Neutrino Oscillations and Decoherence in Short-GRB Progenitors

Short gamma-ray bursts (S-GRBs) are intense flashes of gamma-rays that last less than 2 s in the observer frame. It is widely accepted that S-GRBs originate from the merging of two compact objects, such as a neutron star (NS) and a black hole (BH), or two neutron stars (NS–NS). During the merging phase, angular momentum and energy losses are manifested as gravitational wave emission and electromagnetic radiation. In both cases, the remnant is a BH of a few solar masses. There are different models that try to explain the observed emission of S-GRBs; among others, the Fireball Model (Piran 1999) and the Fireshell Model (Bianco et al. 2008a, 2008b; Bianco & Ruffini 2008; Enderli et al. 2014).

The Fireball Model states that in the case of the NS–NS system an accretion disk is formed around the newly born BH (Berger 2014). In the NS–BH case, the same can occur if the NS is tidally disrupted outside the BH's event horizon. The rapidly rotating BH bends the magnetic field lines forming a double jet perpendicular to the accretion-disk plane. A fraction of the electromagnetic radiation escapes in the form of gamma-rays, while another fraction goes into neutrino–antineutrino emission (Narayan et al. 1992).

In the Fireshell Model scenario, the NS–NS merging leads to a massive NS that exceeds its critical mass and gravitationally collapses to a BH with isotropic energy emission of the order of E_iso ≳ 10⁵² erg. Gravitational waves are produced (Oliveira et al. 2014) together with GeV emission from the accretion onto the Kerr BH (Ruffini et al. 2018). It has been shown (Becerra et al. 2018) that the accretion onto the NS generates neutrino–antineutrino emission in the case of long GRBs, and this emission has been explained as being due to e⁻e⁺-pair annihilation.

In this work, we apply the Fireshell Model to explain neutrino emission in S-GRBs. We describe the conditions under which the neutrino emission takes place, and we analyze the effects of flavor oscillations and decoherence on neutrinos on their way from the source to the observer on Earth.

The work is organized as follows: In Section 2, we describe the model. In Section 3, we derive the expressions for the electron and neutrino number densities, following a statistical treatment. In Section 4, we compute the electron and neutrino fluxes at the source. In Section 5, we analyze the effects of neutrino–flavor oscillations in vacuum, from the moment in which neutrinos are produced up to the time they reach the external crust. In Section 6, we analyze the effects of neutrino–flavor oscillations in matter, as they propagate through the crust and interact with baryons. In Section 7, we introduce the mechanism of decoherence, as neutrinos that leave the crust and propagate through the universe toward the observer interact with background intergalactic particles. We calculate the detected flux on Earth and compare it with the flux at the source. The results are presented and discussed in Section 7.3. Finally, in Section 8 we draw our conclusions.

A typical scenario for S-GRBs within the Fireshell Model is depicted in Figure 1. Two NSs of masses M₁ and M₂, typically of the order of 1.6–2 M_⊙, start spiraling together until they merge giving birth to a BH due to gravitational collapse. Let us consider for simplicity M₁ = M₂. Only the core collapses, leaving a thin crust of a fraction of a solar mass. In the vacuum between the crust and the BH event horizon, a strong electric field is generated due to charge separation. When this field reaches the critical value, E_c, vacuum polarization takes place generating an e⁻e⁺-plasma. Typical densities for the electron–positron plasma are of the order of 10³³ particles cm⁻³. Some of these pairs annihilate, giving neutrinos and antineutrinos that propagate outward, first in vacuum then through the crust formed by e⁻, protons, and neutrons and finally through the intergalactic medium until they reach the observer on Earth (SNO Collaboration 2000; Halzen & Klein 2010). Another fraction of the e⁻e⁺-pairs produce thermal photons. Since at this stage the system is still opaque to radiation, the radiation pressure increases making the plasma expand until it reaches the crust. The whole system continues to expand until it reaches transparency. At this point the thermal photons escape. This is seen as a thermal spike in the spectrum called proper GRB (P-GRB; Ruffini et al. 2001). The remaining material continues to expand while interacting with the circumburst medium producing the prompt emission.

