_{j}$ $3^{\circ}$ Lorentz boost factor $\Gamma$ 300 Redshift $z$ 2
Duration of the burst $t_{\rm dur}$ $100$ s Variability time scale $t_{v}$
$0.5$ s Dissipation efficiency $\varepsilon_{d}$ $\varepsilon_{\rm IS}=0.2$
n/a $\varepsilon_{d}=0.35$ $0.24$ n/a Electron energy fraction
$\varepsilon_{e}$ $0.01$ $0.5$ $0.35-0.36$ n/a Proton energy fraction
$\varepsilon_{p}$ $0.1$ $0.5$ $0.64-0.65$ n/a Electron power-law index $k_{e}$
$2.2$ n/a $2.4-2.6$ n/a Proton power-law index $k_{p}$ $2.2$ $2$ $2.4-2.6$
$2.6$ Magnetization at $R_{\gamma}$ $\sigma$ n/a $45$ $1.35-1.81$ n/a

A useful quantity that allows a comparison among different models is the
radiative efficiency of the jet, defined as the fraction of the total jet
energy which is radiated in photons [117]:

$\eta_{\gamma}=\frac{\tilde{E}_{\gamma,\rm{iso}}}{\tilde{E}_{\rm iso}}\ .$
(3.1)

For example $\eta_{\gamma}=\varepsilon_{\rm{PH}}$ when the dominant radiation
is of photospheric origin or $\eta_{\gamma}=\varepsilon_{\rm
IS}\varepsilon_{e}$ when the radiation is produced at the IS, assuming a fast
cooling regime for electrons.

###  3.2 Spectral energy distribution of protons

For the purposes of this work, it is sufficient to assume that protons and
electrons in the dissipation site are accelerated via Fermi-like
mechanisms333In the reconnection region there are various particle
acceleration sites, see e.g. Ref. [118]. It remains a matter of active
research what is the dominant process responsible for the formation of the
power-law, see e.g. Refs. [102, 103, 118, 119, 120]. The accelerated particles
acquire a non-thermal energy distribution that can be phenomenologically
described as [121]:

$n^{\prime}_{p}(E^{\prime}_{p})=AE^{\prime-k}_{p}\
\exp{\left[-\left(\frac{E^{\prime}_{p}}{E^{\prime}_{p,\rm{max}}}\right)^{\alpha_{p}}\right]}\Theta(E^{\prime}_{p}-E^{\prime}_{p,\rm{min}})\
,$ (3.2)

where
$A=U^{\prime}_{p}\left[\int_{E^{\prime}_{p,\rm{min}}}^{E^{\prime}_{p,\rm{max}}}n^{\prime}_{p}(E^{\prime}_{p})E^{\prime}_{p}dE^{\prime}_{p}\right]^{-1}$
is the normalization of the spectrum (in units of $\rm{GeV}^{-1}\rm{cm}^{-3}$)
and $\Theta$ is the Heaviside function, with
$U^{\prime}_{p}=\varepsilon_{p}\varepsilon_{d}E^{\prime}_{\rm{iso}}$ being the
fraction of the dissipated jet energy that goes into acceleration of protons.
The power-law index is found to be $k\approx 2.3$ in the ultra-relativistic
shock limit in semi-analytical and Monte Carlo simulations, although it is
predicted to be steeper from particle-in-cell simulations [113]. The power-law
index is instead $k=2$ for a non-relativistic shock [122], while it depends on
the jet magnetization for magnetically dominated jets, as we will see later.
The exponential cut-off with $\alpha_{p}$ is due to energy losses of protons
and we adopt $\alpha_{p}=2$ following Ref. [123],
$E^{\prime}_{p,\mathrm{min}}$ is the minimum energy of the protons that are
injected within the acceleration region, and $E^{\prime}_{p,\mathrm{max}}$ is
the maximum proton energy. The latter is constrained by the Larmor radius
being smaller than the size of the acceleration region, or imposing that the
acceleration timescale,

$t^{\prime-1}_{p,\rm{acc}}=\frac{\zeta ceB^{\prime}}{E^{\prime}_{p}}\ ,$ (3.3)

is shorter than the total cooling timescale for protons. Here $\zeta=1$ is the
acceleration efficiency adopted throughout this work. The total cooling
timescale is given by

$t^{\prime-1}_{p,\mathrm{cool}}=t^{\prime-1}_{\mathrm{sync}}+t^{\prime-1}_{p,\mathrm{IC}}+t^{\prime-1}_{p,\mathrm{BH}}+t^{\prime-1}_{p\gamma}+t^{\prime-1}_{p,\mathrm{hc}}+t^{\prime-1}_{p,\mathrm{ad}}\
;$ (3.4)

where $t^{\prime}_{\rm{sync}}$, $t^{\prime}_{p,\rm{IC}}$,
$t^{\prime}_{p,\rm{BH}}$, $t^{\prime}_{p\gamma}$, $t^{\prime}_{p,\rm{hc}}$,
$t^{\prime}_{p,\rm{ad}}$ are the proton synchrotron (sync), inverse Compton
(IC), Bethe-Heitler ($p\gamma\rightarrow pe^{+}e^{-}$, BH), hadronic (hc) and
adiabatic (ad) cooling times, respectively. They are defined as follows [61,
124, 125]:

$\displaystyle t^{\prime-1}_{p,\mathrm{sync}}$ $\displaystyle=$
$\displaystyle\frac{4\sigma_{T}m^{2}_{e}E^{\prime}_{p}B^{\prime
2}}{3m^{4}_{p}c^{3}8\pi}\ ,$ (3.5) $\displaystyle
t^{\prime-1}_{p,\mathrm{IC}}$ $\displaystyle=$
$\displaystyle\frac{3(m_{e}c^{2})^{2}\sigma_{T}c}{16\gamma^{\prime
2}_{p}(\gamma^{\prime}_{p}-1)\beta^{\prime}_{p}}\int_{E^{\prime}_{\gamma,\mathrm{min}}}^{E^{\prime}_{\gamma,\mathrm{max}}}\frac{dE^{\prime}_{\gamma}}{E^{\prime
2}_{\gamma}}F(E^{\prime}_{\gamma},\gamma^{\prime}_{p})n^{\prime}_{\gamma}(E^{\prime}_{\gamma})\
,$ (3.6) $\displaystyle t^{\prime-1}_{p,\mathrm{BH}}$ $\displaystyle=$
$\displaystyle\frac{7m_{e}\alpha\sigma_{T}c}{9\sqrt{2}\pi m_{p}\gamma^{\prime
2}_{p}}\int_{\gamma^{\prime-1}_{p}}^{\frac{E^{\prime}_{\gamma,\mathrm{max}}}{m_{e}c^{2}}}d\epsilon^{\prime}\frac{n^{\prime}_{\gamma}(\epsilon^{\prime})}{\epsilon^{\prime
2}}\left\\{(2\gamma^{\prime}_{p}\epsilon^{\prime})^{3/2}\left[\ln(\gamma^{\prime}_{p}\epsilon^{\prime})-\frac{2}{3}\right]+\frac{2^{5/2}}{3}\right\\}\
,$ (3.7) $\displaystyle t^{\prime-1}_{p\gamma}$ $\displaystyle=$
$\displaystyle\frac{c}{2\gamma^{\prime
2}_{p}}\int_{\frac{E_{\mathrm{th}}}{2\gamma^{\prime}_{p}}}^{\infty}dE^{\prime}_{\gamma}\frac{n^{\prime}_{\gamma}(E^{\prime}_{\gamma})}{E^{\prime
2}_{\gamma}}\int_{E_{\mathrm{th}}}^{2\gamma^{\prime}_{p}E^{\prime}_{\gamma}}dE_{r}E_{r}\sigma_{p\gamma}(E_{r})K_{p\gamma}(E_{r})\
,$ (3.8) $\displaystyle t^{\prime-1}_{\mathrm{hc}}$ $\displaystyle=$
$\displaystyle cn^{\prime}_{p}\sigma_{pp}K_{pp}\ ,$ (3.9) $\displaystyle
t^{\prime-1}_{p,\mathrm{ad}}$ $\displaystyle=$
$\displaystyle\frac{c\Gamma}{R}\ .$ (3.10)

In the definitions above, $\epsilon^{\prime}=E^{\prime}_{\gamma}/m_{e}c^{2}$,
$\gamma^{\prime}_{p}=E^{\prime}_{p}/m_{p}c^{2}$, and ${\alpha=1/137}$ is the
fine structure constant. The cross sections $\sigma_{p\gamma}$ and
$\sigma_{pp}$, for $p\gamma$ and $pp$ interactions respectively, are taken
from Ref. [126]. The function $F(E^{\prime}_{\gamma},\gamma^{\prime}_{p})$ is
provided in Ref. [127], while $K_{p\gamma}$ is the inelasticity of $p\gamma$
collisions [61]:

$K_{p\gamma}(E_{r})=\begin{cases}0.2\,\,\,\,E_{\mathrm{th}}<E_{r}<1\
\mathrm{GeV}\\\ 0.6\,\,\,\,E_{r}>1\ \mathrm{GeV}\ ,\end{cases}$ (3.11)

where
$E_{r}=\gamma^{\prime}_{p}E^{\prime}_{\gamma}(1-\beta^{\prime}_{p}\cos\theta^{\prime})$
is the relative energy between a proton with gamma factor
$\gamma^{\prime}_{p}$ and a photon of energy $E^{\prime}_{\gamma}$, whose
directions form an angle $\theta^{\prime}$ in the comoving system,
$E_{\mathrm{th}}=0.15\,\rm{GeV}$ is the threshold for the photo-hadronic
interaction, $n^{\prime}_{\gamma}(E^{\prime}_{\gamma})$ is the target photon
density field (in units of $\mathrm{GeV}^{-1}\mathrm{cm}^{-3}$), $K_{pp}=0.8$,
and $n^{\prime}_{p}$ is the comoving proton density defined as
$n^{\prime}_{p}=n^{\prime}_{b}/2$, where $n^{\prime}_{b}$ is the baryonic
density defined in Eq. 2.1. As we will see in Sec. 5.3, the proton synchrotron
scenario is such that the properties of the proton distribution (e.g., minimum
energy, power-law slope), as well as the shape of energy distribution itself,
can be directly inferred from the observed GRB prompt spectra.

##  4 Neutrino production in the gamma-ray burst jet

The simultaneous presence of a high density target photon field in the site of
proton acceleration–that can be radiated by co-accelerated electrons, by
protons themselves or have an external origin–leads to an efficient production
of high-energy neutrinos through photo-hadronic interactions. Since the number
of target photons is always much larger than the number density of non-
relativistic (cold) protons in all cases of study, we neglect the $pp$
contribution.

Photo-hadronic interactions lead to charged pion and kaon (as well as neutron)
production, which subsequently cool and decay in muons and neutrinos.
According to the standard picture, pion production occurs through the
$\Delta(1232)$ resonance channel:

$p\,\,+\,\,\gamma\,\,\longrightarrow\,\,\Delta^{+}\,\,\longrightarrow\,\,\begin{cases}n+\pi^{+}\quad
1/3\,\,\text{of all cases}\\\ p+\pi^{0}\quad\,2/3\,\,\,\text{of all
cases}\end{cases}$ (4.1)

followed by the decay chain

$\displaystyle\pi^{+}\rightarrow$ $\displaystyle\mu^{+}+\nu_{\mu}$ (4.2)
$\displaystyle\mu^{+}\rightarrow\bar{\nu}_{\mu}+\nu_{e}\,+e^{+}\ .$ (4.3)

In order to accurately estimate the neutrino spectral energy distribution and
the related neutrino flavor ratio, we rely on the photo-hadronic interaction
model of Ref. [123] (model Sim-B and Sim-C) based on SOPHIA [128]. The latter
includes higher resonances, direct and multi-pion production contributions.
Note that, although we compute the neutrino and antineutrino spectral
distributions separately, in the following we do not distinguish between them
unless otherwise specified.

