# Neutrinos from pre-supernova in the framework of TQRPA method

A. A. Dzhioev1, A. V. Yudin2,3, N. V. Dunina-Barkovskaya2, A. I. Vdovin1
1 Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear
Research, 141980 Dubna, Russia
2 National Research Center Kurchatov Institute, pl. Kurchatova 1, Moscow,
123182, Russia
3 Novosibirsk State University, Pirogova, 2, 630090 Novosibirsk, Russia
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We propose a new method for calculating spectra and luminosities for
(anti)neutrinos produced in the pre-supernova environment by weak processes
with hot nuclei. It is based on the thermal quasiparticle random phase
approximation (TQRPA), that allows microscopic thermodynamically consistent
calculations of the weak-interaction response of nuclei at finite
temperatures. For realistic representative pre-supernova conditions from the
stellar evolution code MESA, we compute (anti)neutrino luminosities and
spectra arising from neutral- and charged-current weak reactions with hot 56Fe
and compare them with the contribution of thermal processes. We find that the
TQRPA approach produces not only a higher total luminosity of electron
neutrinos (mainly born in the electron capture reaction), compared to the
standard technique based on the large-scale shell model (LSSM) weak-
interaction rates, but also a harder neutrino spectrum. Besides, applying the
TQRPA and LSSM, we find that in the context of electron antineutrino
generation, the neutral-current nuclear de-excitation (ND) process via
neutrino-antineutrino pair emission is at least as important as the electron-
positron pair annihilation process. We also show that flavor oscillations
enhance the high-energy contribution of the ND process to the electron
antineutrino flux. This could potentially be important for pre-supernova
antineutrino registration by the Earth’s detectors.

###### keywords:

neutrinos – (stars:) supernovae: general – instrumentation: detectors.

††pubyear: 2023††pagerange: Neutrinos from pre-supernova in the framework of
TQRPA method–Neutrinos from pre-supernova in the framework of TQRPA method

##  1 Introduction

Supernova explosions are magnificent and not yet fully understood phenomena of
nature. For dozens of days the dying star shines like an entire host galaxy.
But despite this, in massive stars that end their lives as a core-collapse
supernova, the main amount of energy during the explosion (of an order of a
few $10^{53}$ erg) is lost by neutrinos. The first and so far the only
detection of neutrinos from a supernova occurred on February 23, 1987; it was
a famous supernova SN1987A that flared up in the Large Magellanic Cloud (see,
e.g., Hirata et al. (1987); Bionta et al. (1987); Alekseev et al. (1987);
Ryazhskaya (2006)).

But even before the explosion, neutrino plays an important role in the life of
a massive star: already at the stage of the neon-oxygen core (a few years
before the collapse), the neutrino luminosity becomes higher than the photon
luminosity (Weaver et al., 1978), and just before the collapse the energy loss
via neutrino emission is many orders of magnitude superior to other processes
(Kato et al., 2017). Such a star, called a pre-supernova, appears to be a
promising source of neutrinos to be detected (Odrzywolek et al., 2004b; Patton
et al., 2017b). Pre-supernova (anti)neutrinos, if registered, can serve as an
alarm signal of an upcoming explosion (Kutschera et al., 2009). Moreover,
since neutrinos can freely propagate through the stellar matter, they carry
direct information about thermodynamic conditions in the core and their
observation would offer a possibility for studying the physical processes that
lead to the core collapse.

Neutrino emission in a stellar core occurs due to a number of thermal
processes and nuclear weak reactions. Thermal neutrinos are mostly emitted via
the electron-positron pair annihilation and plasmon decay, and their
production is entirely determined by the core temperature, density and
electron fraction (Itoh et al., 1996; Kato et al., 2015). In contrast,
neutrino emission via nuclear weak reactions, such as $\beta^{\pm}$-decay and
$e^{\pm}$-capture, has a stronger dependence on the isotopic composition of
the core and its study therefore requires realistic stellar evolution
simulations with a large and accurate nuclear reaction network (Odrzywolek &
Heger, 2010). However, not only isotopic composition but also thermodynamic
conditions in the core affect the neutrino production in nuclear weak
reactions. The reason is that nuclear excited states are thermally populated
in the high-temperature stellar environment and weak reactions on thermally
excited states allow the emission of neutrinos with rather high energy. The
relative contribution of these high-energy neutrinos to the total flux depends
on temperature. It is important to realize that despite the Boltzmann
suppression, the gain in phase space combined with the large matrix element
and increased density of thermally excited states can make this contribution
significant. Eventually, the balance between all these factors depends on
thermodynamic conditions and properties of individual nuclei.

In Patton et al. (2017a); Patton et al. (2017b), the contribution of charged-
current nuclear weak processes to neutrino emissivities was analyzed for
different realistic thermodynamic conditions and isotopic compositions of the
star obtained from the state-of-the-art stellar evolution code MESA. It was
found that, under certain conditions, nuclear processes compete with thermal
processes in their contribution to the (anti)neutrino flux or even dominate in
the energy window relevant for detection, which is mostly due to the decay and
electron capture on $pf$-shell nuclei with $A=50-60$ (iron-group nuclei).
However, it was pointed out that while total emissivities are relatively
robust, the high-energy tails of the neutrino spectra, in the detectable
window, are very sensitive to the details of the calculations. Specifically,
the source of the error lies in the so called single-strength (or effective
$q$-value) approximation (Langanke et al., 2001), which was adopted in Patton
et al. (2017a); Patton et al. (2017b) to generate charged-current neutrino
spectra. An exploratory study of this error was performed in Misch & Fuller
(2016) and it was shown that the specific neutrino energy spectrum obtained
from the single-strength approximation could miss important features when the
most important weak transitions involve thermally excited states.

In Misch & Fuller (2016), (anti)neutrino emission via charged-current
reactions was considered using some $sd$-shell nuclei as an example. To do
this, shell-model calculations were preformed to obtain strength distributions
of Gamow-Teller and Fermi transitions which dominate nuclear weak reactions at
pre-supernova conditions. However, for $pf$-shell nuclei large-scale shell-
model (LSSM) calculations of the GT strength distributions require a huge
dimension of the model space and, therefore, they are still limited by the
nuclear ground and low-lying excited states (Caurier et al., 2005). When
performing shell-model calculations of stellar weak-interaction rates, this
issue is overcome by employing the Brink hypothesis and the method of "back-
resonances" (Langanke & Martínez-Pinedo, 2000). However, even if we could
perform precise LSSM calculations for highly excited states, the state-by-
state consideration of individual contributions would remain computationally
infeasible because of too many thermally populated states at pre-supernova
temperatures $T\approx 1$ MeV ($10^{10}\,\text{K}=0.86$ MeV).

To avoid these shortcomings and compute stellar weak-interaction rates for hot
thermally excited nuclei in a microscopic thermodynamically consistent way,
the thermal quasiparticle random-phase approximation (TQRPA) was proposed in
Dzhioev et al. (2010). The TQRPA approach is based on a statistical
description of the hot nucleus and it enables one to obtain temperature
dependent strength functions for nuclear transitions involved in the
considered weak reaction. The advantage of the TQRPA is that it makes it
possible to treat both endoergic and exoergic weak processes without
application of the Brink hypothesis and the "back-resonance" method and
without violation of the detailed balance principle. Later on, the combination
of the TQRPA and the Skyrme energy density functional methods was successfully
applied to study both charged-current (Dzhioev et al., 2019; Dzhioev et al.,
2020) and neutral-current (Dzhioev et al., 2016; Kondratyev et al., 2019)
stellar weak reactions with $pf$-shell and heavier neutron rich nuclei (see
also recent reviews Dzhioev & Vdovin (2022a, b, c)).

In our recent paper Dzhioev et al. (2023), the TQRPA method was applied for
the first time for computing (anti)neutrino spectrum and energy loss rates due
to nuclear weak processes under specific pre-supernova conditions which
correspond to a realistic pre-supernova model. Using the hot 56Fe as an
example, we have shown that thermodynamically consistent TQRPA calculations
predict an electron neutrino spectrum with an enhanced high-energy fraction.
These high-energy neutrinos are emitted via the exoergic electron capture,
i.e. the process when a thermally excited hot nucleus captures an electron and
then de-excites by energy transfer to the emitted neutrino. Another
interesting process considered in Dzhioev et al. (2023) is the de-excitation
of a hot nucleus via the neutrino-antineutrino pair emission. This neutral-
current process produces $\nu\bar{\nu}$-pairs of all flavors with equal
probability and its potential importance for production of high-energy pre-
supernova (anti)neutrinos was outlined in (Misch & Fuller, 2016; Patton et
al., 2017b). In Dzhioev et al. (2023), performing TQRPA calculations for both
charged- and neutral-current weak-interaction processes with 56Fe generating
electron (anti)neutrinos, we have shown that $\nu\bar{\nu}$-pair emission has
practically no effect on the $\nu_{e}$ spectrum, while this process produces
more high-energy electron antineutrinos than other nuclear weak processes
(positron capture and $\beta^{-}$-decay).

