INTRODUCTION

The interaction of neutrinos with matter is an important problem in various astrophysical phenomena, for example, supernovae, neutron stars mergers, neutron star crusts formation, and so on. In particular, the pressures caused by the neutrino flux and/or magnetic field are considered as an additional key contribution to the formation of a supernova shock wave and a possible mechanism for energy transfer to the entire initially bound matter of the progenitor star.

In the explosion scenario due to neutrino heating, the development of a staled shock wave can be resumed by electron neutrinos and antineutrinos emitted by a cooling proto-neutron star [1, 2]. In addition, multidimensional effects such as convection and plasma instability can contribute to the explosion, as follows from modern simulations of a supernova explosion [3–6]. Indeed, forced convection, which causes magnetorotation instability (MRI) and/or dynamo processes, can lead to a huge increase in magnetic induction with an extremely high field strength of up to tens of teratesla (TT). The corresponding magnetic pressure pumps energy into the stellar material and can be considered as the predominant mechanism of shock wave formation for the scenario of a fast-flowing explosion. Considering that neutrino and/or magnetic pressure makes a significant contribution to the mechanism of supernova explosion, it is necessary to analyze the transport of neutrinos in supernova matter taking into account magnetic effects. Haxton points out for the first time [7] that neutrino reactions on nuclei caused by neutral and charged currents can play an important role in supernova explosions. The presence of a magnetic field leads to a noticeable energy exchange during neutrino scattering on nucleons [8, 9].

The purpose of this work is to analyze additional channels of neutrino-nuclear reactions occurring in the magnetized medium of SNe. It is shown that such channels change the neutrino energy. In particular, we consider the scattering of neutrinos by nucleons in a magnetized hot matter near the neutrino sphere and the corresponding effect in the neutrino energy spectra.

1 NEUTRINO DYNAMICS IN MAGNETIZED SUPERNOVAE

To describe neutrino dynamics we use quite general kinetic equation for phase-space distribution function f(r, p, l)

$$\frac{{{\text{df}}}}{{{\text{d}}l}} = \frac{{\partial {\text{f}}}}{{\partial l}} + z\frac{{\partial {\text{f}}}}{{\partial {\mathbf{r}}}} + \frac{{\partial {\mathbf{p}}}}{{\partial l}}\frac{{\partial {\text{f}}}}{{\partial {\mathbf{p}}}} = {{\Lambda }} + {\text{St}}\left( {\text{f}} \right),~$$
(1)

where a distance passed by neutrino l = ct with speed of light c and time t, ∂r and ∂p represent the partial derivatives with respect to the spatial, r, and momentum coordinates, p = zE/c, with the unit vector z defining the direction of neutrino momentum at energy E. We accounted here that since supernova neutrinos possess typical energies in the MeV range, much larger than the experimental rest-mass limit for active flavors <1 eV, they propagate essentially with the speed of light c. The momentum derivative in Eq. (1), ∂p/∂l accounts for an energy and momentum exchange in the neutrino collisions with environment particles. On the r.h.s. of Eq. (1) Λ stands for all the rates of neutrino production, absorption and annihilation while the term St(f) accounts for fluctuations in scattering processes.

1.1 The Weak Coupling Mode of Neutrinos with Matter

In this article, we focus on the kinetics of neutrinos at the weak coupling of neutrinos with matter for dynamoactive supernovae. Electron-aromatic neutrinos and antineutrinos in the supernova center interact with stellar matter through absorption and emission reactions due to charged current, which significantly contribute to their opacity and lead to an intensive exchange of energy during interaction. The energy spectra of neutrinos flying out of the protoneutron star into the weak coupling region can be parameterized by the following equation

$$\begin{gathered} W\left( {{{E}_{\nu }},{\mathbf{T}}} \right) \\ = E_{\nu }^{2}\int {{\text{d}}\Omega {\text{ f}}\left( {{\mathbf{r}},{\mathbf{p}},l} \right)} \sim ~E_{\nu }^{a}{\text{exp}}\left\{ { - \left( {1~\,\, + ~\,\,\alpha } \right){{{{E}_{V}}} \mathord{\left/ {\vphantom {{{{E}_{V}}} {{{E}_{{{\text{av}}}}}}}} \right. \kern-0em} {{{E}_{{{\text{av}}}}}}}} \right\}. \\ \end{gathered} $$
(2)