Given the photon and proton energy distributions in the comoving frame,
$n^{\prime}_{\gamma}(E^{\prime}_{\gamma})$ and
$n^{\prime}_{p}(E^{\prime}_{p})$, the production rate of secondary particles
is given by [123] (in units of $\rm{GeV}^{-1}\rm{cm}^{-3}\rm{s}^{-1}$)

$Q^{\prime}_{l}(E^{\prime}_{l})=\int_{E^{\prime}_{l}}^{\infty}\frac{dE^{\prime}_{p}}{E^{\prime}_{p}}n^{\prime}_{p}(E^{\prime}_{p})\int_{E_{\mathrm{th}}/2\gamma^{\prime}_{p}}^{\infty}dE^{\prime}_{\gamma}n^{\prime}_{\gamma}(E^{\prime}_{\gamma})cR_{\alpha}(x,y)\
,$ (4.4)

where $x=E^{\prime}_{l}/E^{\prime}_{p}$ is the fraction of proton energy going
into daughter particles, $y=\gamma^{\prime}_{p}E^{\prime}_{\gamma}$, and $l$
stands for $\pi^{+}$, $\pi^{-}$, $\pi^{0}$, and $K^{+}$. Since kaons suffer
less from radiative cooling than charged pions due to their larger mass and
shorter lifetime, their contribution to the neutrino flux becomes important at
high energies [129, 36, 130], whilst it is sub-leading at lower energies,
given the low branching ratio for their production. The “response function”
$R_{l}(x,y)$ contains all the information about the interaction type (cross
section and multiplicity of the products); we refer the interested reader to
Ref. [123] for more details.

Once produced, the charged mesons undergo different energy losses before
decaying into neutrinos. Their energy distribution at decay is

$Q^{\prime\mathrm{dec}}_{l}(E^{\prime}_{l})=Q^{\prime}_{l}(E^{\prime}_{l})\left[1-\exp\left(-\frac{t^{\prime}_{l,\mathrm{cool}}m_{l}}{E^{\prime}_{l}\tau^{\prime}_{l}}\right)\right]\
,$ (4.5)

with $t^{\prime}_{l,{\rm{cool}}}$ being the cooling time scale and
$\tau^{\prime}_{l}$ the lifetime of the meson $l$. The neutrino energy
distribution originating from the decay processes like the one in Eq. 4.2 is

$Q^{\prime}_{\nu_{\alpha}}(E^{\prime}_{\nu_{\alpha}})=\int_{E^{\prime}_{\nu_{\alpha}}}^{\infty}Q^{\prime\mathrm{dec}}_{l}(E^{\prime}_{l})\frac{1}{E^{\prime}_{l}}F_{l\rightarrow\nu_{\alpha}}\left(\frac{E^{\prime}_{\nu_{\alpha}}}{E^{\prime}_{l}}\right)\
,$ (4.6)

where $F_{l\rightarrow\nu_{\alpha}}$ is defined in Ref. [121] for ultra-
relativistic parent particles. The same procedure is followed for
antineutrinos.

The steps above also allow to compute the spectra of charged muons. Again, the
cooled muon spectra are derived as in Eq. 4.5 and the neutrinos generated by
the muon decay are computed following Ref. [121].

The total neutrino injection rate
$Q^{\prime}_{\nu_{\alpha}}(E^{\prime}_{\nu_{\alpha}})$ at the source is
obtained by summing over the contributions from all channels. Finally, the
fluence for the flavor $\nu_{\alpha}$ at Earth from a source at redshift $z$
is (in units of $\mathrm{GeV}^{-1}\mathrm{cm}^{-2}$)

$\Phi_{\nu_{\alpha}}(E_{\nu_{\alpha}},z)=\hat{N}\frac{(1+z)^{2}}{4\pi
d^{2}_{L}(z)}\underset{\beta}{\sum}P_{\nu_{\beta}\rightarrow\nu_{\alpha}}(E_{\nu_{\alpha}})Q^{\prime}_{\nu_{\beta}}\left[\frac{E_{\nu_{\alpha}}(1+z)}{\Gamma}\right]\
,$ (4.7)

where [29]

$\displaystyle P_{\nu_{e}\rightarrow\nu_{\mu}}$
$\displaystyle=P_{\nu_{\mu}\rightarrow\nu_{e}}=P_{\nu_{e}\rightarrow\nu_{\tau}}=\frac{1}{4}\sin^{2}2\theta_{12}\
,$ (4.8) $\displaystyle P_{\nu_{\mu}\rightarrow\nu_{\mu}}$
$\displaystyle=P_{\nu_{\mu}\rightarrow\nu_{\tau}}=\frac{1}{8}(4-\sin^{2}2\theta_{12})\
,$ (4.9) $\displaystyle P_{\nu_{e}\rightarrow\nu_{e}}$
$\displaystyle=1-\frac{1}{2}\sin^{2}2\theta_{12}\ ,$ (4.10)

with $\theta_{12}\simeq 33.5$ degrees [131],
$P_{\nu_{\beta}\rightarrow\nu_{\alpha}}=P_{\bar{\nu}_{\beta}\rightarrow\bar{\nu}_{\alpha}}$,
and $\hat{N}=V^{\prime}_{\rm iso,s}t_{\rm dur}$ [132] being the normalization
factor depending on the volume of the interaction region. The luminosity
distance $d_{L}(z)$ is defined in a flat $\Lambda$CDM cosmology as

$d_{L}(z)=(1+z)\frac{c}{H_{0}}\int_{0}^{z}\frac{dz^{\prime}}{\sqrt{\Omega_{\Lambda}+\Omega_{M}(1+z^{\prime})^{3}}}$
(4.11)

with $\Omega_{M}=0.315$, $\Omega_{\Lambda}=0.685$ and the Hubble constant
$H_{0}=67.4$ km s-1 Mpc-1 [133].

##  5 Results: Gamma-ray burst neutrino emission

Each of the dissipation mechanisms introduced in Sec. 2, according to the
radius at which it takes place, leads to different photon energy
distributions. In Appendix A we report the empirical functions usually adopted
to fit the observed photon spectra. For each of the GRB models considered in
this section, we assume that the spectral energy distribution of photons is
either given by one of the fitting functions or a combination of them. In this
section, we investigate the neutrino production in the prompt phase for each
scenario.

###  5.1 Kinetic dominated jets

####  5.1.1 Jet model with internal shocks

Figure 4: Left: Inverse cooling timescales for protons at the IS radius as
functions of the proton energy in the comoving frame for our benchmark GRB,
see Table 1. The thin solid lines mark the individual cooling processes
introduced in Sec. 3.2; the thick black and red solid lines represent the
total cooling timescale and the acceleration timescale, respectively. The red
star marks the maximum comoving proton energy such that $t^{\prime-1}_{p,{\rm
cool}}=t^{\prime-1}_{p,{\rm acc}}$. Protons are mainly cooled by adiabatic
expansion and $p\gamma$ interactions. Right: Analogous to the left panel, but
for the inverse cooling timescales for pions, muons, and kaons. The dominant
energy losses in this case are adiabatic cooling at low energies and
synchrotron cooling at higher energies.

We focus on the IS model introduced in Sec. 2.1.1 with the photon spectrum
produced at the IS radius and described by the Band function in Eq. A.2. The
radiative efficiency is $\eta_{\gamma}=\varepsilon_{\rm
IS}\varepsilon_{e}\simeq 0.002$ (see Table 1).

In order to establish the relative importance of the various energy loss
processes in this scenario, we compute the proton and the secondary particle
($K^{\pm}$, $\pi^{\pm}$ and $\mu^{\pm}$) cooling times as illustrated in Sec.
3.2. The cooling times are shown in Fig. 4 as functions of the particle energy
in the comoving frame. With the parameters adopted for our benchmark GRB,
protons are mainly cooled by adiabatic expansion up to $E^{\prime}_{p,\rm
max}$ (left panel of Fig. 4), with the second dominant energy loss mechanisms
being photo-hadronic interaction at intermediate energies, and synchrotron
loss at higher energies. For mesons and muons (right panel of Fig. 4)
adiabatic and synchrotron cooling at low and high energies, respectively, are
the two dominant cooling processes.

Following Sec. 4, we compute the neutrino production rate in the comoving
frame at $R_{\rm IS}$ and the correspondent fluence at Earth including flavor
conversions. The results are shown in the right panel of Fig. 5, while the
photon spectrum described by the Band function is shown in the left panel.

Figure 5: Left: Band photon fluence observed at Earth for our benchmark GRB in
the IS model, see Table 1. Right: Correspondent
$\nu_{\alpha}+\bar{\nu}_{\alpha}$ fluence in the observer frame (in red for
the electron flavor and in blue for the muon flavor) in the presence of flavor
conversions. The fluence for the muon flavor peaks at $E^{\rm
peak}_{\nu}=4\times 10^{7}$ GeV.

Our results are in good agreement with analogous estimations reported in the
literature for comparable input parameters, see e.g. Ref. [134]. In this
scenario, the fluence for the muon flavor peaks at $E^{\rm peak}_{\nu}=4\times
10^{7}$ GeV and rapidly declines at higher energies. The effects due to the
cooling of kaons are not visible because, as shown in Fig. 4, the maximum
proton energy is more than one order of magnitude lower than the one at which
kaons cool by synchrotron radiation, and the pion cooling starts around
$\lesssim E^{\prime}_{p,\rm max}/4$.

####  5.1.2 Jet model with a dissipative photosphere and internal shocks

We now explore the model introduced in Sec. 2.1.2 and consider a jet with an
efficient photospheric emission, described by a Band spectrum peaking at the
energy given by Eq. A.4, and undergoing further IS dissipation. At $R_{\rm
IS}$, protons and electrons are efficiently accelerated and turbulent magnetic
fields may build up. In this scenario, electrons cool, other than by emitting
synchrotron radiation, also by Compton up-scattering of the non-thermal
photospheric photons. As we are interested in investigating the case where the
photospheric emission is dominant in the MeV energy range, we consider Case
(I) of Table 1 of Ref. [46], corresponding to the luminosity hierarchy
$L_{\mathrm{PH}}\gg L_{\mathrm{UP}}\gg L_{\mathrm{SYNC}}$, where
$L_{\mathrm{PH}}$, $L_{\mathrm{UP}}$, and $L_{\mathrm{SYNC}}$ stand for the
photospheric luminosity, up-scattered photospheric luminosity of the
accelerated electrons at $R_{\mathrm{IS}}$, and synchrotron luminosity
radiated by the electrons at $R_{\rm IS}$, respectively.