We would also like to mention that to date the only study analyzing the impact
of $\nu\bar{\nu}$-pair emission from nuclear de-excitation on core-collapse
supernova simulations has been performed by Fischer et al. (2013). In
particular, this study showed that $\nu\bar{\nu}$-pair emission has basically
no impact on the global supernova properties and the energy loss during
collapse is dominated by electron neutrinos produced by electron captures on
nuclei. However, it was found that nuclear de-excitation is the leading source
for the production of electron antineutrinos as well as ($\mu,\,\tau$)
(anti)neutrinos during the collapse phase.

In Dzhioev et al. (2023), (anti)neutrino spectra and energy loss rates due to
nuclear processes were computed at sample points inside the stellar core just
before the onset of the collapse. At this moment, for the considered pre-
supernova model, the iron isotope 56Fe dominates the isotopic composition of
the central part of the stellar core. The present work extends our previous
study, and here we apply the TQRPA method at the same pre-supernova conditions
to calculate total (anti)neutrino spectra and luminosities generated in
different nuclear weak processes and integrated over the volume of emission.
In this paper, we also compute (anti)neutrino spectra and luminosities from
thermal processes and take into account oscillation effects.

Our paper is organized as follows: In the next section, we briefly outline the
TQRPA method for computing neutrino spectra produced by hot nuclei in stellar
interior. A comprehensive description of the TQRPA and its application to
study stellar weak-interaction processes is given in the recent reviews
(Dzhioev & Vdovin, 2022a, b, c). Then in Section 3, we describe the structure
of a particular pre-supernova model which has been used to explore the role of
weak nuclear processes in (anti)neutrino production. In Section 4, taking into
account both nuclear and thermal processes, we calculate luminosities of
different (anti)neutrino flavors inside the star and corresponding spectra.
After that, in Section 5, we compare the properties of (anti)neutrino spectra
obtained in the framework of the TQRPA with those obtained using a more
conventional method based on the LSSM calculations and the effective $q$-value
approximation. Before proceeding to a discussion of the detectability of the
calculated antineutrino fluxes (see Section 7 ), in Section 6, we consider the
issue of neutrino oscillation effects. Finally, we provide a short "Summary
and Perspectives" section.

##  2 TQRPA method

In what follows, we assume that under the pre-supernova conditions we are
concerned, weak processes with $pf$-shell nuclei are dominated by Gamow-Teller
transitions and the emitted (anti)neutrinos freely leave the star. Then, the
(anti)neutrino spectra for a single hot nucleus can be expressed in terms of
the temperature-dependent GT strength functions:

  * •

electron (EC) or positron (PC) capture

$\displaystyle\phi^{\text{EC,PC}}(E_{\nu})=\frac{G^{2}_{\text{F}}V^{2}_{\text{ud}}(g^{*}_{A})^{2}}{2\pi^{3}\hbar^{7}c^{6}}\,E^{2}_{\nu}\times$
$\displaystyle\int\limits^{+\infty}_{-E_{\nu}+m_{e}c^{2}}S_{\text{GT}_{\pm}}(E,T)E_{e}p_{e}cf_{e^{\mp}}(E_{e})F(\pm
Z,E_{e})dE,$ (1)

where the upper (lower) sign corresponds to EC (PC) and $E_{e}=E_{\nu}+E$,
$p_{e}c=(E_{e}^{2}-m^{2}_{e}c^{4})^{1/2}$;

  * •

$\beta^{\mp}$-decay

$\displaystyle\phi^{\beta^{\mp}}(E_{\nu})=\frac{G^{2}_{\text{F}}V^{2}_{\text{ud}}(g^{*}_{A})^{2}}{2\pi^{3}\hbar^{7}c^{6}}\,E^{2}_{\nu}\times$
$\displaystyle\int\limits_{-\infty}^{-E_{\nu}-m_{e}c^{2}}S_{\text{GT}_{\mp}}(E,T)E_{e}p_{e}c(1-f_{e^{\mp}}(E_{e}))F(\pm
Z+1,E_{e})dE,$ (2)

where the upper (lower) sign corresponds to the $\beta^{-}$\- ($\beta^{+}$-)
decay and $E_{e}=-E_{\nu}-E$;

  * •

$\nu\bar{\nu}$-pair emission via nuclear de-excitation (ND)

$\displaystyle\phi^{\text{ND}}(E_{\nu})=\frac{G^{2}_{\text{F}}g_{A}^{2}}{2\pi^{3}\hbar^{7}c^{6}}\,E^{2}_{\nu}\int\limits_{-\infty}^{\\-E_{\nu}}S_{\text{GT}_{0}}(E,T)(E+E_{\nu})^{2}dE.$
(3)

Note that the ND process produces the same spectra for $\nu_{e}$ and
$\bar{\nu}_{e}$. Moreover, the spectrum of other (anti)neutrino flavors is
also given by (3).

In the above expressions, $G_{\text{F}}$ denotes the Fermi coupling constant,
$V_{\text{ud}}$ is the up-down element of the Cabibbo-Kobayashi-Maskava quark-
mixing matrix and $g_{A}=-1.27$ is the weak axial coupling constant. Note that
for charged-current reactions we use the effective coupling constant
$g^{*}_{A}$ that takes into account the experimentally observed quenching of
the GT± strength. The function $f_{e^{-}(e^{+})}(E)$ is the Fermi-Dirac
distribution for electrons (positrons) which depends on the chemical potential
$\mu_{e^{-}}$ ($\mu_{e^{+}}=-\mu_{e^{-}}$) and temperature $T$, and the Fermi
function $F(Z,E)$ takes into account the distortion of the charged lepton wave
function by the Coulomb field of the nucleus.

The presented expressions for $\phi^{i}(E_{\nu})$
($i=\text{EC,\,PC,\,ND},\,\beta^{\pm}$) are exact in the sense that no
additional assumptions or approximations are made in their derivation.
Approximations should be made for computing the temperature dependent GT
strength functions defined as

$S_{\mathrm{GT}_{\pm,0}}(E,T)=\sum_{i,f}p_{i}(T)B^{(\pm,0)}_{if}\delta(E-E_{if}).$
(4)

Here $p_{i}(T)=e^{-E_{i}/kT}/Z(T)$ is the Boltzmann population factor for a
state $i$ in the parent nucleus, $B^{\pm,0}_{if}=|\langle
f\|\mathrm{GT}_{\pm,0}\|i\rangle|^{2}/(2J_{i}+1)$ is the reduced transition
probability (transition strength) from the state $i$ to the state $f$ in the
daughter nucleus; $\mathrm{GT}_{0}=\vec{\sigma}t_{0}$ for neutral-current
reactions and $\mathrm{GT}_{\mp}=\vec{\sigma}t_{\pm}$ for charged-current
reactions111The zero component of the isospin operator is denoted by $t_{0}$,
while $t_{-}$ and $t_{+}$ are the isospin-lowering
($t_{-}|n\rangle=|p\rangle$) and isospin-rising ($t_{+}|p\rangle=|n\rangle$)
operators.; $E_{if}$ is the transition energy between the initial and final
states, and it can be both positive and negative. The pre-supernova model
considered in this study has typical temperatures inside the star around
$T_{9}\approx 1-10$ (see the discussion below). Obviously, at such high
temperatures an explicit state-by-state evaluation of the sums in Eq. (4) is
impossible with current nuclear models. We compute the temperature-dependent
strength function (4) applying the TQRPA approach which is a technique based
on the quasiparticle random phase approximation extended to the finite
temperature by the superoperator formalism in the Liouville space (Dzhioev &
Vdovin, 2022a).