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Table 1 shows the values of the parameters of our model.

Table 1.  Parameters of Our Model

Parameter	Symbol	Value
NS radius [km]	RNS	10
NS mass [M⊙]	MNS	2
${e}^{\pm }$ plasma density [part cm−3]	${N}_{e}^{\pm }$	1033
${e}^{\pm }$ plasma temperature [MeV]	kT	2.0
BH radius [cm]	rBH	3.3 × 105
Crust internal radius [cm]	rint	1.69 × 108
Crust external radius [cm]	rext	1.2 × 1010
Source-detector distance [cm]	DL	1028
Mass of the crust [M⊙]	Mcrust	0.1
Density of the crust [g cm−3]	ρcrust	27.47
Proton density in the crust [part cm−3]	Np	0.25 ρcrust
Neutron density in the crust [part cm−3]	Nn	0.25 ρcrust
e−density in the crust [part cm−3]	Ne	0.50 ρcrust

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In order to calculate the number density and energy of the neutrinos created during the merging of the two NSs, we follow a statistical treatment. We treat the neutrinos as a Fermi–Dirac gas in thermodynamical equilibrium at temperature kT =2 MeV (Ruffini et al. 1999). The neutrino emission zone is the same as the one occupied by the e⁻e⁺-plasma, a shell that extends from the BH event horizon (r_BH ≈ 10⁵ cm) to the crust (r_crust ≈ 10¹⁰ cm).

The Fermi–Dirac distribution function for T $\ne$ 0 is given by

$$\begin{eqnarray}&&f(E)=\displaystyle \frac{1}{{e}^{(E-\mu )/{k}_{{\rm{B}}}T}+1},\end{eqnarray} \tag{ 1 }$$

where k_B is the Boltzmann constant and E is the fermion energy. The parameter μ is known as the Fermi level. In the limit $T\to 0$, f(E) becomes a step function θ(E − μ): all the energy levels with E < μ are occupied, while all the others are empty.

3.1. Electrons

In the relativistic case, the energy of the electrons is $E=\sqrt{{p}^{2}{c}^{2}+{m}^{2}{c}^{4}}$, the momentum in terms of the energy is given by $p=\tfrac{1}{c}\sqrt{{E}^{2}-{m}^{2}{c}^{4}}$, and ${dp}=\tfrac{1}{c}\tfrac{{EdE}}{\sqrt{{E}^{2}-{m}^{2}{c}^{4}}}$. The number of particles in the system is given by the expression

$$\begin{eqnarray}&&N=\int f(E)d{\rm{\Omega }}=\displaystyle \frac{4\pi {{Vg}}_{s}}{{\left(2\pi {\hslash }\right)}^{3}}{\int }_{0}^{\infty }\displaystyle \frac{{p}^{2}{dp}}{{e}^{(E-{\mu }_{e})/{k}_{{\rm{B}}}T}+1}.\end{eqnarray} \tag{ 2 }$$

Here,

$$\begin{eqnarray*}&&d{\rm{\Omega }}=\displaystyle \frac{d{\boldsymbol{q}}d{\boldsymbol{p}}}{{\left(2\pi {\hslash }\right)}^{3}}=\displaystyle \frac{4\pi V}{{\left(2\pi {\hslash }\right)}^{3}}{p}^{2}{dp}\end{eqnarray*}$$

is the volume element in phase space and g_s = 2 is the spin degeneracy factor. Changing variables to x = E − mc² yields the number density ρ_e = N/V:

$$\begin{eqnarray}&&{\rho }_{e}=\displaystyle \frac{{\left(2m\right)}^{3/2}}{2{\pi }^{2}{{\hslash }}^{3}}{\int }_{0}^{\infty }\displaystyle \frac{{x}^{1/2}{\left(1+x/2{{mc}}^{2}\right)}^{1/2}(1+x/{{mc}}^{2}){dx}}{1+{e}^{(x+{{mc}}^{2}-{\mu }_{e})/{kT}}}.\end{eqnarray} \tag{ 3 }$$