Following Ref. [46], we define

$x=\frac{\varepsilon_{\rm
IS}\varepsilon_{B}}{\varepsilon_{\mathrm{PH}}}\quad\mathrm{and}\quad
Y=\frac{U^{\prime}_{\mathrm{SYNC}}}{U^{\prime}_{\mathrm{B}}}=\frac{4}{3}\frac{(k_{e}-1)}{(k_{e}-2)}\tau_{T}\gamma^{\prime}_{e,\mathrm{min}}\gamma^{\prime}_{e,\mathrm{cool}}h\
,$ (5.1)

where $Y$ is the Compton parameter, $k_{e}$ is the slope of the electron
energy distribution, $h$ is a function of $\gamma^{\prime}_{e,\mathrm{min}}$
and $\gamma^{\prime}_{e,\mathrm{cool}}$ and depends on the cooling regime,
$\gamma^{\prime}_{e,\mathrm{min}}$ is the minimum Lorentz factor of the
electrons injected in the acceleration region

$\gamma^{\prime}_{e,\mathrm{min}}=\frac{m_{p}}{m_{e}}\frac{k_{e}-2}{k_{e}-1}\mathcal{R}^{-1}\xi^{-1}\varepsilon_{\rm
IS}\varepsilon_{e}\ ,$ (5.2)

with $\xi$ being the fraction of electrons accelerated at the shock and
$\mathcal{R}$ being the number of leptons per baryon. Finally,
$\gamma^{\prime}_{e,\mathrm{cool}}$ is the electron cooling Lorentz factor
obtained from
$\gamma^{\prime}_{e,\mathrm{cool}}m_{e}c^{2}=P(\gamma^{\prime}_{e,\mathrm{cool}})t^{\prime}_{\mathrm{ad}}$
and given by

$\gamma^{\prime}_{e,\mathrm{cool}}(R)\simeq\frac{3m_{e}\mathcal{R}}{4m_{p}\tau_{T}\varepsilon_{\rm
PH}}\frac{1}{x(1+Y)+1}\ ;$ (5.3)

$t^{\prime}_{\mathrm{ad}}$ being the adiabatic cooling timescale and
${P(\gamma^{\prime}_{e,\mathrm{cool}})={4}/{3}\sigma_{\mathrm{T}}c\gamma^{\prime
2}_{e,\mathrm{cool}}(U^{\prime}_{\mathrm{B}}+U^{\prime}_{\mathrm{SYNC}}+U^{\prime}_{\mathrm{PH}})}$
the cooling rate for electrons. The conditions we need to fulfill in order to
satisfy $L_{\mathrm{PH}}\gg L_{\mathrm{UP}}\gg L_{\mathrm{SYNC}}$ are

$\eta<\eta_{\ast}\ ,\quad x\ll 1\ ,\quad xY\ll 1\ ,\quad
Y=\frac{\varepsilon_{\rm IS}\varepsilon_{e}h}{\varepsilon_{\rm{PH}}}\ll 1\ .$
(5.4)

In this way, it is possible to estimate $L_{\mathrm{UP}}=YL_{\mathrm{PH}}$ and
$L_{\mathrm{SYNC}}=xYL_{\mathrm{PH}}$.

We adopt the electron slope $k_{e}=2.2$ and fix $\varepsilon_{\rm IS}$ by
relying on the observations in the optical band; by assuming that the
synchrotron extended emission in this range should not be brighter than what
is typically observed, the following constraint on the flux should hold:
$F^{\mathrm{sync}}_{\nu}(E_{\gamma,\mathrm{opt}})<100$ mJy with $E_{\gamma,\rm
opt}=2$ eV [135]. In our case, $\varepsilon_{\rm IS}=0.2$ satisfies such a
condition. The radiative efficiency of this GRB is
$\eta_{\gamma}=(\tilde{E}_{\rm PH}+\tilde{E}_{\rm SYNC}+\tilde{E}_{\rm
UP})/\tilde{E}_{\rm iso}=\tilde{E}_{\rm PH}(1+Y+xY)/\tilde{E}_{\rm iso}\simeq
0.2$. Since the high-energy photopsheric photons are absorbed by the $e^{\pm}$
pair creation at $R_{\rm PH}$, we use a cut-off for the Band spectrum at
$R_{\rm PH}$, defined in Eq. A.1.

Figure 6: Left: Photon fluence observed at Earth for the IS model with
dissipative photosphere. The photospheric emission (PH, violet line), the
photospheric up-scattered emission (UP, orange line), and the synchrotron
emission of electrons accelerated at $R_{\mathrm{IS}}$ (SYNC, green line) are
plotted together with the total photon fluence (in black). Right:
$\nu_{\alpha}+\bar{\nu}_{\alpha}$ fluence in the observer frame produced at
$R_{\mathrm{IS}}$ with flavor oscillations included (in red for electron and
blue for muon flavors). The astrophysical parameters for this GRB are reported
in Table 1, and $\tilde{E}_{\mathrm{PH}}=6.8\times 10^{53}$ erg,
$\tilde{E}_{\mathrm{SYNC}}\simeq 5.4\times 10^{50}$ erg,
$\tilde{E}_{\mathrm{UP}}\simeq 6.8\times 10^{51}$ erg, $\mathcal{R}=1$,
$\xi$=1. The fluence for the muon flavor peaks at $E^{\rm
peak}_{\nu}=3.2\times 10^{7}$ GeV and its normalization is larger than the one
in Fig. 5, given a higher photon number density in the acceleration region.

We define the total photon energy distribution in the comoving frame at
$R_{\rm IS}$ as

$n^{\prime}_{\gamma,\mathrm{tot}}(E^{\prime}_{\gamma})=\bigg{(}\frac{R_{\rm{PH}}}{R_{\rm{IS}}}\bigg{)}^{2}n^{\prime}_{\gamma,\rm{PH}}(E^{\prime}_{\gamma})+n^{\prime}_{\gamma,\rm{SYNC}}(E^{\prime}_{\gamma})+n^{\prime}_{\gamma,\rm{UP}}(E^{\prime}_{\gamma})$
(5.5)

and compute the cooling processes of protons at the IS. Because of the more
intense photon field at $R_{\rm IS}$ with respect to the simple IS scenario,
$t_{p\gamma}$ and $t_{\rm BH}$ are shorter, while the synchrotron losses are
negligible.

Figure 6 shows the resultant photon (on the left panel) and neutrino (on the
right panel) fluences for the IS shock scenario with a dissipative
photosphere. The black curve in the left panel represents the overall photon
fluence. The total spectrum is consistent with Fermi observations [136], being
the high energy component subdominant with respect to the Band one. From the
right panel of Fig. 6, one can see that the fluence peaks at $E^{\rm
peak}_{\nu}=3.2\times 10^{7}$ GeV and its normalization is only a factor
$\mathcal{O}(10)$ larger than the one of the IS model (see Fig. 5), despite
the larger available photon energy [a factor $\mathcal{O}(100)$]. The reason
for this is the higher $E^{\prime}_{\gamma,\rm peak}$ in the IS scenario with
dissipative photosphere, which affects the ratio of the photon number
densities at $E^{\prime}_{\gamma,\rm peak}$. It should also be noted that here
we do not take into account the anisotropy of the incoming photospheric photon
field at $R_{\rm IS}$, an effect that would cause a further reduction in the
neutrino production efficiency, as pointed out also in Ref [137].

####  5.1.3 Jet model with three emission components

We are interested in a representative GRB of the class of bursts introduced in
Sec. 2.1.3, hence we adopt average values for the spectral index and intensity
of each component. To this purpose, we rely on Refs. [47, 138, 139].

Once the outflow becomes transparent to radiation, a BB component is emitted
at $R_{\rm PH}$, with spectral index $\alpha_{\rm BB}=0.4$ [42] and peaks at
the temperature defined in Eq. 2.4. Subsequently, the kinetic energy of the
outflow is dissipated at the ISs, and the main spectral component (CPL1) is
produced. The latter is described by a CPL with spectral index $\alpha_{\rm
CPL1}=-1$. An additional cut-off power-law (CPL2) begins to appear after a
slight delay with respect to CPL1, with the cut-off shifting to higher
energies until its disappearance. At later times, this additional component is
well described by a simple PL, and we associate it to the beginning of the
afterglow. With this choice, we take into account both interpretations of the
additional energetic component, namely the internal or external origin of
CPL2.

At the deceleration radius $R_{\rm dec}$ (see Eq. 2.7), the external shock
starts accelerating protons and electrons of the wind and the magnetic field
builds up. Motivated by the afterglow modeling [140], we use the following
values for the energy fractions: $\varepsilon_{e}=4\times 10^{-2}$,
$\varepsilon_{B}=10^{-4}$ and
$\varepsilon_{p}=1-\varepsilon_{e}-\varepsilon_{B}$ [112], compatible with our
choice for the prompt efficiency.

We consider a wind type circumburst medium with $A=3\times
10^{34}\,\rm{cm}^{-1}$ [141] and an adiabatic blastwave, with
$\Gamma(t)=\Gamma(t_{\rm dec}/4t)^{1/4}$ [141] and
$R(t)=2\Gamma^{2}(t)ct/(1+z)$ describing the temporal evolution of the Lorenz
factor and the radius of the forward shock after $t_{\rm dec}$, respectively.
The energy of the accelerated particles in the blastwave, at a time $t$ after
the deceleration, is
$\tilde{U}_{p}=4\pi\varepsilon_{p}AR(t)m_{p}c^{2}[\Gamma^{2}(t)-1]$. By
relying on the temporal evolution of the bright GRB investigated in Ref.
[142], we consider a simple PL produced at the forward shock, with power slope
$\alpha_{\rm PL}=-1.8$ (Eq. A.6), and normalize it to $\tilde{E}_{\rm
PL}=\varepsilon_{e}/\varepsilon_{p}\tilde{U}_{p}$. The photon field target for
$p\gamma$ interactions at the forward shock is the sum of the PL, BB and CPL1
components; the latter two being Lorentz transformed in the comoving frame of
the blastwave.

Since we are interested in computing the neutrino fluence emitted at the
forward shock during the prompt phase, we take a representative average radius
$R_{\ast}$ in logarithmic scale between $R_{\rm dec}$ and $R(t_{\rm
dur}-t_{\rm dec})$. The photon energy distribution is

$n^{\prime}_{\rm tot}(E^{\prime}_{\gamma},R_{\ast})=n^{\prime}_{\rm
PL}(E^{\prime}_{\gamma})+\left(\frac{R_{\rm
PH}}{R_{\ast}}\right)^{2}n^{\prime}_{\rm
BB}\left(\frac{E^{\prime}_{\gamma}}{\Gamma_{\rm r}}\right)+\left(\frac{R_{\rm
IS}}{R_{\ast}}\right)^{2}n^{\prime}_{\rm
CPL1}\left(\frac{E^{\prime}_{\gamma}}{\Gamma_{\rm r}}\right)\ ,$ (5.6)

where $\Gamma_{\rm r}$ is the relative Lorenz factor between $\Gamma$ and
$\Gamma_{\ast}\equiv\Gamma(R_{\ast})$.

The BB component is always subdominant, while CPL1 and CPL2 are expected to
vary in absolute and relative intensity from burst to burst; this is true also
in the same GRB, once the temporal evolution is considered. In order to
investigate to what extent the neutrino spectrum may be affected by these
factors, we considered two scenarios of study for the prompt phase: case (I)
such that the energetics of the three components is
$\tilde{E}_{\mathrm{BB}}\simeq 0.1\tilde{E}_{\mathrm{CPL1}}$ and
$\tilde{E}_{\mathrm{CPL1}}=3\tilde{E}_{\mathrm{CPL2}}$ (solid black line in
the top left panel of Fig. 7) and case (II) with
$\tilde{E}_{\mathrm{CPL1}}=1/3\tilde{E}_{\mathrm{CPL2}}$ (dotted black line in
the top left panel of Fig. 7). The three cut-off power-laws (BB, CPL1, and
CPL2) follow Eq. A.5 with peak energies $E_{\rm BB,peak}\simeq 2\times
10^{-5}$ GeV, $E_{\rm CPL1,peak}\simeq 7\times 10^{-5}$ GeV, and $E_{\rm
CPL2,peak}\simeq 2\times 10^{-2}$ GeV, respectively. These values are
consistent with the ones in Refs. [47, 142]. With this set of parameters, the
hierarchy and intensity of the various cooling processes is analogous to the
IS case for the prompt phase (Fig. 4), while adiabatic cooling is the dominant
cooling process by many orders of magnitude at $R_{\ast}$. For what concerns
the forward shock, we assume a differential number density of protons
$n^{\prime}_{p}(E^{\prime}_{p})\propto E^{\prime\ -2.5}_{p}$ [113] injected
between the minimum energy $E^{\prime}_{p,\rm min}=m_{p}c^{2}\Gamma_{\ast}$
and the maximum $E^{\prime}_{p,\mathrm{max}}$, derived from the condition that
the proton acceleration time $t^{\prime}_{\mathrm{acc}}$ is limited by the
adiabatic time $t^{\prime}_{\mathrm{ad}}$ (see Sec. 3.2).