The central concept of the TQRPA is the thermal vacuum $|0(T)\rangle$, a pure
state in the Liouville space, which corresponds to the grand canonical density
matrix operator for a hot nucleus. The time-translation operator in the
Liouville space is the so-called thermal Hamiltonian $\mathcal{H}$ constructed
from the nuclear Hamiltonian after introducing the particle creation and
annihilation superoperators. Within the TQRPA, the strength function (4) is
expressed in terms of the transition matrix elements from the thermal vacuum
to the eigenstates (thermal phonons) of the thermal Hamiltonian
$\mathcal{H}|Q_{i}\rangle=\omega_{i}|Q_{i}\rangle$:

$S_{\mathrm{GT}_{\pm,0}}(E,T)=\sum_{i}\mathcal{B}^{(\pm,0)}_{i}\delta(E-\omega_{i}\mp\Delta_{np}).$
(5)

Here $\mathcal{B}^{(\pm,0)}_{i}=|\langle
Q_{i}\|\mathrm{GT}_{\pm,0}\|0(T)\rangle|^{2}$ is the transition strength to
the $i$th thermal phonon state of a hot nucleus and
$E_{i}^{(\pm,0)}=\omega_{i}\pm\Delta_{np}$ is the respective transition
energy; $\Delta_{np}=0$ for charge-neutral transitions, while for charge-
exchange transitions $\Delta_{np}=\delta\lambda_{np}+\delta M_{np}$, where
$\delta\lambda_{np}=\lambda_{n}-\lambda_{p}$ is the difference between neutron
and proton chemical potentials in the nucleus, and $\delta M_{np}=1.293$ MeV
is the neutron-proton mass splitting. Note that eigenvalues of the thermal
Hamiltonian, $\omega_{i}$, can be both positive and negative. As a result, the
strength functions (5) for upward ($E>0$) and downward ($E<0$) transitions
obey the detailed balance principle:

$S_{\mathrm{GT}_{0}}(-E,T)=\mathrm{e}^{-E/kT}S_{\mathrm{GT}_{0}}(E,T)$ (6)

for charge-neutral GT transitions, and

$S_{\mathrm{GT}_{\mp}}(-E,T)=\mathrm{e}^{-(E\mp\Delta_{np})/kT}S_{\mathrm{GT}_{\pm}}(E,T)$
(7)

for charge-exchange GT transitions. This property makes the TQRPA approach
thermodynamically consistent. Due to negative-energy downward GT transitions,
the $\beta^{\pm}$-decay and $\nu\bar{\nu}$-pair emission become possible for
hot nuclei which are stable in their ground-state. Moreover, due to enhanced
phase space such negative-energy transitions may dominate the EC and PC
processes.

In Dzhioev et al. (2023), (anti)neutrino spectra due to weak processes with
hot 56Fe were computed applying self-consistent TQRPA calculations based on
the SkM* parametrization of the Skyrme effective nucleon-nucleon interaction.
Here "self-consistent" means that both the mean field for protons and neutrons
and the residual interaction are obtained from the same energy density
functional. In the present study, we also use the SkM* parametrization. To
take into account the quenching of the GT± strength in 56Fe, we use the
effective coupling constant $g^{*}_{A}=0.56g_{A}$. Then, the total GT±
strength in 56Fe at $T=0$ becomes close to the experimental one and to the
LSSM results (Caurier et al., 1999).

Before proceeding further, let us briefly recall the main results obtained in
Dzhioev et al. (2023):

  * •

Considering temperature evolution of the GT strength functions in 56Fe, we
have shown that the TQRPA does not support the Brink hypothesis and under
relevant pre-supernova conditions thermal population of nuclear excited states
makes possible (unblocks) negative- and low-energy GT transitions whose
strength increases with temperature. Negative-energy downward transitions
contribute to exoergic weak reactions with hot nuclei.

  * •

Calculated electron neutrino spectra confirm the conclusion of Langanke et al.
(2001) that the effective $q$-value approximation can be applied under stellar
conditions with the electron chemical potential high enough to allow the
excitation of the GT+ resonance by electron capture. Such conditions occur
during the collapse phase. However, our calculations clearly demonstrate that
this approximation can fail in the pre-supernova phase when negative-energy
GT+ transitions from thermally excited states noticeably contribute to
electron capture and the resulting neutrino energy spectrum is double-peaked.
On the whole, thermodynamically consistent calculations of the electron
neutrino spectra performed without assuming the Brink hypothesis indicate that
thermal effects on the GT+ strength function shift the spectrum to higher
energies, and thus make $\nu_{e}$ neutrino detection more likely.

  * •

The inclusion into consideration of the $\nu\bar{\nu}$-pair emission by hot
nuclei shows that this neutral-current process might be a dominant source of
high-energy ($E_{\bar{\nu}}\approx 5-10$ MeV) $\bar{\nu}_{e}$ and
($\mu,\,\tau$) (anti)neutrinos emitted via de-excitation of the GT0 resonance.
Moreover, since the ND process is entirely independent of the electron density
and only depends on temperature, the detection of high-energy pre-supernova
antineutrinos might be a test for thermodynamic conditions in the stellar
interior.

##  3 Pre-sn structure

Figure 1: Top panels: density $\rho$, temperature $T_{9}$ and electron
fraction $Y_{\mathrm{e}}$ profiles along the mass coordinate $m$ for the
25_79_0p005_ml pre-supernova model at the onset of the core collapse. Bottom
panel: the chemical composition at the same conditions.

Let us now consider a characteristic structure of the pre-supernova model we
use in the present work. As in our previous study Dzhioev et al. (2023), we
use the model 25_79_0p005_ml from Farmer et al. (2016), which is a realistic
pre-supernova model with a good mass resolution and the central temperature
which is high enough for our estimates. Its name means that the initial mass
of the model was $25M_{\odot}$, the nuclear reaction network was mesa-79.net,
the maximum mass of a computational cell was $0.005M_{\odot}$, and the mass
loss during the stellar evolution was taken into account (for more details see
Farmer et al. (2016)). The authors of Farmer et al. (2016) employ the stellar
evolution code MESA (Paxton et al., 2015), version 7624. In the output, MESA
produces time-evolving profiles of mass density, temperature, electron
fraction $Y_{\mathrm{e}}$ and chemical composition as a function of the mass
coordinate. The profile that we use corresponds to the onset of the core
collapse, which is defined as the time when the infall velocity exceeds 1000
km/s anywhere in the star.

The detailed structure of this profile is given in Fig. 1. Three top panels
show the density $\rho$, temperature $T_{9}\equiv T/10^{9}~{}K$, and the
electron fraction $Y_{\mathrm{e}}$ as a function of the mass coordinate $m$
(in solar masses $M_{\odot}$) in the central part of the core at $0\leq m\leq
2.5$. The bottom panel shows the dimensionless concentrations of matter
components $Y_{i}=n_{i}/n_{\mathrm{b}}$, where $n_{\mathrm{b}}$ is the
concentration of baryons; $Y_{i}$ are related to commonly used weight
fractions $X_{i}$ as $X_{i}=A_{i}Y_{i}$, where $A_{i}$ is the mass number of
the component. As can be seen, the 56Fe isotope is predominate in the hottest
part of the core up to $m\approx 1\,M_{\odot}$.

##  4 Neutrino generation inside pre-SN

In this section, we consider various processes of neutrino generation under
the pre-supernova conditions described above. First, we describe the
generation processes themselves, then we calculate the corresponding neutrino
luminosities inside the stellar core, and finally, we determine the spectra of
the neutrino flux leaving the star.

###  4.1 Neutrino processes

In what follows we will distinguish neutrino processes according to the type
of neutrinos they produce. In Table 1, in three rows we collect i) the
processes of $\nu_{\mathrm{e}}$ generation, ii) the same for
$\bar{\nu}_{\mathrm{e}}$ and iii) $\nu\bar{\nu}$-processes, where neutrinos of
all three flavors could be born. Among the latter are the ND pair emission by
hot 56Fe, electron-positron pair annihilation (PA), plasmon decay (PL), photo-
neutrino production (PH), and bremsstrahlung (BR). We note that neutrinos from
the recombination have a luminosity more than four orders of magnitude less
than that from other processes and we do not consider them.

Table 1: Different (anti)neutrino generation processes considered in the present study. The $x$ label denotes three neutrino flavors, i.e. $x=\mathrm{e},\,\mu,\,\tau$. label | process
---|---
EC | $\mbox{${}^{56}${Fe}}+e^{-}\rightarrow\mbox{${}^{56}${Mn}}+\nu_{\mathrm{e}}$
$\beta^{+}$ | $\mbox{${}^{56}${Fe}}\rightarrow\mbox{${}^{56}${Mn}}+e^{+}+\nu_{\mathrm{e}}$
PC | $\mbox{${}^{56}${Fe}}+e^{+}\rightarrow\mbox{${}^{56}${Co}}+\bar{\nu}_{\mathrm{e}}$
$\beta^{-}$ | $\mbox{${}^{56}${Fe}}\rightarrow\mbox{${}^{56}${Co}}+e^{-}+\bar{\nu}_{\mathrm{e}}$
ND | $\mbox{${}^{56}${Fe}}^{*}\rightarrow\nu_{x}+\bar{\nu}_{x}$
PA | $e^{-}+e^{+}\rightarrow\nu_{x}+\bar{\nu}_{x}$
PL | $\mathrm{plasmon}\rightarrow\nu_{x}+\bar{\nu}_{x}$
PH | $e^{-}+\gamma\rightarrow e^{-}+\nu_{x}+\bar{\nu}_{x}$
BR | $e^{-}+(A,Z)\rightarrow e^{-}+(A,Z)+\nu_{x}+\bar{\nu}_{x}$

Before proceeding with the calculations of luminosities and spectra for
(anti)neutrinos of different types, let us find the total neutrino
luminosities due to various $\nu\bar{\nu}$ processes listed in Table 1. For
this purpose we use the collection of subroutines by F. X.
Timmes222https://cococubed.com/code_pages/nuloss.shtml, who implemented the
approximations from Itoh et al. (1996). To compute the neutrino luminosity due
to the ND process, we apply the TQRPA method.