By making the substitutions ${\eta }_{e}=({\mu }_{e}-{{mc}}^{2})/{kT}$, w = x/kT, and β = kT/mc², Equation (3) becomes

$$\begin{eqnarray}&&{\rho }_{e}=\displaystyle \frac{{\left(2{mkT}\right)}^{3/2}}{2{\pi }^{2}{{\hslash }}^{3}}[{F}_{1/2}({\eta }_{e})+\beta {F}_{3/2}({\eta }_{e})],\end{eqnarray} \tag{ 4 }$$

where

$$\begin{eqnarray*}&&{F}_{r}={\int }_{0}^{\infty }\displaystyle \frac{{x}^{r}{\left(1+x\beta /2\right)}^{1/2}{dx}}{1+{e}^{x-{\eta }_{e}}}\end{eqnarray*}$$

for r = 1/2, 3/2, 5/2, etc.

A similar expression is obtained for the mean energy of the electrons as a function of temperature and density:

$$\begin{eqnarray}&&\displaystyle \frac{\langle {E}_{e}\rangle }{{\rho }_{e}}={{mc}}^{2}\displaystyle \frac{{F}_{1/2}+2\beta {F}_{3/2}+{\beta }^{2}{F}_{5/2}}{{F}_{1/2}+\beta {F}_{3/2}}.\end{eqnarray} \tag{ 5 }$$

3.2. Neutrinos

Neutrinos have negligible masses compared to their energy (E ≫ mc²), so E ≈ pc. Following the same procedure as in Section 3.1 we find for the neutrino number density

$$\begin{eqnarray}&&{\rho }_{\nu }=\displaystyle \frac{{N}_{\nu }}{V}=\displaystyle \frac{4\pi {g}_{s}{\left({k}_{{\rm{B}}}T\right)}^{3}}{{\left(2\pi {\hslash }c\right)}^{3}}{F}_{2}({\eta }_{\nu }),\end{eqnarray} \tag{ 6 }$$

where

$$\begin{eqnarray*}&&{F}_{n}={\int }_{0}^{\infty }\displaystyle \frac{{x}^{n}{dx}}{1+{e}^{x-{\eta }_{\nu }}}\end{eqnarray*}$$

for n = 1, 2, 3, ... and η_ν = μ_ν/kT.

The neutrino mean energy is given by

$$\begin{eqnarray}&&\langle {E}_{\nu }\rangle =\,\displaystyle \frac{{\left({k}_{{\rm{B}}}T\right)}^{4}}{{\pi }^{2}{\left({\hslash }c\right)}^{3}}{F}_{3}({\eta }_{\nu });\end{eqnarray} \tag{ 7 }$$

thus, the mean energy per neutrino is given by

$$\begin{eqnarray}&&\displaystyle \frac{\langle {E}_{\nu }\rangle }{{\rho }_{\nu }}={kT}\left(\displaystyle \frac{{F}_{3}({\eta }_{\nu })}{{F}_{2}({\eta }_{\nu })}\right).\end{eqnarray} \tag{ 8 }$$

With the parameters given in Table 1 and the formalism presented in Section 3 we have performed a numerical search to determine the electron and neutrino chemical potentials μ_e and μ_ν, and with them the mean energies and spectral functions (Cox & Giuli 1968).

The numerical search gives from Equations (3) and (6) the best values of η_e and η_ν for a given density. The results are η_e = 1.75 and η_ν = 2.0, corresponding to μ_e = 4.01 MeV and μ_ν = 4.00 MeV. Figure 2 shows the occupation numbers f(E) of Equation (1) for a plasma temperature of 2 MeV (Ruffini et al. 1999).

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We have calculated the electron and neutrino fluxes inside the e⁻e⁺-plasma. Each flux is given by the ratio

$$\begin{eqnarray}&&{F}_{e,\nu }=\displaystyle \frac{1}{{\rho }_{e,\nu }}\displaystyle \frac{d\langle {E}_{e,\nu }\rangle }{{dE}}.\end{eqnarray} \tag{ 9 }$$

Figure 3 shows the results of Equation (9) for electron and neutrino fluxes in the region of the e⁻e⁺-plasma.