Figure 7: Top left: Photon fluence in the prompt phase for the model with
three components. It is composed by a thermal BB component (violet dashed
curve), a cut-off power law CPL1 (green dashed curve), and a second cut-off
power law CPL2 (in orange dashed for case (I) such
$\tilde{E}_{\mathrm{CPL1}}=3\tilde{E}_{\mathrm{CPL2}}$; in orange dotted for
case (II) with $\tilde{E}_{\mathrm{CPL1}}=1/3\tilde{E}_{\mathrm{CPL2}}$).
Furthermore $\tilde{E}_{\mathrm{BB}}=0.1\tilde{E}_{\mathrm{CPL1}}$. The total
fluence is plotted in black (solid and dotted lines). Top right: Correspondent
$\nu_{\alpha}+\bar{\nu}_{\alpha}$ fluence in the observer frame with flavor
oscillations included (in red for electron and blue for muon flavors). The
solid line represents the total contribution during the prompt for case (I),
while the dotted one is for case (II). The fluence for the muon flavor peaks
at $E^{\rm peak}_{\nu}=4.2\times 10^{7}$ GeV for the case (I). The low energy
tail is affected by the interaction of protons with CPL2. Bottom left: Photon
fluence for the scenario such that the emission from the forward shock (PL,
blue dash-dotted line) starts during the prompt phase (dashed, BB+CPL1).
Bottom right: Corresponding $\nu_{\alpha}+\bar{\nu}_{\alpha}$ fluence (dashed
curve for the IS emission, dash-dotted for the forward shock, and solid line
for the total).

The neutrino fluence at the IS and forward shock is displayed in Fig. 7. In
the top left panel, the solid line represents the neutrino fluence for case
(I), while the dotted line stands for case (II). The enhancement of the
energetic component CPL2 of almost one order of magnitude leads to a
negligible impact on the neutrino energy distribution, producing only a slight
increase of the fluence at low energies.

The bottom panels of Fig. 7 display the photon fluence and the correspondent
neutrino fluence when the emission from the forward shock starts during the
prompt phase. The dashed lines represent the neutrino fluence produced at the
IS from the interaction of accelerated protons and the photon field (BB+CPL1),
while the dash-dotted line represents the neutrino outcome from the forward
shock at $R_{\ast}$, where we rely on Eq. 5.6 for the photon field. The solid
line describes the total neutrino fluence expected during the prompt phase.
The forward shock contribution is significantly higher than what expected for
the afterglow phase [141]. This is mainly due to a much larger photon number
density in the acceleration region. Furthermore, given the very low magnetic
field and its inefficiency to accelerate particles to very high energies, the
cutoff in the neutrino spectrum occurs at a lower energy compared to the
prompt case (see dot-dashed line). The overall intensity at peak energy of the
neutrino emission in this scenario is slightly larger than in the simple IS
case.

###  5.2 Poynting flux dominated jets

####  5.2.1 ICMART model

Figure 8: Left: Band photon fluence observed at Earth and emitted at $R_{\rm
ICMART}=10^{15}$ cm for the ICMART model. The parameters of this GRB are
reported in Table 1 and $\eta_{\gamma}=0.17$. Right: Correspondent
$\nu_{\alpha}+\bar{\nu}_{\alpha}$ fluence in the observer frame in the
presence of flavor conversions for the ICMART model (in blue and red for the
muon and the electron flavors, respectively). The muon neutrino fluence peaks
at $E^{\rm peak}_{\nu}=1.3\times 10^{7}$ GeV; the high energy tail of the
neutrino distribution shows the double bump structure due to kaon decay.

For the model introduced in Sec. 2.2.1, the typical radius necessary to make
sure that runaway reconnection has enough time to grow is
$R_{\mathrm{ICMART}}\simeq 10^{15}$ cm [48], while the typical width of the
reconnection region is $\Delta=ct_{v}$, where we adopt $t_{v}\simeq 0.5$ s.
The dissipation efficiency may be as high as $0.35$ in this model [143] and
this is the value we adopt for $\varepsilon_{d}$.

It has been shown that a Band-like spectrum may be reproduced in this scenario
by considering an appropriate time dependent injection rate of particles in
the emitting region [144], and this is the spectrum we adopt for this model.
If $\sigma$ is the magnetization parameter at $R_{\mathrm{ICMART}}$, the
magnetic field in the bulk comoving frame can be expressed as [48]:

$B^{\prime}=\left(\frac{2\tilde{L}_{\mathrm{iso}}}{\Gamma^{2}cR^{2}_{\mathrm{ICMART}}}\frac{\sigma}{\sigma+1}\right)^{1/2}\
.$ (5.7)

We use as initial jet magnetization $\sigma_{0}=\sigma_{\rm in}=45$ (see Fig.
2); this choice, as shown in the following, allows for a consistent comparison
with the results of Sec. 5.2.2. By relying on the results from particle-in-
cell simulations, we assume $k_{p}(\sigma_{0})\simeq 2$ (see Eq. 2.15).
Furthermore, we set $\varepsilon_{p}=0.5$ and $\varepsilon_{e}=0.5$ (see Eq.
2.13). With this choice, the radiative efficiency of our benchmark GRB turns
out to be $\eta_{\gamma}=\varepsilon_{d}\varepsilon_{e}\simeq 0.17$. The
photon number density is normalized to $\eta_{\gamma}E^{\prime}_{\rm iso}$.
Since $R_{\rm IS}\simeq R_{\rm ICMART}$, we obtain similar trends for the
cooling times as in Fig. 4, except for the synchrotron loss that starts to
dominate at $E^{\prime}_{p}\sim 10^{8}$ GeV and a slightly increased rate of
$p\gamma$ interactions due to a larger photon number density in the
dissipation region.

The neutrino fluence is displayed in the right panel of Fig. 8. The fluence
for the muon flavor peaks at $E^{\rm peak}_{\nu}=1.3\times 10^{7}$ GeV. Note
that the double bump due to kaon decay is clearly visible in the high-energy
tail of the energy distribution. This is due to $E^{\prime}_{p,\rm max}\sim
E^{\prime}_{K,\rm max}\sim 2\times 10^{9}$ GeV (while $E^{\prime}_{\pi,\rm
max}~{}\sim 1.2\times 10^{7}$ GeV). This feature is determined by the stronger
magnetic field in the acceleration region ($B^{\prime}\simeq 9$ kG, while for
example $B^{\prime}\simeq 1$ kG in the IS model).

####  5.2.2 Magnetized jet model with gradual dissipation

For the model introduced in Sec. 2.2.2, we follow Ref. [49] and assume that
the energy which is dissipated in the optically thick region is reprocessed
into quasi-thermal emission, leading to a black-body-like emission from the
photosphere. In the optically thin region, the synchrotron radiation from
electrons is the dominant emission mechanism and it represents the non-thermal
prompt emission. The energy emitted at the photosphere is obtained by
integrating the energy dissipation rate (Eq. 2.11) up to $R_{\mathrm{PH}}$ and
considering that only the fraction $(R/R_{\mathrm{PH}})^{4/9}$ of the energy
dissipated at $R$ remains thermal at $R_{\mathrm{PH}}$. In the optically thin
region, electrons are always in the fast cooling regime and
$E^{\prime}_{\gamma,\mathrm{ssa}}\gg E^{\prime}_{\gamma,\mathrm{cool}}$ for
$R_{\mathrm{ph}}<R<R_{\mathrm{sat}}$ with our choice of parameters. Here
$E^{\prime}_{\gamma,\mathrm{ssa}}$ is the synchrotron self-absorption energy
[50]:

$E^{\prime}_{\gamma,\mathrm{ssa}}\sim\left(\frac{h^{3}}{8\pi
m_{p}}\frac{\xi\tilde{L}_{\mathrm{iso}}}{4\pi\Gamma_{\rm
sat}}\frac{1}{R^{2}\Gamma(R)}\right)^{1/3}\ .$ (5.8)

The shape of the synchrotron spectrum follows Eq. A.10, but we replace
$E^{\prime}_{\mathrm{cool}}$ with $E^{\prime}_{\gamma,\mathrm{ssa}}$ and use
$\alpha_{\gamma}=1$ for $E^{\prime}_{\gamma}<E^{\prime}_{\gamma,\mathrm{ssa}}$
[50]. Furthermore, only a fraction $\xi=0.2$ [49] of electrons is accelerated.

For our reference GRB we adopt $\lambda=4\times 10^{8}$ cm [49]. The terminal
Lorenz factor of the outflow is $\Gamma_{\rm sat}\simeq\Gamma_{0}\sigma_{0}$.
We choose $\Gamma_{\rm sat}=300$ and the initial jet magnetization is
$\sigma_{0}=\Gamma^{2/3}_{\mathrm{sat}}\sim 45$.

For what concerns protons, we assume that they are accelerated into a power-
law distribution starting from a minimum Lorentz factor

$\gamma^{\prime}_{p,\rm
min}=\mathrm{max}\bigg{[}1,\frac{k_{p}-2}{k_{p}-1}\frac{\varepsilon_{p}}{2}\sigma(R)\bigg{]}$
(5.9)

and extending up to a maximum value determined by balancing the energy gain
and loss rates, as described in Sec. 3.2. We also assume that the power of the
proton distribution is the same as the one of electrons, namely $k_{p}=k_{e}$
(see Eq. 2.15). The latter assumption is motivated by particle-in-cell
simulations of magnetic reconnection for $\sigma\gg 1$ [103], but it has to be
yet demonstrated for $\sigma\sim 1$ [101].

We compute the neutrino production assuming that dissipation and particle
acceleration start at $R_{1}=aR_{\rm PH}$ with $a=3$. Being an arbitrary
choice for the starting radius, we explore the effects of $a$ on the neutrino
fluence in Appendix B. For illustration purposes, we compute the neutrino
production rate at three radii ($R_{1}$, $R_{2}$, and $R_{3}$) equally
distanced in logarithmic scale. We make this choice in order to establish the
qualitative trend of the neutrino production during the evolution of the
outflow in the optically thin region. The photon and proton distributions are
normalized at each radius $R_{i}$ along the jet to the energy dissipated
between $R_{i-1}$ and $R_{i}$, where $R_{0}=R_{\mathrm{PH}}$ and
$R_{3}=R_{\mathrm{sat}}$. At each $R_{i}$, the photon field coming from
$R_{j}$ is Lorentz transformed through the relative Lorentz factor

$\Gamma_{\mathrm{rel},ij}=\frac{1}{2}\left(\frac{\Gamma_{i}}{\Gamma_{j}}+\frac{\Gamma_{j}}{\Gamma_{i}}\right)\
,$ (5.10)

that holds as long as $\Gamma_{i},\Gamma_{j}\gg 1$. The total photon number
density used as input at each radius $R_{i}$ for producing neutrinos is thus

$n^{\prime}_{\rm
tot}(E^{\prime}_{\gamma},R_{i})=\sum_{j=0}^{i}\bigg{(}\frac{R_{j}}{R_{i}}\bigg{)}^{2}n^{\prime}_{j}\bigg{(}\frac{E^{\prime}_{\gamma}}{\Gamma_{\mathrm{rel},ij}}\bigg{)}\frac{1}{\Gamma_{\mathrm{rel},ij}}$
(5.11)

where $n^{\prime}_{j}(E^{\prime}_{\gamma})$ is the photon energy distribution
at $R_{j}$ (in units of $\rm{GeV}^{-1}\rm{cm}^{-3}$).