Figure 2: Total (anti)neutrino luminosities inside the central part of the
star for the pair production processes listed in Table 1.

Figure 2 shows the _total_ (i.e. $\nu$ and $\bar{\nu}$ of all flavors)
luminosities in the central part of the star as a function of the mass
coordinate $m$. As seen from the figure, only PA and ND neutrinos are
important under the considered conditions. Therefore, in the subsequent
discussion we account for only these pair production processes. It should be
noted that ND neutrinos have the highest luminosity in the very central part
of the star, but the growth of their luminosity is almost completely halted
after $m\approx 0.5\,\text{M}_{\odot}$. The reason for this is that the very
central part of the star has a sufficiently high temperature $T_{9}\approx
9.0-10.0$ to enable the emission of high-energy $\nu\bar{\nu}$-pairs via the
ND process. These high-energy pairs are produced by the de-excitation of the
thermally excited GT0 resonance (see Fig. 4 in Dzhioev et al. (2023) and its
discussion). In 56Fe, the GT0 resonance is located at about $10$ MeV and its
thermal population rapidly decreases as we move into the colder parts of the
star. Because of this, the emission of high-energy (anti)neutrinos due to the
ND process stops, which leads to stagnation in the luminosity. In contrast,
the luminosity of PA neutrinos continues to grow up to $m\approx
1.7\,\text{M}_{\odot}$ and ends up being more intense (see Misiaszek et al.
(2006), where the problem with the PA luminosity and spectrum calculations is
discussed).

###  4.2 $\nu_{\mathrm{e}}$-type neutrino

Now we are ready to calculate neutrino luminosities and spectra for specified
processes. First, let us consider the processes that produce electron-type
neutrinos, i.e. EC, $\beta^{+}$, ND and PA (see Table 1).

Figure 3: Upper panel: $\nu_{\mathrm{e}}$-type neutrino luminosities as
functions of the mass coordinate $m$. Lower panel: spectrum of
$\nu_{\mathrm{e}}$-type neutrinos, leaving the star.

In Fig. 3, the $\nu_{\mathrm{e}}$ neutrino luminosity
$L_{\nu}[\mathrm{erg}/\mathrm{sec}]$ is shown as a function of the mass
coordinate in the inner part of a star, $0\leq m\leq 2$ (upper panel). As is
evident from the figure, EC neutrinos are several orders of magnitude more
luminous than neutrinos from other processes. Moreover, the luminosity of EC
neutrinos greatly exceeds the luminosity of $\bar{\nu}_{e}$ and ($\mu,\,\tau$)
(anti)neutrinos produced in various nuclear and thermal processes. Thus, it
appears that EC on 56Fe is the dominant energy loss process for the considered
pre-supernovae conditions, and the present study confirms the conclusion made
by Fischer et al. (2013) about the role of electron capture in core-collapse
supernovae.

The lower panel of Fig. 3 shows the energy luminosity spectrum
$S_{\nu}[\mathrm{erg}/\mathrm{sec}/\mathrm{MeV}]$ (which is connected to the
total luminosity as $L_{\nu}=\int S_{\nu}dE_{\nu}$) of $\nu_{\mathrm{e}}$
neutrinos leaving the star. It is evident that EC neutrinos not only have a
much higher total luminosity than $\nu_{e}$ from other processes but also a
higher average energy (see the discussion in Section 5).

###  4.3 $\bar{\nu}_{\mathrm{e}}$-type neutrino

Regardless of the fact that $\nu_{\mathrm{e}}$-type neutrinos, primarily from
EC, are the most important energy loss source in a star,
$\bar{\nu}_{\mathrm{e}}$ neutrinos are also important. This is due to the fact
that most detectors are especially sensitive to the inverse beta-decay
reaction initiated namely by $\bar{\nu}_{\mathrm{e}}$ (see the discussion in
Section 7).

Figure 4: The same as in Fig. 3 but for $\bar{\nu}_{\mathrm{e}}$-type
neutrinos.

Figure 4 shows the same quantities (luminosity and spectrum) as in Fig. 3 but
for $\bar{\nu}_{\mathrm{e}}$. As obvious from the figure, the essential
difference compared to the previously considered case is that there is no one
dominant process that produces the $\bar{\nu}_{\mathrm{e}}$ flux. The PC and
$\beta^{-}$ contributions are small and, in the very center of the star, the
ND antineutrinos are most luminous. But as discussed above, due to the
temperature decrease as we move from the center and a strong temperature
dependence of the ND process, the growth of their luminosity almost stops
completely after $m\approx 0.5\,\text{M}_{\odot}$. On the other hand,
$L_{\nu}$ from the PA process grows up till $m\approx 1.7\,\text{M}_{\odot}$
and the resultant luminosity is several times higher than that from the ND
process. However, while the spectrum of PA antineutrinos is peaked at low
energies, $\bar{\nu}_{\mathrm{e}}$ from the ND process have a higher energy
peak in the spectrum at $E_{\nu}\approx 6.0$ MeV, and their luminosity even
exceeds the PA luminosity in the energy range $E_{\nu}\approx 6-9$ MeV. We
will discuss the implications of this in Section 7.

###  4.4 $\nu_{\mu,\tau}$\- and $\bar{\nu}_{\mu,\tau}$-type neutrino

Despite the fact that emission of heavy-lepton type (anti)neutrinos has
basically no impact on the energy loss during the pre-supernova phase, this
process might be important from the point of view of the detection, which is
possible due to flavor oscillations. So we have calculated the luminosities
and spectra for ($\mu,\,\tau$) (anti)neutrinos as well and put them in Fig. 5.

Figure 5: The same as in Fig. 3 but for $\nu_{x}$\- and $\bar{\nu}_{x}$-type
neutrinos, where $x$ stands for $\mu$ or $\tau$.

Referring to the figure, the luminosities and spectra of $\mu,\,\tau$-type
neutrinos and antineutrinos produced by the pair annihilation are very close,
while for the ND process they are exactly the same. Therefore, further we will
not distinguish $L_{\nu}$, $S_{\nu}$ between $\nu_{\mu,\,\tau}$ and
$\bar{\nu}_{\mu,\,\tau}$. We also see that the final (i.e. at
$m=2.0\,\text{M}_{\odot}$) PA and ND luminosities are almost the same, but the
spectra are quite different. As in the case of $\bar{\nu}_{e}$, the PA
spectrum of ($\mu,\,\tau$) (anti)neutrinos is peaked at low energies, while
the ND spectrum has a high-energy peak at $E_{\nu}\approx 6.0$ MeV. Recall
that nuclear de-excitation produces the same spectra for all (anti)neutrino
flavors. When Fig. 5 is compared with Fig. 4, it is apparent that production
of heavy-lepton flavor (anti)neutrinos in the PA process is lower than that of
$e$-type (anti)neutrinos. This is because pair annihilation into
$\nu_{e}\bar{\nu}_{e}$ proceeds via both neutral and charged currents, while
$\nu_{x}\bar{\nu}_{x}$ ($x=\mu,\,\tau$) pairs are produced only via neutral
current $Z^{0}$ boson exchange (Misiaszek et al., 2006). For this reason, the
ND process appears to be more important for $\mu$ and $\tau$ (anti)neutrino
production than for $\nu_{e}$ and $\bar{\nu}_{e}$.

##  5 Characteristic features of the TQRPA method

Before we proceed to discussing the issue of detecting pre-supernova neutrinos
on Earth, let us compare our method with that used in the MESA code. The MESA
code is based on the pre-calculated weak-interaction reaction tables (for EC,
PC and $\beta^{\pm}$ processes), which for 56Fe are obtained applying large-
scale shell-model calculations (Langanke & Martínez-Pinedo, 2000). In the
present study, the electron (anti)neutrino spectra and luminosities from the
LSSM weak-interaction rates are computed on the basis of extended tables
333https://theory.gsi.de/~gmartine/rates/ by applying the effective $q$-value
method (Langanke et al., 2001; Patton et al., 2017a). Namely, it is assumed
that a weak process on a state in the parent nucleus leads to a single state
in the daughter nucleus and the energy difference, $q$, between the initial
and final states is the same for all excited states in the parent nucleus
(Brink’s hypothesis). Then, for a single nucleus the neutrino spectra from
charge-exchange weak reactions take the form

$\displaystyle\phi^{\text{EC,PC}}(E_{\nu})=N_{\text{EC,PC}}\frac{E^{2}_{\nu}(E_{\nu}-q)^{2}}{1+\exp\left(\dfrac{E_{\nu}-q\mp\mu_{e}}{kT}\right)}\Theta(E_{\nu}-q-m_{e}c^{2}),$
$\displaystyle\phi^{\beta^{\pm}}(E_{\nu})=N_{\beta^{\pm}}\frac{E^{2}_{\nu}(E_{\nu}-q)^{2}}{1+\exp\left(\dfrac{E_{\nu}-q\mp\mu_{e}}{kT}\right)}\Theta(q-E_{\nu}-m_{e}c^{2}),$
(8)

where the upper (lower) sign refers to EC and $\beta^{+}$ (PC and $\beta^{-}$)
and $\mu_{e}$ is the electron chemical potential. The effective $q$-value and
normalization factors $N_{i}$ are fitting parameters and they are adjusted to
the average (anti)neutrino energy and weak reaction rates