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As soon as they are created, neutrinos start to propagate outward at nearly the speed of light from the region close to the event horizon toward the crust. This region is opaque to radiation, but nothing prevents neutrinos from escaping. Because of the geometry of the source (see Figure 1) we shall consider propagation and oscillations in vacuum in the inner region between the BH and the crust.

Neutrinos oscillate because the flavor states in which they are created are a superposition of mass eigenstates. Because they have different masses they evolve with different phases.

The Hamiltonian in the mass basis is given by

$$\begin{eqnarray}{H}_{m}=\left[\begin{array}{lll}{m}_{1} & 0 & 0\\ 0 & {m}_{2} & 0\\ 0 & 0 & {m}_{3}\end{array}\right].\end{eqnarray} \tag{ 10 }$$

The mass hierarchies are denoted, as usual: m₁ ≤ m₂ ≪ m₃ (normal hierarchy), m₁ ≪ m₂ ≤ m₃ (inverted hHierarchy), or m₁ ≈ m₂ ≈ m₃ (degenerate hierarchy). Adopting the normal hierarchy and setting m₁ = 0 yields m₂ = 0.00858 eV and m₃ = 0.0506 eV.

The neutrino mass Hamiltonian ${H}_{m}=\mathrm{diag}({m}_{1},{m}_{2},{m}_{3})$ is transformed to the flavor basis by applying upon it the mixing matrix (Bilenky 2000; Kersten & Smirnov 2016)

$$\begin{eqnarray}&&\begin{array}{l}U\\ =\left[\begin{array}{lcr}{c}_{12}{c}_{13} & {s}_{12}{c}_{13} & {s}_{13}{e}^{-\delta }\\ -{s}_{12}{c}_{23}-{c}_{12}{s}_{23}{s}_{13}{e}^{\delta } & {c}_{12}{c}_{23}-{s}_{12}{s}_{23}{s}_{13}{e}^{\delta } & {s}_{23}{c}_{13}\\ {s}_{12}{s}_{23}-{c}_{12}{c}_{23}{s}_{13}{e}^{\delta } & -{c}_{12}{s}_{23}-{s}_{12}{c}_{23}{s}_{13}{e}^{\delta } & {c}_{23}{c}_{13}\end{array}\right],\end{array}\end{eqnarray} \tag{ 11 }$$

where c_ij (s_ij) is the cosine (sine) of the mixing angles θ_ij. Therefore,

$$\begin{eqnarray}&&{H}_{f}={{UH}}_{m}{U}^{-1}.\end{eqnarray} \tag{ 12 }$$

The parameters entering U and H_m are listed in Table 2. In the present analysis we have taken, for the Dirac CP-violating phase, the value δ/π = 0. Furthermore, and in order to separate the electronic flavor from linear combinations of the μ and τ ones, we apply to the flavor Hamiltonian the decoupling matrix (Kersten & Smirnov 2016)

$$\begin{eqnarray}D=\left[\begin{array}{ccc}1 & 0 & 0\\ 0 & 1/\sqrt{2} & 1/\sqrt{2}\\ 0 & -1/\sqrt{2} & 1/\sqrt{2}\end{array}\right].\end{eqnarray} \tag{ 13 }$$

Table 2.  Neutrino–Oscillation Parameters, for the Normal (NH) and Inverted (IH) Mass Hierarchies

${\sin }^{2}({\theta }_{12})=0.297$

${\sin }^{2}({\theta }_{13})=0.0215$

${\sin }^{2}({\theta }_{23})=0.425$ (NH)

${\sin }^{2}({\theta }_{23})=0.589$ (IH)

${\delta }_{\mathrm{atm}}^{2}={m}_{3}^{2}-{m}_{1}^{2}=2.56\times {10}^{-3}$ eV2

${\delta }_{\mathrm{solar}}^{2}={m}_{2}^{2}-{m}_{1}^{2}=7.37\times {10}^{-5}$ eV2

Note. Solar (${\delta }_{\mathrm{solar}}^{2}$) and atmospheric (${\delta }_{\mathrm{atm}}^{2}$) squared mass differences and mixing angles (θ_ij) are listed in the table. The values are taken from http://pdg.lbl.gov/2017/reviews/rpp2017-rev-neutrino-mixing.pdf.