Here, Ω denotes the solid angle of vector z, \({{E}_{{{\text{av}}}}}\) is an average energy and α a numerical parameter describing the amount of spectral pinching; the value α = 2 corresponds to a Maxwell–Boltzmann spectrum, and α = 2.3 to a Fermi–Dirac distribution with zero chemical potential. Beyond the protoneutron star surface in neutrino sphere region it is impossible to maintain both chemical equilibrium between neutrinos and stellar matter and diffusion. However, noticeable energy exchange between neutrinos and strongly magnetized stellar material can affect neutrino spectra.

Neutrinos corresponding to a heavy lepton are less energetically coupled to stellar plasma, mainly due to pair formation reactions such as nucleon bremsstrahlung, electron positron, and neutrino-antineutrino annihilation. However, the total opacity is mainly determined by the scattering of neutrinos on nucleons. Thus, heavy-leptonic neutrinos are decoupled from thermal equilibrium in the energy sphere, which is located much deeper inside the nascent protoneutron star than the transport sphere located next to the neutrino sphere, where the transition from diffusion to free flow is disrupted. The corresponding scattering atmosphere heavy-lepton neutrinos still often collide with neutrons and protons. As shown in Sections 1.2 and 1.3, magnetic effects in this case noticeably enhance the energy exchange in neutrino-nucleon scattering due to neutral current.

Matter in the neutrino sphere region corresponds to a moderate density n ~ 0.1−10 Tg cm–3 (1 Tg = 1012 g) and temperature T ~ 5−10 MeV. We assume strong fluctuations of temperature T and density n in this region since it meets strong convection of matter and corresponds to a vicinity of bifurcation point for stellar material between a collapse to central compact object and supernova ejecta. Figure 1a shows the Fermi energy of nucleons \(~E_{{\text{F}}}^{{\text{N}}}\) and electrons \(E_{{\text{F}}}^{{\text{e}}}\) versus a portion of electrons Ye at density n = 1 Tg/cm3. One sees that at realistic numbers of beta-equilibrium parameter Ye ~ 0.2–0.3 these values for nucleons and electrons are small and large as compared to temperature, respectively (see Fig. 1a). Therefore, nucleon components, with \(~E_{{\text{F}}}^{{\text{N}}}\) \( \ll \) T, represent non-degenerate gas while an electron gas, with \(E_{{\text{F}}}^{{\text{e}}}\) \( \gg \) T, is strongly degenerated. As a consequence neutrino-electron scattering cross section is strongly suppressed because of Pauli principle. Such a blocking effect also leads to actual termination of charged current component in neutrino- nucleon scattering. Magnetization gives rise to effectively increasing Fermi energy and further diminution of respective scattering. So that corresponding mean free path (mfp) rises up to 10th km at considered densities. Therefore, we neglect hereafter by the r.h.s. of Eq. (1). On the contrary neutrino-nucleon scattering due to neutral current component can be considered as an independed process with corresponding mfp lf = (NNσGT0)–1 ~ 100 m. Here Ni = ni/mi represents the number density of ith nuclear particle (N denotes nucleon) with mass mi and contribution ni to the total mass density n, σGT0 denotes respective cross section σGT0 \( \approx \) 10−40 cm2 (Eν/37 MeV)2 (see [3]).

Fig. 1.
figure 1

Neutrino sphere properties: (a) The Fermi Energy at density n = 1 Tg cm–3 versus beta-equilibrium parameter Ye, (b) The neutrino energy dependence of an average transferred energy \(\left\langle {\delta E} \right\rangle \) in inelastic scattering at temperatures T = 5 and 10 MeV corresponding to the curves 1 and 2, see Eq. (3).