$\displaystyle\langle
E_{\nu,\bar{\nu}}\rangle=\frac{\int^{\infty}_{0}(\phi^{\text{EC,PC}}+\phi^{\beta^{\pm}})E_{\nu}dE_{\nu}}{\int^{\infty}_{0}(\phi^{\text{EC,PC}}+\phi^{\beta^{\pm}})dE_{\nu}},$
(9)
$\displaystyle\lambda^{i}=\int^{\infty}_{0}\phi^{i}(E_{\nu})dE_{\nu}~{}~{}~{}i=\text{EC,~{}PC,}~{}\beta^{\pm}.$
(10)

The values of $\langle E_{\nu,\bar{\nu}}\rangle$ and $\lambda^{i}$ are listed
in the LSSM rate tabulations for a grid of temperature/density points and can
be easily interpolated in between (Fuller et al., 1985). Note that for 56Fe
under the considered pre-supernova conditions, the effective $q$-value method
results in a negative $q$ for the $\nu_{e}$ spectrum. This means, that
according to Eq. (5), only electron captures contribute to $\nu_{e}$
production in the charge-exchange channel. In contrast, for the
$\bar{\nu}_{e}$ spectrum we have $q>m_{e}c^{2}$, i.e. both PC and
$\beta^{-}$-decay contribute to $\bar{\nu}_{e}$ emission.

To compute ND (anti)neutrino spectra within the shell-model method, we use the
ground-state GT0 strength distribution $S_{\text{GT}_{0}}(E)$ for 56Fe derived
from the LSSM calculations (see Fig. 1 in Sampaio et al. (2002)). Then, we
apply the same method as in Fischer et al. (2013) to obtain the GT0 strength
function for pair emission. Namely, adopting Brink’s hypothesis and exploiting
the detailed balance relation (6), we get

$\displaystyle\phi^{\text{ND}}(E_{\nu})=\frac{G^{2}_{\text{F}}g_{A}^{2}}{2\pi^{3}\hbar^{7}c^{6}}\,E^{2}_{\nu}\int\limits^{\infty}_{E_{\nu}}S_{\text{GT}_{0}}(E)\exp\left(-\frac{E}{kT}\right)(E-E_{\nu})^{2}dE.$
(11)

Combining this expression with the effective $q$-value method for charge-
changing processes, we obtain the LSSM (anti)neutrino spectra from nuclear
processes.

Figure 6: Comparison of the TQRPA and LSSM (anti)neutrino spectra from nuclear
processes. In both methods, the $\nu_{e}$ spectra shown in the upper panel are
determined almost entirely by the electron capture (see Fig. 3). In the middle
panel, labels "TQRPA" and "LSSM" represent the $\bar{\nu}_{e}$ spectra that
include contributions from PC, $\beta^{-}$-decay and ND, while spectra
obtained without the ND contribution are marked as "TQRPA*" and "LSSM*". The
$\nu_{x}$ and $\bar{\nu}_{x}$ ($x=\mu,\,\tau$) spectra from the ND process are
the same in the lower panel. In the middle and lower panels we also show the
spectra from the pair annihilation thermal process.

In Fig. 6, we compare the TQRPA and LSSM (anti)neutrino energy luminosity
spectra from nuclear processes integrated over the emission region $m\leq
2\,\text{M}_{\odot}$. Focusing on the upper panel of the figure, we notice
that both the TQRPA and LSSM calculations give a low-energy peak in the
$\nu_{e}$ spectrum at $E_{\nu}\approx 4.0$ MeV. In both methods, EC dominates
$\nu_{e}$ production and the observed low-energy peak is mainly caused by the
GT+ resonance excitation. The TQRPA approach also produces a second higher
energy peak, which makes the spectrum harder. This high-energy peak correspond
to $\nu_{e}$ emitted after downward negative-energy transitions from thermally
excited states. We would like to emphasise that the absence of the second peak
in the LSSM spectrum is the result of the effective $q$-value approximation.
In the LSSM calculations of stellar weak-interaction rates, downward
transitions are taken into account using the back-resonance method (Langanke &
Martínez-Pinedo, 2000), and they produce a high-energy peak in the neutrino
spectrum (see, e.g., Fig. 2 in Langanke et al. (2001)).

The role of the ND process in the $\bar{\nu}_{e}$ spectrum is depicted in the
middle panel of Fig. 6, where the TQRPA and LSSM spectra with and without the
ND contribution are compared with the PA spectrum. As illustrated, the TQRPA
and LSSM spectra without the ND contribution have a double-bump structure,
reflecting the contribution of PC and $\beta^{-}$-decay. Both methods predict
that in the absence of the ND process, emission of pre-supernova
$\bar{\nu}_{e}$ is entirely dominated by pair annihilation. Taking nuclear de-
excitation into account increases the role of nuclear processes in
$\bar{\nu}_{e}$ emission. In both methods, the high-energy ND contribution
around $E_{\nu}\approx 7-9$ MeV is comparable or even exceeds the PA
contribution. For heavy-lepton flavor (anti)neutrino emission the role of the
ND process is even more important (see the lower panel of Fig. 6).

Comparing the ND spectra from the TQRPA and LSSM calculations, we notice that
the LSSM produces a single-peak spectrum, which is different from the double-
peak spectrum produced by the TQRPA. In both methods, the high-energy peak
corresponds to (ant)neutrinos emitted via de-excitation of the GT0 resonance.
The TQRPA calculations of the GT0 strength distribution in 56Fe predict a
somewhat higher centroid energy of the GT0 resonance as compared to the shell-
model strength distribution (see Fig. 1 in Dzhioev et al. (2016)). As a
result, the high-energy peak in the TQRPA spectrum is slightly shifted to
higher energies. The appearance of a low-energy peak in the TQRPA spectrum is
related to thermally unblocked low-energy GT0 transitions which, owing to the
detailed balance principle, significantly increase the strength of inverse
negative-energy transitions (Dzhioev et al., 2016). Due to the application of
Brink’s hypothesis, no such transitions appear in the GT0 strength
distribution obtained from the ground-state LSSM calculations. For this
reason, within the TRQPA, the average energy of emitted ND (anti)neutrinos
($\langle\mathcal{E}^{\nu\bar{\nu}}\rangle_{\text{TQRPA}}=2.97$ MeV) is lower
than that obtained from the LSSM calculations
($\langle\mathcal{E}^{\nu\bar{\nu}}\rangle_{\text{LSSM}}=4.07$ MeV). However,
the luminosities of ND (anti)neutrinos in both methods are nearly the same
($L^{\nu\bar{\nu}}_{\text{TQRPA}}=4.9$, $L^{\nu\bar{\nu}}_{\text{LSSM}}=4.6$).
It is also interesting to note that
$\langle\mathcal{E}^{\nu\bar{\nu}}\rangle_{\text{TQRPA}}$ is close to the
values of $\langle\mathcal{E}^{\nu\bar{\nu}}\rangle$ obtained in Fischer et
al. (2013) (see Table II) at $T=0.7\,\text{and}\,1.0$ MeV.

Figure 7: Total (i.e. nuclear plus thermal) (anti)neutrino spectra obtained
from the TQRPA and LSSM calculations. For $\nu_{e}$ neutrinos only electron
captures are important and contributions from other nuclear and thermal
processes are negligible. Note also that the $\bar{\nu}_{e}$ spectra without
the ND contribution can be considered with good accuracy as the PA ones (see
the middle panel in Fig. 6). The neutrino spectra should be multiplied by the
number shown in parenthesis.

The similarities and differences between the expected total (i.e. nuclear plus
thermal) TQRPA and LSSM spectra as well as the role of the ND process in the
pre-supernova (anti)neutrino emission are summarized in Fig. 7. The comparison
between the TQRPA and LSSM spectra for $\nu_{e}$ and the discussion of the
reasons for their differences has been made above. Here we just mention that
within the TQRPA the energy luminosity $L_{\nu_{e}}=\int
S_{\nu_{e}}(E_{\nu})dE_{\nu}$ is about twice larger than that predicted by the
LSSM. The reasons for this are discussed below.

As for the total $\bar{\nu}_{e}$ spectra, we remind that in both methods the
role of PC and $\beta^{-}$-decay is negligible and without the ND contribution
the spectra are dominated by the PA process. As shown in the middle panel of
Fig 7, the ND process only slightly enhances the emission of low-energy
$\bar{\nu}_{e}$, but at higher energies $E_{\nu}=7-9$ MeV its contribution is
comparable or even bigger than the PA one. It should be emphasised that the
increased role of the ND process in the emission of high-energy
$\bar{\nu}_{e}$ is confirmed by both the TQRPA and LSSM methods.