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The "decoupled" flavor Hamiltonian reads

$$\begin{eqnarray}&&{H}_{f}^{{dec}}={{DH}}_{f}{D}^{-1},\end{eqnarray} \tag{ 14 }$$

and from the diagonalization of this Hamiltonian we obtain the eigenvalues and eigenvectors for electron–neutrinos ν_e and nonelectronic neutrinos ${\nu }_{x}=\tfrac{1}{\sqrt{2}}({\nu }_{\mu }\pm {\nu }_{\tau })$.

Following the model sketch in Figure 1, neutrinos oscillate in vacuum until they reach the internal radius of the crust. At this point, they interact with matter. We assume the crust is formed by electrons, protons, and neutrons in the amounts given in Table 1. A matter Hamiltonian must be added to the flavor Hamiltonian in vacuum. For the matter Hamiltonian we consider a diagonal one:

$$\begin{eqnarray}&&{H}_{\mathrm{mat}}=\mathrm{diag}({V}_{m},0,0),\end{eqnarray} \tag{ 15 }$$

where ${V}_{m}=\sqrt{2}{G}_{{\rm{F}}}({N}_{e}+{N}_{p}+{N}_{n})$ is the matter potential, G_F = 8.963 × 10⁻⁴⁴ MeV cm³ is the Fermi constant, and N_e, N_p and N_n are the electron, proton, and neutron densities in the crust, respectively.

Because of the thickness of the crust (≈10² cm) the interactions with matter are negligible despite the values of the baryon densities. Therefore, we shall not take these interaction into account in our analysis.

Once the neutrinos arrive at the external radius of the crust they continue their way to the detector on Earth. As the distance that they have to travel is of the order of D_L ≈ 10²⁸ cm, corresponding to typical S-GRB redshifts (Ruffini et al. 2016), decoherence effects (Schlosshauer 2007) may take place due to interactions of the source neutrinos with neutrinos in the cosmic background. Decoherence effects are relevant in the reconstruction of the sequence of events starting from the primordial production of neutrinos and ending at their detection. What we would like to evaluate quantitatively is the difference between the composition of neutrinos of the source, as dictated by the neutrino–oscillation mechanism, and their time evolution governed by decoherence. The decoherence mechanism we have in mind in not kinematic and, as we said before, it is due to interactions with other particles like neutrinos that fill the space between the source and the detector. In order to achieve this goal we shall proceed to:

• 1.  
  calculate the density matrix from the diagonalization of the flavor Hamiltonian (Equation (14));
• 2.  
  construct the time evolution matrix that determines the time dependence of the density matrix; and
• 3.  
  calculate the probability of detecting neutrinos of a given flavor on Earth.

In what follows, we present the corresponding theoretical details.

7.1. Flavor Eigenstates at $t=0$

The density matrix for electron–neutrinos leaving the crust is

$$\begin{eqnarray}&&{\rho }_{{\nu }_{e}}=| {\phi }_{{\nu }_{e}}\rangle \otimes \langle {\phi }_{{\nu }_{e}}{| }_{(t=0)}.\end{eqnarray} \tag{ 16 }$$

With the amplitudes of the electron–neutrino eigenvalue obtained by the diagonalization of H_f, Equation (16) is readily calculated.

The density matrix (Equation (16)) is that of a pure state, that is, ρ² = ρ, and its diagonalization yields the survival probabilities of the electron–, muon–, and tau–neutrino channels, respectively.