Full size image

1.2 Energy Exchange in Neutrino-Nuclear Reactions

The change in neutrino energy during scattering on magnetized nucleons ν + N → ν' + N' was considered by Kondratyev et al. in [8, 9]. Recall that the interaction of the field H with spin magnetic moments splits the spin-up and spin-down energy levels of nucleons (or with moments directed along and opposite to the direction of the magnetic field) by the value Δ = |gα|μNH ≡ |gα| ωL. Here µN represents the nuclear magneton, ωL = μNH is the Larmour frequency, and gα denotes the nucleon g-factor well known for protons gp ≈ 5.586 and neutrons gn ≈ –3.826. Consequently, when scattering due to the components of the neutral current of the Gamow–Teller interaction (GT0) on nucleons occupying the spin levels up and down, the neutrino undergoes endo- and exo-energetic transitions, respectively [8, 9]. These mechanisms are originated by the transition operator GT0 (GT0 = σt0) with the transfer of spin and parity Jπ = 1+, which induces a spin flip during the effective scattering process. In one effective collision a neutrino loses or gains energy Δ.

In a case of multiple scattering the energy exchange can be quantified using the energy transfer cross section. This value is defined as \(S_{1}^{i}\) = −\(\int {d\epsilon \epsilon } \)(\(d{{\sigma }}_{{\nu \to \nu {\kern 1pt} '}}^{i}\)/d\(~\epsilon \)) with the energy transfer \(\epsilon \) and the differential cross section for neutrino scattering on the ith nucleus. At a temperature T the effective GT0 neutrino scattering in a magnetized nucleon gas the energy transfer cross section has the form [8, 9]

$${{S}_{1}} \approx {{\sigma }_{{{\text{GT0}}}}}\Delta (2{{\delta }_{E}} - {{\left. {(1 + \delta _{E}^{2}){\text{ th}}({{{{\delta }_{T}}} \mathord{\left/ {\vphantom {{{{\delta }_{T}}} 2}} \right. \kern-0em} 2}))} \right|}_{{\Delta < E,T}}},$$
(3)
$$ \approx {{\sigma }_{{{\text{GT0}}}}}\Delta (2{{\delta }_{E}} - {{{{\delta }_{T}}} \mathord{\left/ {\vphantom {{{{\delta }_{T}}} 2}} \right. \kern-0em} 2}).$$
(4)

Here δT = Δ/T, δE = Δ/Eν and th(x) is the hyperbolic tangent. The dependence of this value on the energy of the incoming neutrino Eν is determined by the temperature T, and the product of the splitting Δ and the scattering cross section σGT0 in the nucleon gas determines the corresponding intensity of energy exchange.

The average value of the energy exchange, i.e. the energy transfer cross section S1 related to the scattering cross section σGT0, is shown in Fig. 1b. It is seen that this value changes from a positive value (i.e., exoenergetic neutrino scattering leading to acceleration) for a hot nucleon gas to a negative value (i.e., an endoenergetic collision leading to neutrino deceleration) for a cold system. Such a transition from the stopping mode to the acceleration mode occurs under the conditions Eν ≈ 4T. The physical reason for such a transition is obviously a decrease in the thermal population of the upper split energy level of the nucleon, which leads to the suppression of the contribution of GT0 transitions from this level to the underlying level. The condition of this change from one mode to another is well described by the relation given above and does not depend on the magnitude of the splitting and, consequently, on the geometry of magnetic induction [6].

1.3 The Effect of Energy Exchange in the Energy Spectra of Neutrinos

Modification of the neutrino energy spectra during the quasi-free mode of evolution is illustrated in Fig. 2 for realistic parameters of the SNe. As can be seen, a noticeable part of the accelerated neutrinos arises due to such dynamics, weakly coupled to the magnetized matter of the star in the vicinity of the neutrino sphere.