Referring to the lower panel of Fig 7, the dominating role of ND
(anti)neutrinos in the $\nu_{\mu,\tau}$ and $\bar{\nu}_{\mu,\tau}$ spectra is
predicted by both the TQRPA and LSSM calculations, and both methods
demonstrate the increased contribution of the ND process to the emission of
high-energy ($\mu,\,\tau$) (anti)neutrinos. Thus, the present study
generalises the results of Fischer et al. (2013) and we can conclude that
nuclear de-excitation via $\nu\bar{\nu}$-pair emission might be a leading
source for the production of heavy-lepton type (anti)neutrinos not only during
the collapse phase but also during the pre-supernova phase. Once again, we
stress that this result is confirmed by both TQRPA and LSSM calculations.
Considering that detection of $\bar{\nu}_{e}$ is the main detection channel
and taking into account that flavor oscillations can swap the $\bar{\nu}_{e}$
and $\bar{\nu}_{\mu,\tau}$ spectra, it is clear that this result might be
important for experiments. We will return to this problem in the next two
sections.

Table 2 lists the average (anti)neutrino energies
$\langle\mathcal{E}_{\nu}\rangle=\int S_{\nu}(E)dE/\int S_{\nu}(E)E^{-1}dE$
and the luminosities $L_{\nu}=\int S_{\nu}(E)dE$ computed from the total
spectra shown in Fig. 7. As seen from the table, both the TQRPA and LSSM
calculations predict that at pre-supernova phase the most luminous and
energetic are electron neutrinos. In both methods, the ND process increases
the energy luminosities $L_{\bar{\nu}_{e},\,\nu_{x},\bar{\nu}_{x}}$
($x=\mu,\,\tau$), and this effect is significant for heavy-lepton type
(anti)neutrinos. The ND process also slightly increases the average energy for
$\bar{\nu}_{e}$, $\nu_{x}$ and $\bar{\nu}_{x}$. This increase is most
pronounced for
$\langle\mathcal{E}_{\nu_{x},\bar{\nu}_{x}}\rangle_{\text{LSSM}}$. In the
TQRPA, the ND process simultaneously enhances both the low- and high-energy
parts of the spectrum (see the lower panel of Fig. 7), which is why
$\langle\mathcal{E}_{\nu_{x},\bar{\nu}_{x}}\rangle_{\text{TQRPA}}$ remains
almost unchanged.

Table 2: Average energy $\langle\mathcal{E}_{\nu}\rangle$ (in MeV) and luminosity $L_{\nu}$ (in erg/sec) of the (anti)neutrinos produced by all processes listed in Table 1, for the considered pre-supernova model. Columns marked with a star symbol show data computed without the ND contribution. | TQRPA | TQRPA* | LSSM | LSSM*
---|---|---|---|---
$\langle\mathcal{E}_{\nu_{e}}\rangle$ | 4.68 | 4.68 | 4.04 | 4.04
$L_{\nu_{e}}$ | $8.52\times 10^{50}$ | $8.52\times 10^{50}$ | $4.67\times 10^{50}$ | $4.67\times 10^{50}$
$\langle\mathcal{E}_{\bar{\nu}_{e}}\rangle$ | 2.49 | 2.41 | 2.58 | 2.41
$L_{\bar{\nu}_{e}}$ | $2.93\times 10^{47}$ | $2.44\times 10^{47}$ | $2.85\times 10^{47}$ | $2.39\times 10^{47}$
$\langle\mathcal{E}_{\nu_{x},\bar{\nu}_{x}}\rangle$ | 2.82 | 2.68 | 3.21 | 2.68
$L_{\nu_{x},\bar{\nu}_{x}}$ | $9.78\times 10^{46}$ | $4.88\times 10^{46}$ | $9.50\times 10^{46}$ | $4.88\times 10^{46}$
Figure 8: Electron capture rate
$|dY_{e}/dt|=Y_{{}^{56}\text{Fe}}\lambda^{\text{EC}}$ (left panel) and average
energy $\langle E_{\nu}\rangle$ of emitted EC neutrinos (right panel) as
functions of the mass coordinate. The TQRPA results are compared with the LSSM
calculations. To demonstrate the importance of thermal effects, we also show
the results of the QRPA calculations, which correspond to a cold ($T=0$) 56Fe
nucleus. The upper $x$-axis shows the chemical potential $\mu_{e}$.

The TQRPA and LSSM results in Table 2 are in reasonable agreement with each
other except the electron neutrino luminosity $L_{\nu_{e}}$. Namely, the TQRPA
result is about twice the luminosity calculated from the LSSM. We remind that
in both methods electron captures on 56Fe are the main source of $\nu_{e}$. It
must be emphasized that the $L_{\nu_{e}}$ value obtained in the LSSM is not a
result of the $q$-value approximation, but it can be calculated directly from
the LSSM weak-interaction rate tables
444https://theory.gsi.de/~gmartine/rates/. To understand the reason for the
discrepancy, we note two facts. First, at pre-supernova temperatures
$T_{9}\approx 2-10$, the TQRPA rates for electron capture by 56Fe generally
exceed the shell-model ones (see Fig. 3 in Dzhioev et al. (2019)). A
comparison of the TQRPA and LSSM electron capture rates
$|dY_{e}/dt|=Y_{{}^{56}\text{Fe}}\lambda^{\text{EC}}$ along the mass
coordinate, shown in the left panel of Fig. 8, confirms this observation. A
detailed analysis performed in Dzhioev et al. (2010) reveals that this
disagreement stems from a larger strength of thermally unblocked low- and
negative-energy Gamow-Teller transitions predicted by the TQRPA. Second, a
larger strength of thermally unblocked low- and negative-energy transitions
favours the emission of high-energy EC neutrinos. This can be clearly seen
(see the right panel of Fig. 8) if we compare the TQRPA and LSSM average
energy $\langle E_{\nu}\rangle$ of emitted EC neutrinos. According to the LSSM
calculations, $\langle E_{\nu}\rangle$ is significantly lowered when we move
from the center of the star decreasing the chemical potential $\mu_{e}$,
whereas it varies rather weakly around $\langle E_{\nu}\rangle\approx 4.7$ MeV
within the TQRPA. This stability is a result of the increased fraction of
high-energy neutrinos emitted due to thermally unblocked transitions, which
compensate for the decrease in available electron energy. It is clear that
both these facts enhance the EC neutrino luminosity.

To demonstrate the importance of thermal effects, we compute electron capture
rates and average neutrino energies within the QRPA, i.e. for a cold nucleus
56Fe. Referring to Fig. 8, without thermal effects, i.e. without thermally
unblocked low- and negative-energy GT+ transitions, EC rates and average
neutrino energies are significantly smaller than the one obtained in finite
temperature TQRPA and LSSM calculations. We also see that the importance of
thermal effects increases as the chemical potential $\mu_{e}$ reduces.

Obviously, due to larger electron capture rates the core should radiate more
energy away by neutrino emission, keeping the core on a trajectory with lower
temperature and entropy. Moreover, larger electron capture rates predicted by
the TQRPA for iron-group nuclei should basically result in a faster
deleptonization of the progenitor star core at the pre-collapse and early
collapse phases. We note, however, that the difference between the TQRPA and
LSSM rates is much smaller than that (Langanke & Martínez-Pinedo, 2000)
between the LSSM rates and electron capture rates estimated by Fuller, Fowler
and Newman on the basis of the independent particle model (Fuller et al.,
1982) . Besides, a larger strength of thermally unblocked negative-energy
transitions accelerates the total $\beta^{-}$-decay rate, which, for a certain
range of electron-to-baryon ratios $Y_{e}$, can exceed the electron capture
rate (Martínez-Pinedo et al., 2000) and counteracts the reduction of $Y_{e}$.
Anyway, definite conclusions about the influence of the TQRPA rates on
(pre-)supernova models might be drawn only on the basis (pre)-supernova
computer simulations employing these new rates.

##  6 Oscillations

In principle, the calculated spectra of neutrinos and antineutrinos are not
yet sufficient to estimate the effect produced in the Earth’s detectors. In
fact, we should also take into account the influence of neutrino flavor
oscillations. To do this, we will follow a simple approach (Kato et al.,
2015), which allows us to at least qualitatively assess the impact of this
effect. We will focus on $\bar{\nu}_{\mathrm{e}}$-type neutrinos, as being of
primary interest in terms of detection (see below Section 7). So we will
assume that the electron antineutrino spectrum after oscillation can be
written as

$S_{\bar{\nu}_{e}}=pS_{\bar{\nu}_{e}}^{0}+(1{-}p)S^{0}_{\bar{\nu}_{x}},$ (12)

where index $0$ stands for the original unoscillated spectra and
$x=\mu\,\text{or}\,\tau$. In (12), the quantity $p$ is the survival
probability which is connected to the mixing parameters as
$p=\cos^{2}\theta_{12}\cos^{2}\theta_{13}\approx 0.676$ for the normal
hierarchy (NH) and $p=\sin^{2}\theta_{13}\approx 0.024$ for the inverted
hierarchy (IH) (see, e.g., Kolupaeva et al. (2023)). Thus, the NH provides
moderate mixing while the IH almost suppresses the original
$\bar{\nu}_{\mathrm{e}}$ flux and replaces it by that for
$\bar{\nu}_{\mathrm{\mu}}$ or $\bar{\nu}_{\mathrm{\tau}}$, both of which have
the same spectrum. Let us remind that according to Table 1 and the discussion
in Section 4, the original $\bar{\nu}_{\mathrm{e}}$ flux is produced by the
PC, $\beta^{-}$, PA and ND processes, while the
$\bar{\nu}_{\mathrm{\mu,\tau}}$ flux is formed in the last two processes.