Once the photon distributions are set, we evaluate the proton cooling times at
each radius. In all the three cases, dominant losses are due to the adiabatic
cooling up to $\simeq 10^{5}$ GeV, and $p\gamma$ interactions for
$10^{5}\,\mathrm{GeV}\lesssim E^{\prime}_{p}\lesssim E^{\prime}_{p,\rm max}$.
Synchrotron losses become relevant around $10^{7}$ GeV. Given the very strong
magnetic field (see Table 2), the secondaries suffer strong synchrotron
losses; this considerably affects the resulting neutrino spectrum, which is
damped at energies much lower than in all the other models investigated so far
in this work. A summary of the input parameters at the three $R_{i}$ is
reported in Table 2.

Table 2:  Summary table for the input parameters adopted at the radii $R_{1}$,
$R_{2}$ and $R_{3}$ in the magnetic model with gradual dissipation: the radius
($R$), the comoving magnetic field ($B^{\prime}$), the Lorentz factor
($\Gamma$), the maximum energy of protons ($E^{\prime}_{p,\rm max}$), pions
($E^{\prime}_{\pi,\rm max}$), muons ($E^{\prime}_{\mu,\rm max}$), and kaons
($E^{\prime}_{K,\rm max}$), as well as the power-law slope ($k_{e}=k_{p}$) of
electrons and protons.

$R\,[\mathrm{cm}]$ $B^{\prime}\,[\mathrm{kG}]$ $\Gamma$ $E^{\prime}_{p,\rm
max}\,[\mathrm{GeV}]$ $E^{\prime}_{\pi,\rm max}\,[\mathrm{GeV}]$
$E^{\prime}_{\mu,\rm max}\,[\mathrm{GeV}]$ $E^{\prime}_{K,\rm
max}\,[\mathrm{GeV}]$ $k_{e}$ $R_{1}$ $7.1\times 10^{12}$ $1.7\times 10^{3}$
$176$ $1.1\times 10^{8}$ $6.5\times 10^{4}$ $3.6\times 10^{3}$ $7.5\times
10^{6}$ $2.4$ $R_{2}$ $1.6\times 10^{13}$ $5.4\times 10^{2}$ $230$ $1.7\times
10^{8}$ $2\times 10^{5}$ $1.1\times 10^{4}$ $2.3\times 10^{7}$ $2.5$ $R_{3}$
$3.6\times 10^{13}$ $1.7\times 10^{2}$ $300$ $2.5\times 10^{8}$ $6.4\times
10^{5}$ $3.5\times 10^{4}$ $7.3\times 10^{7}$ $2.6$

Figure 9: Left: Photon fluence in the observer frame for the GRB model
invoking continuous magnetic dissipation for the parameters reported in Tables
1 and 2. The total photon energy distribution is shown in black, and its
components at $R_{\rm PH}$, $R_{1}$, $R_{2}$, and $R_{3}$ are plotted in
violet, orange, green, and coral respectively. Right: Correspondent
$\nu_{\alpha}+\bar{\nu}_{\alpha}$ fluence at Earth after flavor oscillations
in the left panel (in blue for the muon flavor and in red for the electron
one). The fluence for the muon flavor peaks at $E^{\rm peak}_{\nu}=7.2\times
10^{5}$ GeV. An unusual spectral structure is clearly visible.

The (photon) neutrino fluence at Earth is shown in the (left) right panel of
Fig. 9. The slope of the three non-thermal synchrotron components and their
distribution peaks decrease as the distance from the source increases. The
high energy cut-off of each spectral component is given by Eq. A.1. Notably,
the dominant component comes from the smallest radius, while the contribution
coming from larger radii gets lower and lower ($67\%$, $26\%$, and $7\%$ from
$R_{1},R_{2}$ and $R_{3}$, respectively). The significant drop in the neutrino
flux between $R_{1}$ and $R_{3}$ is mainly due to the decrease of the proton
power slope (see Table 2), which causes a more pronounced drop in proton
number density in the energy range of interest. This is a peculiar feature of
this model, which predicts parameters depending on the jet magnetization, and
thus changing with the radius.

The neutrino fluence for the muon flavor peaks at $E^{\rm
peak}_{\nu}=7.2\times 10^{5}$ GeV, which is about $\mathcal{O}(10-100)$ GeV
smaller than in the models presented in the previous sections, although
roughly comparable in intensity at peak. This is due to the fact that, in this
case, the main contribution to the neutrino flux comes from the interaction of
protons with thermal photons, whose energy peaks at $\sim 22$ keV. The second
bump visible in the spectrum is instead represented by the kaon contribution.
Apart from the ICMART model, this is the only other case out of the ones
studied in this work in which this feature is clearly identified at higher
energies. The reason is the very strong magnetic field in these two magnetic
models. Another peculiar feature of this model is the low-energy tail of the
neutrino distribution, which is higher than in previous cases. This is due to
a combination of the larger number density [$\mathcal{O}(10^{3}-10^{4})$] of
protons at low energies in the acceleration region and the extended photon
field at higher energies.

###  5.3 Proton synchrotron model

In order to estimate the neutrino production in the proton synchrotron model
(see Sec. 2.3), we need to evaluate the fraction of the proton energy which
goes into $p\gamma$ interactions. We consider the photon spectral fit as in
Eq. A.10 and follow Ref. [40], which provides the cooling energy
$E_{\gamma,\mathrm{cool}}$, the peak energy (or minimum injection energy)
$E_{\gamma,\mathrm{peak}}\equiv E_{\gamma,\mathrm{min}}$, and the energy flux
at the cooling energy ($F_{\gamma,\mathrm{cool}}$).

Another inferred quantity is the cooling timescale of the radiating particles,
$t_{\mathrm{cool}}\sim 1$ s. The cooling time $t_{\rm cool}$ is related, after
Lorentz transforming, to the comoving magnetic field $B^{\prime}$ and
$\gamma^{\prime}_{\rm cool}$ by means of Eq. A.7. The variability timescale is
assumed to be $t_{v}=0.5$ s; the duration of the burst, as well as the
redshift information, is extracted from the GRB catalog [145]. These
observables can be used to constrain the source parameters, such as
$B^{\prime}$, $\gamma^{\prime}_{\mathrm{min}}$, $R_{\gamma}$, $\Gamma$, and
$E^{\prime}_{\gamma,\mathrm{bol,iso}}$ through the following relations [40]:

$\displaystyle E_{\gamma,\mathrm{peak}}$ $\displaystyle=$
$\displaystyle\frac{3}{2}\frac{\hbar eB^{\prime}\gamma^{\prime
2}_{\mathrm{min}}}{m_{p}c}\frac{\Gamma}{1+z}\ ,$ (5.12) $\displaystyle
E_{\gamma,\mathrm{cool}}$ $\displaystyle=$
$\displaystyle\left(\frac{m_{p}}{m_{e}}\right)^{5}\frac{54\pi^{2}\hbar
em_{e}c}{\sigma^{2}_{T}B^{\prime 3}t^{2}_{\mathrm{cool}}}\frac{1+z}{\Gamma}\
,$ (5.13) $\displaystyle F_{\gamma}$ $\displaystyle=$ $\displaystyle
F_{\gamma,\mathrm{cool}}\left(\frac{E_{\gamma,\mathrm{cool}}}{h}\right)\left[\frac{3}{4}+2\sqrt{\frac{E_{\gamma,\mathrm{peak}}}{E_{\gamma,\mathrm{cool}}}}-2+\frac{2}{k_{p}-2}\sqrt{\frac{E_{\gamma,\mathrm{peak}}}{E_{\gamma,\mathrm{cool}}}}\right]\
,$ (5.14) $\displaystyle E^{\prime}_{\gamma,\mathrm{bol,iso}}$
$\displaystyle=$ $\displaystyle\frac{4\pi
d^{2}_{L}(z)F_{\gamma}t_{\mathrm{dur}}}{\Gamma(1+z)}\ ,$ (5.15) $\displaystyle
R_{\gamma}$ $\displaystyle=$ $\displaystyle\frac{2ct_{v}\Gamma^{2}}{(1+z)}\ .$
(5.16)

where $F_{\gamma}=\tilde{L}_{\gamma,\rm bol,iso}/4\pi d_{L}^{2}(z)$ is the
bolometric isotropic radiative flux (in units of $\rm GeV\,cm^{-2}\,s^{-1}$),
$\tilde{L}_{\gamma,\rm bol,iso}$ being the bolometric isotropic luminosity of
the burst over the whole energy range. Using these relations we can infer
$B^{\prime},\gamma^{\prime}_{\rm{min}},R_{\gamma}$ and $E^{\prime}_{\gamma,\rm
bol,iso}$ as functions of $\Gamma$.

Figure 10: Histogram of $Y_{p}$ (see Eq. 5.17) for a subset of GRBs analyzed
in Ref. [40] for which redshift information is available; $\Gamma=300$ and
$t_{v}=0.5$ s are adopted. The parameter $Y_{p}$ quantifies the relative
importance between the proton synchrotron emission and $p\gamma$ interactions.
The very low values of $Y_{p}$ for most of GRBs in the sample suggest a
negligible neutrino production of this class of GRBs.

In order to figure out the relative importance between proton synchrotron and
$p\gamma$ cooling for the sample of GRBs studied in Ref. [40], we introduce
the following parameter [146]:

$Y_{p}\equiv\frac{L^{\prime}_{\mathrm{p},p\gamma}}{L^{\prime}_{\mathrm{p},\mathrm{syn}}}\approx\frac{\sigma_{p\gamma}}{\sigma_{\text{p,T}}}\frac{U^{\prime}_{p,\mathrm{syn}}}{U^{\prime}_{B}}=\frac{\sigma_{p\gamma}}{\sigma_{\text{p,T}}}\frac{E^{\prime}_{\gamma,\mathrm{tot,iso}}}{V^{\prime}_{\mathrm{iso}}}\frac{8\pi}{B^{\prime
2}}=\frac{\sigma_{p\gamma}}{\sigma_{\text{p,T}}}\frac{8\pi
F_{\gamma}d^{2}_{L}(z)}{\Gamma^{2}R^{2}_{\gamma}cB^{\prime 2}}\ ,$ (5.17)

where $L^{\prime}_{p,p\gamma}$ and $L^{\prime}_{p,\mathrm{sync}}$ are the
proton energy loss rates for $p\gamma$ interactions and synchrotron emission
respectively, and $\sigma_{\text{p,T}}=\sigma_{\text{T}}(m_{e}/m_{p})^{2}$. By
relying on Eqs. 5.13 and 5.16, $Y_{p}$ can be estimated as a function of the
bulk Lorentz factor.

For our bencnhmark $\Gamma=300$, we compute $Y_{p}$ for the GRBs studied in
Ref. [40] for which redshift information is available. The histogram of
$Y_{p}$ is shown in Fig. 10. We can see that $Y_{p}$ spreads over almost three
orders of magnitude, with very low typical values. Hence, assuming proton
synchrotron radiation as the main emission mechanism, we expect this class of
GRBs to be poor emitters of high energy neutrinos. To show this
quantitatively, we compute the neutrino fluence for our representative GRB.