7.2. Time Dependence of the Density Matrix

To calculate the time dependence of the density matrix for neutrinos leaving the crust we add to the flavor Hamiltonian the interaction of the electron–neutrinos with the environment. For this, we follow the formalism presented in Bes & Civitarese (2017) and Schlosshauer (2007). Accordingly, we construct the matrix

$$\begin{eqnarray}&&A={H}_{f}^{\mathrm{dec}}+\mathrm{diag}(B{\lambda }_{\mathrm{coup}}/2,0,-B{\lambda }_{\mathrm{coup}}/2),\end{eqnarray} \tag{ 17 }$$

where λ_coup is the coupling constant and B is a constant field acting on the neutrinos. The diagonalization of A leads to the eigenvalues and eigenvectors needed to construct the evolution matrix U(t) (Schlosshauer 2007), which is defined by the expression

$$\begin{eqnarray}&&U(t)=V\,\mathrm{diag}({e}^{i{{\rm{\Omega }}}_{n}t})\,{V}^{-1},\end{eqnarray} \tag{ 18 }$$

where V is the matrix of eigenvectors of A, Ω_n, n = 1, 2, 3, the associated eigenvalues, being both V and Ω_n functions of the strength B. In writing Equation (18), we use ${\hslash }=1$. In this picture the density matrix ${\rho }_{{\nu }_{e}}$ of Equation (16) evolves with time as

$$\begin{eqnarray}&&\rho (t,B)=U(t)\,{\rho }_{{\nu }_{e}}\,{U}^{-1}(t).\end{eqnarray} \tag{ 19 }$$

If the strength B is distributed like a Gaussian around B = 0 with standard deviation σ, we integrate the matrix ρ(t, B) in B so that its elements [ρ(t)]_ij are

$$\begin{eqnarray}&&{[\rho (t)]}_{{ij}}=\int {[\rho (t,B)]}_{{ij}}\displaystyle \frac{{e}^{-{B}^{2}/2{\sigma }^{2}}}{\sqrt{2\pi }\sigma }{dB}.\end{eqnarray} \tag{ 20 }$$

A last diagonalization of ρ(t) for t is sufficiently large, of the order of L/c being L the distance from the source to the detector and c the speed of light, c, and leads to the survival probabilities of neutrinos of a given flavor, in this case, of electron–neutrinos. These probabilities are needed in order to renormalize the neutrino flux at the Earth, as explained below.

7.3. Results for $\rho (t=0)$ and $\rho (t)$

For the normal hierarchy and masses m₁ = 0, m₂ =0.00866 eV, m₃ = 0.0495 eV, and δ = 0 in Equation (11), we get

$$\begin{eqnarray}{\rho }_{{\nu }_{e}}(t=0)=\left[\begin{array}{ccc}0.680 & -0.124 & 0.449\\ -0.124 & 0.022 & -0.082\\ 0.449 & -0.082 & 0.297\end{array}\right].\end{eqnarray} \tag{ 21 }$$

The diagonalization of ρ(t = 0) gives

$$\begin{eqnarray}&&{P}_{{\nu }_{e}\to {\nu }_{e}}(t=0)=1,{P}_{{\nu }_{e}\to {\nu }_{x}}(t=0)=0.\end{eqnarray} \tag{ 22 }$$

To illustrate the effect of decoherence we calculate the time evolution given by Equation (20) with λ_coup = 1.0 and σ = 20.

For a sufficiently large number of oscillations in the presence of the interactions due to the background and for the chosen parameterization, the density matrix is given by

$$\begin{eqnarray}\begin{array}{rcl}\rho (t) & = & {\rho }_{({Re})}(t)+i{\rho }_{({Im})}(t)\\ & = & \left[\begin{array}{ccc}0.680 & -0.0005 & 0.00002\\ -0.0005 & 0.022 & -0.0002\\ 0.00002 & -0.0002 & 0.296\end{array}\right]\\ & + & i\left[\begin{array}{ccc}0 & 0.005 & -0.0008\\ -0.005 & 0 & -0.0003\\ 0.0008 & 0.0003 & 0\end{array}\right],\end{array}\end{eqnarray} \tag{ 23 }$$

which is no longer the density matrix of a pure state, as ρ² $\ne$ ρ.

Diagonalization of this matrix gives the survival probabilities

$$\begin{eqnarray}&&{P}_{{\nu }_{e}\to {\nu }_{e}}(t\to \infty )=0.67871,\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&{P}_{{\nu }_{e}\to {\nu }_{x}}(t\to \infty )=0.32129.\end{eqnarray} \tag{ 25 }$$

Figure 4 shows the elements of the density matrix as a function of time, as the system evolves to the pointer states.The fact that the matrix ρ(t) loses its pure-state nature due to decoherence is better illustrated by the results shown in Figure 5, where the final pointer states are identified by means of the probabilities ${P}_{{\nu }_{e}\to {\nu }_{e}}$ and ${P}_{{\nu }_{e}\to {\nu }_{x}}$.