Fig. 2.
figure 2

Neutrino energy spectra in arbitrary units for α = 2, Eav = 10 MeV and el = 1 (i.e. l = 0)—curves 1, 2—curves 2, and 3—curves 3 at T = 5 MeV (a) and 10 MeV (b). Curves 4 corresponds to respective single effective collision event at ∆ = 2 MeV.

Full size image

1.3.1. A single effective collision event. In a single collision, the relationship between the corresponding exo- and endoenergetic modes is determined by the ratio of the filling of the respective nucleon levels and the phase volume of neutrinos in the output channel, i.e. exp{δT} (1−δE)2θ(1 – δE)/(1 + δE)2), with a step function θ(x). When this ratio is less than 1, the number of endoenergetic collisions is greater than enexoenergetic collisions, and vice versa. Therefore, for neutrino dynamics in a magnetized nucleon gas, the change of acceleration and stopping modes preferably corresponds to the condition δT = −2 ln{(θ(1 – δE)(1 – δE)/(1 + δE)}|δE,δT < 1 ≈ 4δE. The same ratio of the initial neutrino energy Eν and the temperature of the nucleon gas T for switching dynamic modes is obtained also in the case of multiple effective collisions considered in Eqs. (3) and (4) and discussion therein.

1.3.2. Multiple effective collision events. Using Eq. (4), we define the energy transfer coefficient as

$$\frac{{\partial {{E}_{\nu }}}}{{\partial {\text{l}}}} = \mathop \sum \limits_i {{N}_{i}}{{S}_{i}} \approx {{E}_{\nu }}{{\left( {1 - \frac{{{{E}_{v}}}}{{4T}}} \right)} \mathord{\left/ {\vphantom {{\left( {1 - \frac{{{{E}_{v}}}}{{4T}}} \right)} {{{l}_{{\text{t}}}}}}} \right. \kern-0em} {{{l}_{{\text{t}}}}}},$$
(5)

where the average energy transfer length is \(l_{{\text{t}}}^{{ - 1}}\) = \(2\sum\nolimits_i {\sigma _{{{\text{G}}{{{\text{T}}}_{{\text{0}}}}}}^{i}{{N}_{i}}\delta _{{{{E}_{i}}}}^{2}} \). As is justified in Section 1.1, the magnetized gas of nucleons makes a predominant contribution to the neutrino-matter energy exchange in the neutrino sphere region. Then, given Eq. (3) the length of the energy transfer is estimated as lt ≈ 100 m (3 MeV/Δav)2 (10 Tg cm−3)/n, with the average splitting value \(\Delta _{{{\text{av}}}}^{2} = \sum\nolimits_i {{{{{N}_{i}}{{\Delta }}_{i}^{2}} \mathord{\left/ {\vphantom {{{{N}_{i}}{{\Delta }}_{i}^{2}} N}} \right. \kern-0em} N}} \). Neglecting the right hand side of Eq. (1) for a uniform flow z∂f/∂r = 0, the solution of Eq. (1) is given by replacing Eν by the solution of Eq. (5), i.e. Eν → elEν(el + (1 – el) Eν/4T)–1 with el = exp(l/lt).

Figure 2 shows the effect of energy exchange in the neutrino energy spectra during evolution in the vicinity of the neutrino sphere. The Maxwell–Boltzmann distribution corresponding to α = 2 and Eav = 10 MeV in Eq. (2) is taken as the initial one. It is seen that the effect of energy transfer in a magnetized nucleon gas leads to an increase in the neutrino energy at the maximum of the distribution. When the neutrino path l approaches the average energy transfer length lt, we get a spread in the distribution W(E) with an increase in energy at the maximum point almost linearly with an increase in el. This acceleration is especially effective at higher gas temperatures.