Figure 11: Left: Photon fluence in the observer frame for the proton
synchrotron model; see Table 1 for the model parameters, in addition
$\tilde{E}_{\mathrm{\gamma,\rm iso}}=5\times 10^{53}$ ergs, $t_{\rm cool}=0.5$
s, $\gamma_{\rm min}/\gamma_{\rm cool}=12$, $E_{\gamma,\mathrm{cool}}=7$ keV.
Right: Correspondent $\nu_{\alpha}+\bar{\nu}_{\alpha}$ fluence (in red and
blue for the electron and muon flavors, respectively). The peak in the
neutrino distribution ($E^{\rm peak}_{\nu}=3.5\times 10^{4}$ GeV), due to the
cooling energy break $E_{\gamma,\rm cool}$, is shifted to lower energies with
respect to the other analized models. The damping at high energies is due to
the very strong magnetic field in the emitting region ($B^{\prime}\simeq
8.5\times 10^{6}$ G).

We adopt the following GRB parameters: ${\gamma_{\rm min}/\gamma_{\rm
cool}=12}$, $E_{\gamma,\mathrm{cool}}=7$ keV, $\Gamma=300$, $z=2$
$E_{\gamma,\mathrm{peak}}=(\gamma_{\rm min}/\gamma_{\rm cool})^{2}\
E_{\gamma,\mathrm{cool}}$, which result in $\tilde{E}_{\gamma,\rm
bol,iso}\simeq 7\times 10^{53}$ erg. These values are compatible with the
average ones inferred from the sample considered in Ref. [40], namely
$\langle\tilde{E}_{\gamma,\rm bol,iso}\rangle\simeq 7.4\times 10^{53}$ erg,
$\langle E_{\gamma,\rm cool}\rangle\simeq 6.4$ keV, and $\langle\gamma_{\rm
min}/\gamma_{\rm cool}\rangle\simeq 11.9$. Furthermore, we choose $k_{p}=2.6$
for the slope of the injection proton spectrum
$Q^{\prime}(\gamma^{\prime}_{p})$, that reproduces the typical value of the
high energy photon index $\beta\sim-2.3$; note that $k_{p}$ is almost never
constrained for the sample in Ref. [40].

As for the radiated energy, this fiducial GRB is comparable to the ones
analyzed in the previous sections, except for the total energetics. In fact,
given the very high magnetic field, $B^{\prime}\simeq 8.5\times 10^{6}$ G, the
total isotropic energy is $\tilde{E}_{B,\rm iso}\propto
R^{2}_{\gamma}\Gamma^{2}B^{\prime 2}\sim\mathcal{O}(10^{60})$ erg, much larger
than the typical energy that a GRB jets is able to release (spin down of
magnetars or through the Blandford-Znajek mechanism [10]). Since the
synchrotron radiation dominates by many orders of magnitude over all the other
proton energy loss mechanisms, we assume $U^{\prime}_{p,\rm iso}\simeq
E^{\prime}_{\gamma,\rm bol,iso}$, where $U^{\prime}_{p}$ is the total
isotropic proton energy in the comoving frame. Such a jet turns out to be
highly inefficient in radiating energy, given that $\tilde{E}_{\rm
iso}\sim\tilde{E}_{B,\rm iso}\gg\tilde{E}_{\gamma,\rm bol,iso}$.

By considering the proton energy distribution as in Eq. A.10 and normalizing
it to $U^{\prime}_{p}$, we compute the neutrino fluence and show it in the
right panel of Fig. 11. The left of the same figure shows the total
synchrotron photon fluence. Analogously to the model in Sec. 5.2.2, the peak
in the neutrino distribution ($E^{\rm peak}_{\nu}=3.5\times 10^{4}$ GeV) is
due to the cooling energy break $E_{\gamma,\rm cool}$ and it is shifted to
lower energies. The neutrino spectrum is furthermore strongly damped at high
energies due to the synchrotron cooling of mesons in the jet. Our estimation
of the neutrino emission results to be in agreement with the one reported in
the independent work of Ref. [52], for GRBs with similar parameters.

The proton synchrotron model, besides requiring unreasonable total jet
energies, predicts the smallest neutrino fluence among all models considered
in this work. We note that with the choice made of parameters, our
representative GRB has $Y_{p}\sim\mathcal{O}(10^{-4})$; hence, our estimation
may be considered an optimistic one, given the distribution of $Y_{p}$ shown
in Fig. 10. We refer the interested reader to Ref. [52] for additional details
and discussion on this model.

##  6 Discussion

In this work, we have computed the neutrino fluence for a class of models
adopted to describe the prompt phase of long GRBs, all having the same
$\tilde{E}_{\mathrm{iso}}$. Because of the diversity of electromagnetic GRB
data and the uncertainties inherent to the models (e.g., jet composition,
energy dissipation mechanism, particle acceleration, and radiation
mechanisms), an exhaustive theoretical explanation of the mechanism powering
GRBs is still lacking. To compare the neutrino production across models, we
have selected fiducial input parameters for a benchmark GRB motivated by
observations. In addition, the modeling of the dissipative and acceleration
efficiencies, as well as the properties of the accelerated particle
distributions have been guided by the most recent simulation findings. A
summary of our input parameters is reported in Table 1. In this section, we
compare the energetics of the GRB models explored in this work, discuss the
the detection prospects of stacked neutrino fluxes as well as the variation of
the quasi-diffuse neutrino flux due to the uncertainties in the jet
parameters.

###  6.1 Energetics

A summary of our findings is reported in Table 3, where the radiative
efficiency of the jet (Eq. 3.1) is listed for the six GRB models investigated
in this paper together with the isotropic photon and neutrino (for six
flavors) energies, as well as the ratio of the latter two. As already
discussed in Sec. 5, the least efficient model in converting
$\tilde{E}_{\mathrm{iso}}$ in $\tilde{E}_{\gamma,\mathrm{iso}}$ is the proton
synchrotron model, whilst the most efficient one is the model which considers
a dissipative photosphere as the main source of prompt emission. This is
mainly due to the high dissipative efficiency suggested by recent three-
dimensional simulations [87]. Note that the radiative efficiency is an input
parameter of each model, since we do not compute the radiation spectra self-
consistently.

Table 3:  Summary of the derived quantities for the models considered in this
work and our benchmark parameters value (see Table 1). The radiative
efficiency of the jet (Eq. 3.1), the isotropic photon energy in the
$1\,\mathrm{keV}$–$10\,\mathrm{MeV}$ energy range, the isotropic neutrino
energy for neutrinos and antineutrinos of all flavors, the ratio between the
isotropic total neutrino and photon energies, the neutrino energy at the
fluence peak, and the maximum proton energy are listed. The model with the
smallest radiative efficiency is the proton synchrotron model; this model has
also the smallest $\tilde{E}_{\nu,\mathrm{iso}}$. The most radiatively
efficient model is the one with a dissipative photosphere.

Model $\eta_{\gamma}\,(\%)$ $\tilde{E}_{\gamma,\rm iso}\,[\rm erg]$
$\tilde{E}_{\nu,\rm iso}\,[\rm erg]$ $\tilde{E}_{\nu,\rm
iso}/{\tilde{E}_{\gamma,\rm iso}}$ $E^{\rm peak}_{\nu_{\mu}}\,[\rm GeV]$
$E_{p,\rm max}\,[\rm GeV]$ IS $0.2$ $6.8\times 10^{51}$ $2.3\times 10^{48}$
$3.4\times 10^{-4}$ $4\times 10^{7}$ $1.2\times 10^{11}$ PH-IS $20$ $6.9\times
10^{53}$ $7.2\times 10^{49}$ $1.1\times 10^{-4}$ $3.2\times 10^{7}$ $7.5\times
10^{10}$ 3-COMP $0.3$ $8.7\times 10^{51}$ $5.2\times 10^{48}$ $6\times
10^{-4}$ $4.2\times 10^{7}$ $1.2\times 10^{11}$ ICMART $17.5$ $6\times
10^{53}$ $1.8\times 10^{51}$ $3\times 10^{-3}$ $1.3\times 10^{7}$ $1.7\times
10^{11}$ MAG-DISS $8$ $2.7\times 10^{53}$ $5.2\times 10^{50}$ $2\times
10^{-3}$ $7.2\times 10^{5}$ $2.5\times 10^{10}$ p-SYNCH $2\times 10^{-5}$
$4.8\times 10^{53}$ $7.2\times 10^{49}$ $1.4\times 10^{-4}$ $3.5\times 10^{4}$
$6.9\times 10^{9}$

Among the models considered in this work, all with identical
$\tilde{E}_{\mathrm{iso}}$, neutrinos carry the largest amount of energy in
the ICMART model, followed by the model invoking magnetic dissipation; among
the kinetic dominated jet scenarios, the case with a dissipative photosphere
is the most efficient one in terms of neutrino production. It is worth noting
that, although in the model with three components $E_{\gamma,\rm iso}$ is just
$30\%$ higher than in the IS one, $E_{\nu,\rm iso}$ is a factor $2.3$ larger.
The reason for this lies in the fact that protons interact with a high-energy
photon component comparable in intensity to the one in the $\gamma$-ray range
(i.e., $1$ keV–$10$ MeV) in the three component model, while the number
density of photons above $10$ MeV is negligible in the IS model. This also
explains the trend for
$\tilde{E}_{\nu,\mathrm{iso}}/\tilde{E}_{\gamma,\mathrm{iso}}$ reported in
Table 3 (note that $\tilde{E}_{\gamma,\rm iso}$ in Table 3 is estimated over
the energy range $1$ keV–$10$ MeV; hence, this ratio, when defined with the
bolometric photon energy used for neutrino production, should be slightly
smaller than the one reported for the IS model with a dissipiative
photosphere, the model with three components, the magnetic one with gradual
dissipation, and the proton synchrotron model).

One last remark should be done on our results for the IS and ICMART models. In
Ref. [137], the ICMART scenario predicts the least neutrino flux, given the
larger emission radius than $R_{\rm IS}$. This is not the case in our work for
two reasons: first, the chosen representative variability timescale $t_{v}$
provides emission radii comparable in the two scenarios; second, the
microphysics parameter that we adopt for the IS case are less favorable in
terms of radiative efficiency and neutrino production efficiency, if compared
to the parameters adopted in Ref. [137], which result to be the same for all
their cases of study.

###  6.2 Detection perspectives

In order to compare the neutrino detection perspectives for our six models, we
compute the all-sky quasi-diffuse flux for neutrinos and antineutrinos. We
assume that our benchmark GRB at $z=2$ yields a neutrino emission that is
representative of the entire GRB population. For ${\rm\dot{N}\simeq 667}$ yr-1
long GRBs per year [32], the stacking flux for the muon flavor over the whole
sky is defined as

$F_{\nu_{\mu}}(E_{\nu})=\frac{1}{4\pi}{\rm{\dot{N}}}\Phi_{\nu_{\mu}}(E_{\nu},z=2)\
.$ (6.1)

Figure 12: Model comparison of the expected all-sky quasi-diffuse fluxes for
the six GRB models considered in this work for the benchmark jet parameters
listed in Table 1. The quasi-diffuse flux has been computed by relying on Eq.
6.1 for $\nu_{\mu}+\bar{\nu}_{\mu}$; all models have identical
$\tilde{E}_{\mathrm{iso}}$. For comparison, the IceCube staking limits
(combined analysis for $1172$ GRBs) [32] and the expected sensitivity for
IceCube-Gen2 (based on a sample of $1000$ GRBs) [147] are reported (solid and
dashed black lines). By relying on the most up-to-date best-fit GRB
parameters, all models predict a quasi-diffuse flux that lies below the
sensitivity curves; however, a large spread in energy and shape of the
expected neutrino fluxes is expected for different jet models.