Figure 9: The total $\bar{\nu}_{e}$ spectra obtained with the normal (NH) and
inverted (IH) hierarchy are compared with the original unoscillated spectrum
(0). On each plot we also show the ND contribution to the spectra.

In Figure 9, we present the total (nuclear plus thermal) $\bar{\nu}_{e}$
spectra for the normal and inverted hierarchies computed within the TQRPA and
LSSM approaches and compare them with the unoscillated spectra. On each plot
we also show the ND contribution which does not depend on oscillations since
$S_{\bar{\nu}_{e}}^{0,\text{ND}}=S_{\bar{\nu}_{x}}^{0,\text{ND}}$. From the
TQRPA and LSSM plots one can see that NH oscillations have a small effect on
the shape of the original unoscillated spectrum and just decrease its absolute
value. Therefore, the average energy of $\bar{\nu}_{e}$ remains almost
unchanged (see Table 2) while their total luminosity $L_{\bar{\nu}_{e}}$ is
suppressed by a factor of $\sim 0.7$. In contrast, replacing the original
$\bar{\nu}_{\mathrm{e}}$ flux by the $\bar{\nu}_{\mathrm{x}}$ one caused by IH
oscillations results in a significant suppression of the spectrum except the
energy range where the ND contribution dominates. As a result,
$L_{\bar{\nu}_{e}}$ is lowered by a factor of more than three, while
$\langle\mathcal{E}_{\bar{\nu}_{e}}\rangle_{\text{LSSM}}$ rises from from 2.6
MeV to 3.2 MeV, and $\langle\mathcal{E}_{\bar{\nu}_{e}}\rangle_{\text{TQRPA}}$
is practically unchanged.

Figure 10: The ratio $R^{\text{NH,IH}}$ of the spectra obtained with the
normal and inverted hierarchies to the unoscillated one. The ratios computed
without the ND contribution are marked with a star symbol.

To demonstrate the energy dependence of the $\bar{\nu}_{e}$ spectrum
suppression due to oscillations, in Fig. 10 we plot the ratio
$R^{\text{NH,IH}}(E_{\nu})=S_{\bar{\nu}_{e}}^{\text{NH,IH}}(E_{\nu})/S_{\bar{\nu}_{e}}^{\text{0}}(E_{\nu})$.
Referring to the figure, without the ND contribution, the ratio
$R^{\text{NH}}$ is close to the survival probability $p=0.676$, while the
value of $R^{\text{IH}}$ is determined by the ratio
$S^{\text{0,PA}}_{\bar{\nu}_{x}}/S^{\text{0,PA}}_{\bar{\nu}_{e}}$. The latter
increases from 0.1 at low energies up to 0.3 at $E_{\nu}\approx 15$ MeV. The
inclusion of the ND process changes a monotonic energy dependence of
$R^{\text{NH,IH}}$. Namely, both the TQRPA and LSSM calculations demonstrate
that for high-energy ($E_{\nu}\approx 5-10$ MeV) antineutrinos the ND process
reduces spectrum suppression and this effect is most pronounced for IH
oscillations. Within the TRQPA approach, the ND process also increases the
fraction of low-energy neutrinos in the oscillated $\bar{\nu}_{e}$ spectrum,
which, as discussed earlier in Section 5, is due to temperature-induced
increase in the strength of low-energy GT0 transitions in 56Fe.

##  7 Detection

The dominant detection process for electron antineutrinos from (pre-)supernova
is the inverse beta-decay:

$\bar{\nu}_{\mathrm{e}}+p\rightarrow n+e^{+}.$ (13)

Of course, other reactions can be important as well (see, e.g., Manukovskiy et
al. (2022)), but here we will focus on the most standard one. The cross-
section of (13) can be estimated as (see, e.g., Odrzywolek et al. (2004a))

$\sigma_{\mathrm{IBD}}\sim p_{e^{+}}E_{e^{+}},$ (14)

where $E_{e^{+}}=E_{\bar{\nu}}-(m_{\mathrm{n}}{-}m_{\mathrm{p}})c^{2}$ is the
emitted positron energy and $p_{e^{+}}$ is its momentum. So the minimum
threshold energy required to induce this reaction is
$E^{\mathrm{min}}_{\bar{\nu}}=(m_{\mathrm{n}}{-}m_{\mathrm{p}}{+}m_{\mathrm{e}})c^{2}\approx
1.8$ MeV. If the detector employed has the detection efficiency 100% above the
threshold $E_{\mathrm{th}}\geq 1.8$ MeV, then the number of detected electron
antineutrinos with the energy spectrum $S(E_{\bar{\nu}})$ is expressed as

$N(E_{\mathrm{th}})\sim\int\limits_{E_{\mathrm{th}}}^{\infty}\sigma_{\mathrm{IBD}}(E_{\bar{\nu}})S(E_{\bar{\nu}})\frac{dE_{\bar{\nu}}}{E_{\bar{\nu}}}.$
(15)

Applying this relation and using the TQRPA and LSSM spectra shown in Fig. 9,
we constructed Fig. 11 to compare three cases: NH, IH and unoscillated one.

Figure 11: The relative number $N(E_{\mathrm{th}})/N_{0}$ of events in the
detector and the detection rate enhancement factor $D(E_{\text{th}})$ as a
function of the threshold energy $E_{\mathrm{th}}$.

In Fig. 11, with the solid curves with filled symbols we denote a relative
number of events in the detector, $N(E_{\mathrm{th}})/N_{0}$, as a function of
the threshold energy. Here the normalisation constant $N_{0}$ is
$N(E_{\mathrm{th}}=1.8\,\text{MeV})$ for the unoscillated TQRPA spectrum. We
note that $N^{\text{TQRPA}}_{0}$ and $N^{\text{LSSM}}_{0}$ are rather close to
each other. As indicated in the figure, NH oscillations lead to a moderate
reduction of the $N(E_{\mathrm{th}})/N_{0}$ ratio, whereas IH oscillations
reduce the number of detected events by about half. Besides, the ratio
$N(E_{\mathrm{th}})/N_{0}$ demonstrates the importance of the detector’s
threshold parameter $E_{\mathrm{th}}$ and how drastically it can affect the
number of registered pre-supernova $\bar{\nu}_{e}$. For example, a rather
moderate value of $E_{\mathrm{th}}\approx 5$ MeV reduces the number of
registered events by almost half.

In order to show the effect of the ND process on neutrino detection, in Fig.
11 we also plot with the dashed curves with empty symbols the quantity

$D(E_{\text{th}})=\left(1-\frac{N^{\text{ND}}(E_{\text{th}})}{N(E_{\text{th}})}\right)^{-1},$
(16)

which is a detection rate enhancement factor due to ND neutrinos (right axes).
As expected, ND neutrinos play an important role in the case of IH
oscillations. Namely, for $E_{\text{th}}\approx 5-8$ MeV they enhance the
TQRPA and LSSM detection rates by factors 5.5 and 3.5, respectively. At low
threshold energies the enhancement factor is $D\approx 3.0$. An increase in
the survival probability $p$ diminishes the role of ND neutrinos, but even in
the case of unoscillated spectra, they still contribute substantially to
$\bar{\nu}_{e}$ neutrino detection for $E_{\text{th}}\approx 5-8$ MeV.

Thus, we can conclude that from the viewpoint of detection of pre-supernova
electron antineutrinos, the neutral-current de-excitation of 56Fe is an
important process especially in the moments immediately preceding the collapse
when the core temperature is high enough for the thermal population of the GT0
resonance.

##  8 Summary and perspectives

Let us summarize the main results of the presented research. With the TQRPA
approach we are able to take into account thermal effects which influence
various nuclear weak-interaction processes contributing to (anti)neutrino
generation under the conditions characteristic for a hot (pre-)supernova
matter. We focus our attention on (anti)neutrino production by a hot 56Fe,
which dominates the isotopic composition of the stellar core at the onset of
the collapse. The realistic pre-supernova profile we use was obtained with the
help of the famous MESA evolutionary code. In accordance with our previous
estimates (Dzhioev et al., 2023), we have found that the TQRPA approach
produces not only a higher energy luminosity of electron neutrinos (mainly
born in the EC process), compared to a more conventional LSSM approach, but
also a harder neutrino spectrum as compared with those produced with the
effective $q$-value approximation. Both these effects are due to a larger
strength of low- and negative-energy GT+ transitions in the thermally excited
56Fe predicted by the TQRPA.