1.4 Fluctuation Effects in Energy Spectra

Significant fluctuations near the neutrino sphere and the bifurcation point cause large fluctuations in the properties of the corresponding stellar material. Let’s average the results of changes in energy spectra over fluctuations. For temperature T we assume a uniform distribution in the range from 5 to 10 MeV regardless of density fluctuations. As can be seen in Fig. 3a, the maximum of the distribution W(E) is shifted towards high energies, approaching the region of 10–20 MeV. The properties of such an averaged energy distribution resemble the results for the temperature T = 10 MeV ensuring thereby an efficiency of the acceleration mechanism at higher temperatures.

Fig. 3.
figure 3

(a) Neutrino energy spectra in arbitrary units for α = 2, Eav = 10 MeV and el = 1 (i.e. l = 0)—curve 1, 2—curve 2, and 3—curve 3 average over T from 5 to 10 MeV. (b) The average neutrino energy depending on the path length l for α = 2, Eav = 10 MeV, T = 5 MeV—curve 1 and 10 MeV—curve 2. Curve 3 corresponds to averaging over temperature T in the range from 5 to 10 MeV.

Full size image

2 ENHANCEMENT OF SPECTRUM HARDNESS AND SENSITIVITY OF LARGE NEUTRINO DETECTORS

Sharply variable transient particle fluxes can be detected using large-volume neutrino telescopes KM3NeT [10], Baikal [11], etc. Sensitivity to neutrinos on a scale of 10 MeV can be achieved by observing a collective increase in the rate of counting coincidences using multiple detectors. At the same time, increasing hardness of the spectrum increases the efficiency of registration [12]. The spectrum hardness can be characterized quantitatively by the average energy of the particles \(\left\langle E \right\rangle \)

$$\left\langle E \right\rangle = \int {dEW\left( E \right)E} \approx {{e}_{{\text{l}}}}{{E}_{{{\text{av}}}}}{{(1 + ({{e}_{{\text{l}}}}-1){{{{E}_{{{\text{av}}}}}} \mathord{\left/ {\vphantom {{{{E}_{{{\text{av}}}}}} {4T}}} \right. \kern-0em} {4T}})}^{{ - 1}}}.$$
(6)

In Eq. (6) we used the Saddle Point method justified at el ~ 1. As can be seen in Fig. 3b, the value \(\left\langle E \right\rangle \) grows almost linearly with increasing path length l.

CONCLUSIONS

We have considered the energy transfer during neutrino scattering on nucleons in strong magnetic fields, presumably arising in supernovae, and corresponding effects in the neutrino energy spectra. It is shown that nuclear magnetization leads to the appearance of new reaction channels induced by neutral current, which result in additional noticeable mechanisms in the dynamics of neutrinos weakly coupled to matter. It is found that the transfer coefficient in the energy space in kinetic equations changes from positive to negative values with increasing neutrino energy. For a magnetized non-degenerate nucleon gas such a switching of acceleration and deceleration modes occurs when the neutrino energy exceeds about four times the gas temperature, while the Larmour frequency of nucleons is quite small. Such a change in dynamic properties occurs due to the principle of detailed balance and the difference in the volume of the phase space for neutrinos in the final channel at scattering on nucleons with spin up and spin down and does not depend on the magnitude of level splitting in magnetic fields. Consequently, such a property is insensitive to the magnetization geometry. The corresponding acceleration and/or deceleration rates are determined by the product of splitting ∆ and the scattering cross section σGT0 in the nucleon gas. At realistic properties of stellar material such effects of nuclear neutrino collisions lead to an increase in the hardness of the neutrino energy spectra. Since electron neutrinos decouple from matter in the neutrino sphere and then experience several (on average, a single) effective collisions, the corresponding acceleration effect is relatively small. Outside the energy sphere the dynamics of heavy-leptonic neutrinos is mainly determined by collisions with nucleons. In the scattering atmosphere (up to the neutrino sphere) these collisions are frequent enough to maintain the spatial diffusion of heavy-leptonic neutrinos. Accordingly, a significant traveled path l in the magnetized region of the star leads to a significant acceleration effect in the case of a heavy-flavor component. Finally, we notice that such strong magnetization also occurs at neutron star mergers.