Figure 12 shows the resultant all-sky quasi-diffuse fluxes for the muon flavor
for the six GRB models as functions of the neutrino energy (colored curves).
For comparison, we also show the GRB staking limits of IceCube [32] and the
projected ones for IceCube-Gen2 [147] (black curves). In agreement with the
non-detection of high-energy neutrinos from targeted GRB searches [32], our
forecast for the neutrino fluxes lies below the experimental limits and is in
agreement with the upper limits reported by the ANTARES Collaboration [31] and
with the ones expected for KM3NeT [148]. The ICMART and the magnetic model
with gradual dissipation predict comparable neutrino flux at peak energy. The
models invoking ISs (IS, PH-IS, 3-COMP) are the ones with the lowest photon
and neutrino yield. This is due to the microphysics parameters adopted in this
work; we refer the reader to Appendix C for details on the differences with
respect to standard assumptions commonly used in the literature.

An important aspect to consider in targeted GRB searches is the large spread
in energy and shape of the expected neutrino fluxes for different jet models.
It is evident from Fig. 12 that the neutrino flux peak energy ranges from
$\mathcal{O}(10^{4})$ GeV for the proton synchrotron model to
$\mathcal{O}(10^{8}-10^{9})$ GeV for the IS model with a dissipative
photosphere. As such, targeted searches assuming one specific GRB model, such
as the IS one, as benchmark case for the GRB neutrino emission may lead to
biased results.

Another caveat of our modeling is that the spectral energy distributions of
photons and the ones of the secondary particles produced through $p\gamma$
interactions are not computed self-consistently; this may affect the overall
expected emission, see e.g. Refs. [149, 150, 151, 152, 153] for dedicated
discussions. In addition, since none of the considered jet models can account
for all observational constraints, a population study [43] may further affect
the expected quasi-diffuse emission.

###  6.3 Uncertainties in the jet parameters

In this work, we have chosen one benchmark GRB as representative of the whole
population and have relied on the best fit values of the input parameters.
However, there are intrinsic uncertainties of the characteristic jet
parameters, which can lead to variations of the expected neutrino flux, see
e.g. Refs. [36, 74]. Hence, we now investigate the impact of the variation of
two of the most uncertain parameters, the variability timescale $t_{v}$ and
the Lorentz boost factor $\Gamma$, on the quasi-diffuse neutrino flux.

Consistently with dedicated analysis [115, 116], we adopt $\Gamma_{\rm
min}=100$ and $\Gamma_{\rm max}=1000$ as the minimum and maximum values of the
Lorentz factor, respectively. The resulting neutrino fluxes are displayed in
the top panel of Fig. 13. A variation up to five orders of magnitude for the
proton synchrotron case is observable. We note that a band for the model with
magnetic dissipation case is missing. This is due to the fact that the outflow
saturates below the photosphere for $\Gamma_{\rm sat}=100$ ($R_{\rm
sat}>R_{\rm PH}$ for $\Gamma_{\rm sat}\gtrsim 121$; since we focus on the
production of neutrinos in the optically thin region above the photosphere, we
do not provide information about the $\Gamma_{\rm sat}=100$ case); we instead
show the case with $\Gamma=1000$ (see Appendix B for a discussion on the
dependence of the neutrino emission on the input parameters in the magnetized
model with gradual dissipation). The neutrino flux for the ICMART case is
shifted to higher energies for larger boost factors; this is determined by a
compensation effect due to the fact that the distance of the emitting region
from the central engine is assumed to be constant and around $\sim 10^{15}$ cm
in this model, thus being completely independent on the bulk Lorentz factor.

Figure 13: Same as Fig. 12, but for extreme values of boost factor $\Gamma$
and the variability timescale $t_{v}$. Top: The bands for the quasi-diffuse
neutrino flux are displayed for $\Gamma_{\rm min}=100$ and $\Gamma_{\rm
max}=1000$. Bottom: The bands for the quasi-diffuse neutrino flux are
displayed for $t_{v}=10$ ms and $t_{v}=10$ s.

In the bottom panel of Fig. 13, the variation of the quasi-diffuse neutrino
flux is shown as a function of the variability timescale. According to the
analysis performed on a wide sample of GRB lightcurves [31], we choose
$t_{v,\rm min}=10$ ms and $t_{v,\rm max}=10$ s as representative extreme
values for the variability time. For the simple IS model, the model with three
components, the proton-synchrotron model, and the ICMART model, $t_{v}$ is
directly connected to the size of the emitting region. For the IS model with
dissipative photosphere and the magnetic model with gradual dissipation
instead, $t_{v}$ is not related to any observable erratic behavior in the
electromagnetic signal; this explains why no band is considered for the
magnetic model with gradual dissipation, while the band in the case of the IS
model with dissipative photosphere comes from simply varying $R_{\rm IS}$.

It is worth noting that while the quasi-diffuse fluxes shown in Fig. 12 sit
below the IceCube stacking limits and expected sensitivity of IceCube-Gen2,
once taking into account the variability ranges of $t_{v}$ and $\Gamma$, the
quasi-diffuse fluxes for the different models can hit the expected sensitivity
of IceCube-Gen2; this hints that it may be possible to constrain extreme
configurations responsible for the prompt neutrino emission.

##  7 Conclusions

Long duration gamma-ray bursts (GRBs) are subject of investigation since long
time, being among the most mysterious transients occurring in our universe. In
the attempt of explaining the observed electromagnetic GRB emission, various
models have been proposed. The main goal of this work is to show that the
neutrino emission strongly depends on the chosen jet model, despite the fact
that different jet models may be equally successful in fitting the observed
electromagnetic spectral energy distributions.

To this purpose, we choose a benchmark GRB and compute the neutrino emission
for kinetic dominated jets, i.e. in the internal shock model, also including a
dissipative photosphere as well as three spectral components. We also consider
Poynting flux dominated jets: a jet model invoking internal-collision-induced
magnetic reconnection and turbulence (ICMART) and a magnetic jet model with
gradual dissipation. A jet model with dominant proton synchrotron radiation in
the keV-MeV energy range is also taken into consideration. In particular, the
neutrino production for the latter two models has been investigated for the
first time in this work.

Defining the radiative efficiency as the ratio of isotropic gamma-ray energy
to the total isotropic energy of the jet, we find that the least radiatively
efficient model is the proton synchrotron one, while the most efficient one is
the model with a dissipative photosphere. However, the model predicting the
largest amount of isotropic-equivalent energy going into neutrinos is the
ICMART one.

In the context of targeted searches, it should be noted that the expected
quasi-diffuse neutrino flux can vary up to $3$ orders of magnitude in
amplitude and peak at energies ranging from $10^{4}$ to $10^{8}$ GeV. The
predicted spectral shape of the neutrino distribution is also strongly
dependent on the adopted jet model. A summary of our findings is reported in
Table 3 and Fig. 12.

This work highlights the great potential of neutrinos in pinpointing the GRB
emission mechanism in the case of successful neutrino detection. In
particular, it suggests the need to rely on a wide range of jet models in
targeted stacking searches.

Note added: The modeling of the neutrino emission for the proton synchrotron
model is also presented in the independent work of Ref. [52]. Our paper
focuses on the comparison of the neutrino production across different GRB
models for the prompt emission, while Ref. [52] investigates the plausibility
of the proton synchrotron interpretation.

## Acknowledgments

We are grateful to Daniele Caprioli, Jochen Greiner, Gor Oganesyan for
insightful discussions and Damien Bégué, Kohta Murase and Walter Winter for
useful comments on the manuscript. This project has received funding from the
Villum Foundation (Project No. 13164), the Carlsberg Foundation (CF18-0183),
the Knud Højgaard Foundation, the Deutsche Forschungsgemeinschaft through
Sonderforschungbereich SFB 1258 “Neutrinos and Dark Matter in Astro- and
Particle Physics” (NDM), and the MERAC Foundation.

##  Appendix A Spectral energy distributions of photons: fitting functions

In order to describe the electromagnetic emission, in this appendix we
introduce the main spectral functions used to fit the electromagnetic data:
the Band function, the cut-off power-law, a simple power-law, and a double
broken power-law usually representing the synchrotron emission from a
marginally fast cooling particle population. The various spectral functions
introduced here are then employed to model the GRB emission in Sec. 5.

When the energy distribution does not present an intrinsic cut-off at high
energies (e.g., Band function and synchrotron spectrum), we define
$E^{\prime}_{\gamma,\rm cutoff}$ as the energy at which the opacity to photon-
photon pair production becomes unity

$\tau_{\gamma\gamma}(E^{\prime}_{\gamma,\rm cutoff})\simeq
0.1\sigma_{T}E^{\prime}_{\ast}n^{\prime}_{\gamma}(E^{\prime}_{\ast})\frac{R_{\gamma}}{2\Gamma}=1$
(A.1)

where $E^{\prime}_{\ast}=m^{2}_{e}c^{4}/E^{\prime}_{\gamma,\rm cutoff}$ and
$\tau_{\gamma\gamma}(E^{\prime}_{\gamma,\rm cutoff})$ is the opacity for the
photons with energy $E^{\prime}_{\gamma,\rm cutoff}$ and number density
distribution $n^{\prime}_{\gamma}(E^{\prime}_{\gamma})$.

###  A.1 Band function

The Band function [154] is the most used empirical function to fit the time-
integrated electromagnetic spectra. Despite fitting well the data, it is still
lacking a clear physical meaning. It consists of a smoothly joint broken
power-law:

$n^{\rm{Band}}_{\gamma}(E_{\gamma})=C\begin{cases}\left(\frac{E_{\gamma}}{100\
\mathrm{keV}}\right)^{\alpha_{\gamma}}\exp\left[-\frac{(\alpha_{\gamma}+2)E_{\gamma}}{E_{\gamma,\rm{peak}}}\right]\quad&E_{\gamma}<E_{\gamma,c}\\\
\left(\frac{E_{\gamma}}{100\
\mathrm{keV}}\right)^{\beta_{\gamma}}\exp(\beta_{\gamma}-\alpha_{\gamma})\left(\frac{E_{\gamma,c}}{100\
\mathrm{KeV}}\right)^{\alpha_{\gamma}-\beta_{\gamma}}\quad&E_{\gamma}\geq
E_{\gamma,c}\end{cases}$ (A.2)

where

$E_{\gamma,c}=\left(\frac{\alpha_{\gamma}-\beta_{\gamma}}{\alpha_{\gamma}+2}\right)E_{\gamma,\rm
peak}\ ,$ (A.3)

$C$ is a normalization constant (in units of
$\mathrm{GeV}^{-1}\mathrm{cm}^{-3}$), $\alpha_{\gamma}$ and $\beta_{\gamma}$
are the low-energy and high-energy power-law photon indices, $E_{\gamma,c}$
represents the energy where the low-energy power-law with an exponential
cutoff ends and the pure high energy power-law begins. The peak energy
$E_{\gamma,\rm peak}$ is chosen to satisfy the Amati relation [155]:

$\tilde{E}_{\gamma,\rm
peak}=80\left(\frac{\tilde{E}_{\gamma,\mathrm{iso}}}{10^{52}\
\mathrm{erg}}\right)^{0.57}\ \mathrm{keV}\ .$ (A.4)

The typical spectral parameters inferred from observations are:
$\alpha_{\gamma}\simeq-1.1$, $\beta_{\gamma}\simeq-2.2$, and $E_{\gamma,\rm
peak}\simeq 300$ keV [2].

###  A.2 Cut-off power-law

Although the Band spectrum is the best fitting function for most GRBs, it has
been shown that in some cases a cut-off power-law (CPL) can represent the
preferred model [156, 157, 158]. The CPL is a power-law model with a high
energy exponential cut-off :

$n^{\rm{CPL}}_{\gamma}(E_{\gamma})=C\left(\frac{E_{\gamma}}{100\
\mathrm{keV}}\right)^{\alpha_{\gamma}}\exp\left[-\frac{(\alpha_{\gamma}+2)E_{\gamma}}{E_{\gamma,\rm
peak}}\right]\ ,$ (A.5)

where $\alpha_{\gamma}$ is the photon index and $E_{\gamma,\rm peak}$ the peak
energy, whose value will be specified later. In an optically thick thermal
scenario, $\alpha_{\gamma}=1$ in the Rayleagh-Jeans limit, $\alpha_{\gamma}=2$
in the Wien limit, $\alpha_{\gamma}=0.4$ for a non-dissipative photosphere in
the coasting phase and $\alpha_{\gamma}<0$ for all non-thermal emissions [42].