We also have found that the nuclear neutral-current de-excitation process via
$\nu\bar{\nu}$-pair emission is at least as important as the electron-positron
pair annihilation in the context of the electron antineutrino generation. Our
calculations of $\bar{\nu}_{e}$ spectra based on the TQRPA and LSSM
distributions of the GT0 strength in 56Fe have shown that, although
$\bar{\nu}_{e}$ from the ND process have a lower total luminosity compared to
the PA process, their luminosity in the high-energy region $E_{\nu}=7-9$ MeV
is comparable or even exceeds the PA contribution. Moreover, both the TQRPA
and LSSM calculations predict that the ND process is crucial for generation of
heavy-lepton flavor (anti)neutrinos under pre-supernova conditions. The latter
is of extreme importance because of the problem of pre-supernova neutrino
registration by the Earth’s detectors. We have also explored the effect of
neutrino oscillations and found that ND antineutrinos are important in all
cases but especially in the case of the inverted mass hierarchy when they
increase the detection rate by several times.

The main goal of the present study is to draw attention to the importance of
the careful treatment of neutrino processes with hot nuclei. Our research can
be continued and extended in several aspects. First, it is interesting to
study within the TQRPA the effects of nuclear composition on the neutrino
luminosities and spectra. For example, according to Patton et al. (2017b),
there is no one dominating nucleus in the center of the star but rather a
bunch of iron-group nuclei such as 50-51Ti, 54Cr etc. Probably this is due to
a different and more extended nuclear reaction network than the one we use. We
are planning to address this issue in forthcoming studies.

In addition, the diversity of pre-supernovae may be caused by another factor:
a pre-supernova may be part of a close binary system with mass exchange
(Laplace et al., 2021) which can lead to significant differences in stellar
profiles. And it should be remembered that not only a core-collapse supernova
but also a type Ia thermonuclear supernova (Odrzywolek & Plewa, 2011) can be a
source of neutrinos that can be detected the on Earth.

## Acknowledgements

AY and NDB are grateful to S.I. Blinnikov for useful discussions on neutrino
spectra from supernova progenitors on the late stages of their evolution. AY
thanks RSF 22-12-00103 grant for support. All authors thank the anonymous
reviewer for constructive comments and important suggestions that helped
improve this manuscript.

## Data Availability

Data generated from computations are reported in the body of the paper.
Additional data can be made available upon reasonable request.

## References

  * Alekseev et al. (1987) Alekseev E. N., Alekseeva L. N., Volchenko V. I., Krivosheina I. V., 1987, Soviet Journal of Experimental and Theoretical Physics Letters, 45, 589
  * Bionta et al. (1987) Bionta R. M., et al., 1987, Phys. Rev. Lett., 58, 1494
  * Caurier et al. (1999) Caurier E., Langanke K., Martínez-Pinedo G., Nowacki F., 1999, Nuclear Physics A, 653, 439
  * Caurier et al. (2005) Caurier E., Martínez-Pinedo G., Nowacki F., Poves A., Zuker A. P., 2005, Rev. Mod. Phys., 77, 427
  * Dzhioev & Vdovin (2022a) Dzhioev A. A., Vdovin A. I., 2022a, Physics of Particles and Nuclei, 53, 885
  * Dzhioev & Vdovin (2022b) Dzhioev A., Vdovin A. I., 2022b, Physics of Particles and Nuclei, 53, 939
  * Dzhioev & Vdovin (2022c) Dzhioev A. A., Vdovin A. I., 2022c, Physics of Particles and Nuclei, 53, 1051
  * Dzhioev et al. (2010) Dzhioev A. A., Vdovin A. I., Ponomarev V. Y., Wambach J., Langanke K., Martínez-Pinedo G., 2010, Phys. Rev. C, 81, 015804
  * Dzhioev et al. (2016) Dzhioev A. A., Vdovin A. I., Martínez-Pinedo G., Wambach J., Stoyanov C., 2016, Phys. Rev. C, 94, 015805
  * Dzhioev et al. (2019) Dzhioev A. A., Vdovin A. I., Stoyanov C., 2019, Phys. Rev. C, 100, 025801
  * Dzhioev et al. (2020) Dzhioev A. A., Langanke K., Martínez-Pinedo G., Vdovin A. I., Stoyanov C., 2020, Phys. Rev. C, 101, 025805
  * Dzhioev et al. (2023) Dzhioev A. A., Yudin A. V., Dunina-Barkovskaya N. V., Vdovin A. I., 2023, Particles, 6, 682
  * Farmer et al. (2016) Farmer R., Fields C. E., Petermann I., Dessart L., Cantiello M., Paxton B., Timmes F. X., 2016, ApJS, 227, 22
  * Fischer et al. (2013) Fischer T., Langanke K., Martínez-Pinedo G., 2013, Phys. Rev. C, 88, 065804
  * Fuller et al. (1982) Fuller G. M., Fowler W. A., Newman M. J., 1982, ApJ, 252, 715
  * Fuller et al. (1985) Fuller G. M., Fowler W. A., Newman M. J., 1985, ApJ, 293, 1
  * Hirata et al. (1987) Hirata K., et al., 1987, Phys. Rev. Lett., 58, 1490
  * Itoh et al. (1996) Itoh N., Hayashi H., Nishikawa A., Kohyama Y., 1996, ApJS, 102, 411
  * Kato et al. (2015) Kato C., Delfan Azari M., Yamada S., Takahashi K., Umeda H., Yoshida T., Ishidoshiro K., 2015, ApJ, 808, 168
  * Kato et al. (2017) Kato C., Nagakura H., Furusawa S., Takahashi K., Umeda H., Yoshida T., Ishidoshiro K., Yamada S., 2017, ApJ, 848, 48
  * Kolupaeva et al. (2023) Kolupaeva L., Gonchar M., Ol’shevskii A., Samoylov O., 2023, Phys. Usp., 66, 753
  * Kondratyev et al. (2019) Kondratyev V. N., Dzhioev A. A., Vdovin A. I., Cherubini S., Baldo M., 2019, Phys. Rev. C, 100, 045802
  * Kutschera et al. (2009) Kutschera M., Odrzywołek A., Misiaszek M., 2009, Acta Physica Polonica B, 40, 3063
  * Langanke & Martínez-Pinedo (2000) Langanke K., Martínez-Pinedo G., 2000, Nuclear Phys. A, 673, 481
  * Langanke et al. (2001) Langanke K., Martínez-Pinedo G., Sampaio J. M., 2001, Physical Review C, 64, 055801
  * Laplace et al. (2021) Laplace E., Justham S., Renzo M., Götberg Y., Farmer R., Vartanyan D., de Mink S. E., 2021, A&A, 656, A58
  * Manukovskiy et al. (2022) Manukovskiy K. V., Yudin A. V., Agafonova N. Y., Malgin A. S., Ryazhskaya O. G., 2022, Soviet Journal of Experimental and Theoretical Physics, 134, 277
  * Martínez-Pinedo et al. (2000) Martínez-Pinedo G., Langanke K., Dean D. J., 2000, The Astrophysical Journal Supplement Series, 126, 493
  * Misch & Fuller (2016) Misch G. W., Fuller G. M., 2016, Phys. Rev. C, 94, 055808
  * Misiaszek et al. (2006) Misiaszek M., Odrzywołek A., Kutschera M., 2006, Phys. Rev. D, 74, 043006
  * Odrzywolek & Heger (2010) Odrzywolek A., Heger A., 2010, Acta Phys. Polon. B, 41, 1611
  * Odrzywolek & Plewa (2011) Odrzywolek A., Plewa T., 2011, A&A, 529, A156
  * Odrzywolek et al. (2004a) Odrzywolek A., Misiaszek M., Kutschera M., 2004a, Astroparticle Physics, 21, 303
  * Odrzywolek et al. (2004b) Odrzywolek A., Misiaszek M., Kutschera M., 2004b, Acta Physica Polonica B, 35, 1981
  * Patton et al. (2017a) Patton K. M., Lunardini C., Farmer R. J., 2017a, ApJ, 840, 2
  * Patton et al. (2017b) Patton K. M., Lunardini C., Farmer R. J., Timmes F. X., 2017b, ApJ, 851, 6
  * Paxton et al. (2015) Paxton B., et al., 2015, ApJS, 220, 15
  * Ryazhskaya (2006) Ryazhskaya O. G., 2006, Physics Uspekhi, 49, 1017
  * Sampaio et al. (2002) Sampaio J., Langanke K., Martinez-Pinedo G., Dean D., 2002, Physics Letters B, 529, 19
  * Weaver et al. (1978) Weaver T. A., Zimmerman G. B., Woosley S. E., 1978, ApJ, 225, 1021