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The neutrino flux, for neutrinos emitted at the source (see Figure 3), should then be renormalized to account for the evolution of ρ from pure to pointer states. This is done by multiplying the curves of Figure 3 by the probabilities (24) and (25). The results are shown in Figure 6.

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We shall calculate the characteristic wavelengths of flavor oscillations to compare them with the size of the regions where the effects take place.

In the case of flavor oscillations, the amplitude of the electron flavor survival ${A}_{{\nu }_{e}\to {\nu }_{e}}$ is given by

$$\begin{eqnarray}&&{Re}({A}_{{\nu }_{e}\to {\nu }_{e}})=| {U}_{\nu 1}{| }^{2}+\displaystyle \sum _{j=2,3}| {U}_{\nu j}{| }^{2}\cos ({{\rm{\Delta }}}_{1j})\end{eqnarray} \tag{ 26 }$$

and

$$\begin{eqnarray}&&{Im}({A}_{{\nu }_{e}\to {\nu }_{e}})=\displaystyle \sum _{j=2,3}| {U}_{\nu j}{| }^{2}\sin ({{\rm{\Delta }}}_{1j}),\end{eqnarray} \tag{ 27 }$$

for the real and imaginary parts of the amplitude, respectively, with

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{1j}=\displaystyle \frac{({m}_{j}^{2}-{m}_{1}^{2}){c}^{4}t}{2E{\hslash }}=\displaystyle \frac{{\delta }_{1j}^{2}\,t}{2E{\hslash }}.\end{eqnarray} \tag{ 28 }$$

To make a rough estimation of the period of oscillations, we take the squared mass difference ${\delta }_{12}^{2}$ in the normal hierarchy and write for the period T_osc (Kersten & Smirnov 2016)

$$\begin{eqnarray}&&{T}_{\mathrm{osc}}=\displaystyle \frac{8\pi E{\hslash }c}{{\delta }_{12}^{2}\,{c}^{4}}.\end{eqnarray} \tag{ 29 }$$

The corresponding wavelength for flavor oscillations will then be

$$\begin{eqnarray}&&{\lambda }_{\mathrm{osc}}={{cT}}_{\mathrm{osc}}.\end{eqnarray} \tag{ 30 }$$

For the neutrino mean energy obtained in our calculations, $\langle E\rangle =3.98\,\mathrm{MeV}$, we have λ_osc = 1.98 × 10⁸ cm. This is much larger than the distance from the event horizon to the external part of the crust (≈10⁵ cm), therefore confirming the pure-state nature of the density matrix (Equation (16)) for neutrinos leaving the source.

7.4. About the Observability of the Emitted Neutrinos

In this work, we have investigated the processes leading to the emission of neutrinos in short-GRB progenitors. Following the discussions advanced in the literature (Bianco & Ruffini 2008) we have modeled the system so that the e⁻e⁺-plasma is the main source of neutrinos. These neutrinos travel through the region between the BH event horizon and the crust, their density matrix being that of pure states described by neutrino flavor-oscillations. Because of the astronomical scale of the distance between the source and the detector on Earth, decoherence effects due to interaction with the cosmic background may become important. We have calculated these effects by adopting a Gaussian model to incorporate the cosmic background.

The present calculations give a mean neutrino energy that falls in the range of SN neutrinos. The value of the predicted neutrino flux is still far away from observation, but considering the continuous advances in detector technology, it could be reachable by future generations of experiments.

Further work is in progress concerning the time delay between neutrino and photon emission in GRBs.

The authors would like to thank Dr. A. Marinelli for useful discussions. This work has been partially supported by the National Research Council of Argentina (CONICET) by the grant PIP 616, and by the Agencia Nacional de Promoción Científica y Tecnológica (ANPCYT) PICT 140492. A.V.P and O.C. are members of the Scientific Research career of the CONICET.