###  A.3 Power law

In the cases of faint bursts or narrow detector bandpass, the whole GRB
spectrum, or one of its components, can be fitted with a simple power-law
[159] defined as

$n^{\mathrm{PL}}_{\gamma}(E_{\gamma})=C\left(\frac{E_{\gamma}}{100\
\mathrm{keV}}\right)^{\alpha_{\gamma}}\ ,$ (A.6)

where $C$ is the normalization and $\alpha_{\gamma}$ is the power-law photon
index.

###  A.4 Double broken power law

This is a spectral model that is commonly adopted to describe the synchrotron
emission of a fast cooling population of particles that are being injected
into the emitting region with a power-law distribution at a rate
$Q(\gamma)\propto\gamma^{-k}$ with
$\gamma_{\mathrm{min}}<\gamma<\gamma_{\mathrm{max}}$. During an emission
period $t$, the charged particles of mass $m$ loose most of their energy above
the characteristic value $\gamma_{\rm cool}$:

$\gamma_{\mathrm{cool}}(t)=\frac{6\pi
mc}{\sigma_{\mathrm{T}}\beta^{2}B^{2}t}\left(\frac{m}{m_{e}}\right)^{2}\ ,$
(A.7)

where $m_{e}$ is the electron mass. Considering a constant injection rate of
particles in the emitting region which radiate in the fast cooling regime
($\gamma_{\mathrm{min}}>\gamma_{\mathrm{cool}}$) at a rate
$\propto\gamma^{2}$, after a time $t$ the emitting particle distribution has
the following shape [69]:

$n(\gamma,t)\propto\begin{cases}0&\gamma<\gamma_{\mathrm{cool}}\ \ \rm{and}\ \
\gamma>\gamma_{\mathrm{\mathrm{max}}}\\\
\gamma^{-2}&\gamma_{\mathrm{cool}}<\gamma<\gamma_{\mathrm{min}}\\\
\gamma^{-(k+1)}&\gamma_{\mathrm{min}}<\gamma<\gamma_{\mathrm{max}}\
.\end{cases}$ (A.8)

Given that each particle radiates photons with a characteristic synchrotron
energy

$E_{\gamma}(\gamma)=\frac{3}{2}\frac{\hbar e}{mc}\gamma^{2}B,$ (A.9)

the particle distribution in Eq. A.8 emits the following synchrotron spectrum:

$n^{\mathrm{sync}}_{\gamma}(E_{\gamma})=C\begin{cases}\left(\frac{E_{\gamma}}{E_{\gamma,\mathrm{cool}}}\right)^{-\frac{2}{3}}\quad&E_{\gamma}<E_{\gamma,\mathrm{cool}}\\\
\left(\frac{E_{\gamma}}{E_{\gamma,\mathrm{cool}}}\right)^{-\frac{3}{2}}\quad&E_{\gamma,\mathrm{cool}}<E_{\gamma}<E_{\gamma,\mathrm{min}}\\\
\left(\frac{E_{\gamma,\mathrm{min}}}{E_{\gamma,\mathrm{cool}}}\right)^{-\frac{3}{2}}\left(\frac{E_{\gamma}}{E_{\gamma,\mathrm{min}}}\right)^{-\frac{k+2}{2}}&E_{\gamma,\mathrm{min}}<E_{\gamma}<E_{\gamma,\mathrm{max}}\
,\\\ \end{cases}$ (A.10)

where $E_{\gamma,\mathrm{cool}}$, $E_{\gamma,\mathrm{min}}$, and
$E_{\gamma,\mathrm{max}}$ correspond to the characteristic photon energies
mainly emitted by particles with gamma factors $\gamma_{\mathrm{cool}}$,
$\gamma_{\mathrm{min}}$, and $\gamma_{\mathrm{max}}$, respectively.

##  Appendix B Magnetized jet model with gradual dissipation: dependence of
the neutrino emission on the input parameters

One of the main, but less certain, parameters of the jet model with gradual
magnetic dissipation is the initial magnetization $\sigma_{0}$. This, in turn,
determines the photospheric radius, the saturation Lorentz factor, the energy
dissipation rate, and other parameters. For this reason, we investigate the
impact of $\sigma_{0}$ on the photon and neutrino fluences by considering a
case with $\sigma_{0}=100$. All the other parameters, like $\tilde{E}_{\rm
iso}$, are identical to the ones adopted in Sec. 5.2.2. We follow the same
procedure to calculate the neutrino flux as outlined in Sec. 5.2.2.

In Fig. 14, we show snapshots of the photon fluence (left panel) and neutrino
fluence (right panel) for $\sigma_{0}=100$ at three indicative radii. A
comparison between the $\sigma_{0}=45$ and $\sigma_{0}=100$ cases is shown in
Fig. 15.

Figure 14: Similar to Fig. 9, but for $\sigma_{0}=100$. The fluence for the
muon flavor peaks at $E^{\rm peak}_{\nu}=3.3\times 10^{5}$ GeV. In addition,
$E_{\nu,\rm iso}=8.6\times 10^{50}$ erg, $E_{\gamma,\rm iso}=2.3\times
10^{53}$ erg, $\Gamma_{\rm sat}=1000$, $\sigma_{0}=100$, $R_{\rm PH}=7.1\times
10^{11}$ cm, $R_{1}=2.1\times 10^{12}$ cm, $R_{2}=3\times 10^{13}$ cm,
$R_{3}=4\times 10^{14}$ cm, $\Gamma_{1}=180,\Gamma_{2}=422,\Gamma_{3}=1000$,
$\tilde{E}_{\nu,\rm iso}/\tilde{E}_{\gamma,\rm iso}=3.7\times 10^{-3}$,
$\eta_{\gamma}=7\%$.

Figure 15: Left: Total photon fluence in the observer reference frame,
obtained as the sum of the components produced at
$R_{\rm{PH}},\,R_{1},\,R_{2}$ and $R_{3}$ for the $\sigma_{0}=45$ (solid line)
and $\sigma_{0}=100$ (dashed line) cases, respectively. Right: Correspondent
$\nu_{\alpha}+\bar{\nu}_{\alpha}$ fluence (in red and in blue for the electron
and muon flavors, respectively). For parameters used, see captions of Figs. 9
and 14.

A noticeable difference is appreciable between the photon spectral energy
distributions. As the initial magnetization increases, the saturation Lorentz
factor increases, namely $\Gamma_{\rm sat}=\sigma^{3/2}_{0}=1000$ for
$\sigma_{0}=100$. The energy is dissipated at a rate $\dot{E}\propto R^{1/3}$,
while the photosphere occurs at a smaller distance from the source ($R_{\rm
PH}\propto 1/\Gamma_{\rm sat}$). As a result, less energy is dissipated during
the optically thick regime (most of the energy is dissipated at $R>R_{\rm
PH}$) and the photospheric emission becomes dimmer (compare dashed and solid
lines at $E_{\gamma}\sim 10^{-5}$–$10^{-4}$ GeV in Fig. 15). The
characteristic synchrotron energy $E_{\gamma,\rm
min}\propto\Gamma\gamma^{\prime 2}_{\rm min}B^{\prime}$ (Eq. A.9) decreases
with the radius (see, e.g., dashed colored curves in the left panel of Fig.
14), while the normalization of the synchrotron photon spectra increases with
respect to the case of $\sigma_{0}=45$ (see dashed curve in the left panel of
Fig. 15) because of the higher dissipation rate (Eq. 2.11). For a higher
$\sigma_{0}$, particle acceleration begins at smaller radii and so does the
production of neutrinos. Moreover, the power slopes of the electron and proton
distributions (accelerated via reconnection) are harder [102, 103] because of
the higher magnetization in the acceleration region. Because of the larger
saturation radius ($R_{\rm sat}\propto\Gamma_{\rm sat}^{2}$) found for higher
$\sigma_{0}$, the dissipated energy up to $R_{1}=aR_{\rm ph}\propto\Gamma_{\rm
sat}^{-1}$ that is available for relativistic particles is less than in the
case of lower initial magnetizations.

Figure 16: Left: Total photon fluence in the observer reference frame,
obtained as the sum of the components produced at
$R_{\rm{PH}},\,R_{1},\,R_{2}$ and $R_{3}$ for the $\sigma_{0}=45$ and $a=13$
case. Right: Correspondent $\nu_{\alpha}+\bar{\nu}_{\alpha}$ fluence (in red
and in blue for the electron and muon flavors, respectively).

The combination of a smaller amount of dissipated energy up to a given radius,
smaller volume and harder proton power slope leads to a neutrino flux at peak
(whose main contribution comes from $R_{1}$) comparable to the one with
$\sigma_{0}=45$ (see right panel of Fig. 15).

For the case of $\sigma_{0}=100$, the second bump in the neutrino spectrum has
a fluence that is comparable to the one of the first bump at $\sim 10^{6}$
GeV. On the contrary, the second bump is barely visible in the neutrino energy
distribution with $\sigma_{0}=45$ (compare solid and dashed lines in the right
panel of Fig. 15). This is because in the $\sigma_{0}=100$ case, pions suffer
stronger synchrotron losses, hence the neutrino intensity resulting from the
decay of pions decreases to the level of the one produced by kaons. This is
also the reason for a slight shift in the neutrino flux peak to lower
energies. Another noticeable feature is the low energy tail. The latter turns
out to be higher in the $\sigma_{0}=100$ case, given the higher number density
of photons at higher energies.

Finally, in order to explore the effects of the arbitrary choice of the
parameter $a$, we considered the case with $\sigma_{0}=45$ and $a=13$, where
$R_{1}\sim R_{\rm sat}$, see Fig. 16. Since most of the energy is dissipated
within $R_{1}$, the neutrino contribution from $R_{1}$ is dominant, although
lower by a factor $\mathcal{O}(10)$ if compared to the case with $a=3$.
However, since the case $a=13$ represents an extreme case, such that all the
energy is locally dissipated near the saturation radius, the fluence of
$\mathcal{O}(10^{-6})\,\rm{GeVcm}^{-2}$ should be considered as the lower
limit for the neutrino production from a GRB described by the magnetic model
with gradual dissipation for the specific set of parameters adopted in this
work.

##  Appendix C Quasi-diffuse neutrino flux for standard internal shock
parameters

In this work, we have adopted input parameters inspired by the results of
recent PIC simulations of mildly relativistic shocks for the IS models (see
Sec. 3.1). However, in the literature, under the assumption that GRBs are the
main sources of ultra-high-energy cosmic rays, the following parameters are
often adopted: $\varepsilon_{p}=10/12$,
$\varepsilon_{e}=\varepsilon_{B}=1/12$, and $k_{p}=2$, see e.g. [95]. Fig. 17
shows the quasi-diffuse neutrino emission for these input parameters, in order
to facilitate a comparison with the existing literature. One can see that the
flux normalization of the simple IS model, the IS model with dissipative
photosphere and the IS model with three components is indeed larger than what
is shown in Fig. 12, and roughly at the same level of the ICMART model, the
proton synchrotron model and magnetized jet model with gradual dissipation.

Figure 17: Quasi-diffuse neutrino flux for the six models considered in this
work and computed as in Fig. 12, but with the classically adopted microphysics
parameters for the Internal shocks:
${\varepsilon_{p}=10/12,\varepsilon_{e}=\varepsilon_{B}=1/12}$, and $k_{p}=2$.

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