# Multimessenger search for electrophilic feebly interacting particles from
supernovae

Pedro De la Torre Luque<EMAIL_ADDRESS>The Oskar Klein
Centre, Department of Physics, Stockholm University, Stockholm 106 91, Sweden
Shyam Balaji<EMAIL_ADDRESS>Laboratoire de Physique Théorique et
Hautes Energies (LPTHE), UMR 7589 CNRS & Sorbonne Université, 4 Place Jussieu,
F-75252, Paris, France  Institut d’Astrophysique de Paris, UMR 7095 CNRS &
Sorbonne Université, 98 bis boulevard Arago, F-75014 Paris, France  Pierluca
Carenza<EMAIL_ADDRESS>The Oskar Klein Centre, Department of
Physics, Stockholm University, Stockholm 106 91, Sweden

###### Abstract

We study MeV-scale electrophilic Feebly Interacting Particles (FIPs), that may
be abundantly produced in Supernova (SN) explosions, escape the star and decay
into electrons and positrons. This exotic injection of leptons in the Milky
Way leaves an imprint in both photon and cosmic-ray fluxes. Specifically,
positrons lose energy and annihilate almost at rest with background electrons,
producing photons with $511$ keV energy. In addition, electrons and positrons
radiate photons through bremsstrahlung emission and upscatter the low-energy
galactic photon fields via the inverse Compton process generating a broad
emission from X-ray to $\gamma$-ray energies. Finally, electrons and positrons
are directly observable in cosmic ray experiments. In order to describe the
FIP-induced lepton injection in full generality, we use a model independent
parametrization which can be applied to a host of FIPs such as axion-like
particles, dark photons and sterile neutrinos. Theoretical predictions are
compared to experimental data to robustly constrain FIP-electron interactions
with an innovative multimessenger analysis.

##  I Introduction

Explosions of massive stars, called Supernovae (SN), produce a copious amount
of light Feebly Interacting Particles (FIPs). Popular and well studied
examples of such new particles are axions [1, 2, 3, 4] and axion-like
particles (ALPs) [5, 6, 7], light CP-even scalars [8, 9, 10, 11, 12], sterile
neutrinos [13, 14, 15], dark photons (DPs) [16], dark flavored particles [17]
and unparticles [18] (see Ref. [19] for a broad discussion on FIPs).
Therefore, SN play a fundamental role in understanding the existence and
properties of FIPs. For typical SN core temperatures $\mathcal{O}(30)$ MeV,
FIPs with masses up to $\mathcal{O}(100)$ MeV can be abundantly emitted. The
weak interactions between FIPs and ordinary matter allow efficient subtraction
of energy from the stellar core, thereby shortening the neutrino burst signal
[20, 21]. This powerful physical argument has been used to exclude different
new physics scenarios based on the SN 1987A neutrino observation [1].

FIPs don’t only indirectly affect the SN neutrino burst, but they also leave
various signatures depending on their interactions or decays outside the SN.
For example, very light ALPs that interact with photons can be converted into
$\gamma$-rays in the galactic magnetic field. This would lead to a distinctive
$\gamma$-ray emission in coincidence with the SN explosion. This approach has
been used to search for ALPs from SN 1987A [22, 23] and to predict their
signals from future galactic or extra-galactic SN [24, 25, 26, 27], as well as
from the diffuse $\gamma$-ray background [28, 29, 30]. A similar idea can be
applied to MeV-scale ALPs, decaying into photons after escaping the SN and
generating an unexpected $\gamma$-ray emission [31, 32, 23, 33, 34].

In this work we focus on electrophilic FIPs with masses in the MeV-range,
particularly interesting for phenomenology [35, 36, 37, 38] and experimental
searches [39]. We explore the multimessenger signals induced by FIPs decaying
into electron-positron pairs, showing that they can be used to constrain a
variety of well-motivated FIP models, such as ALPs, DPs and sterile neutrinos.
FIPs produced in a SN and decaying into electrons and positrons, contribute to
the total galactic background of electrons and positrons generated from
astrophysical sources (mainly pulsar wind nebulas and SN remnants) [40, 41]
and cosmic ray (CR) interactions with the interstellar gas [42].

An observable strongly affected by this exotic lepton injection is the $511$
keV signal generated from positron interactions with interstellar gas. We
exploit observations at these energies to set limits on the total injected
flux of electrons and positrons, agnostic to the exact FIP model. The proposed
recipe can be readily applied to any electrophilic FIP efficiently produced in
SN. In the pioneering work Ref. [43] (also see Sec. 12.5.1 of Ref. [44]), the
authors examined the $511$ keV photon constraint for decaying heavy neutrinos,
using early data on the $511$ keV observations. The same bound for DPs was
recently explored in Ref. [45]. The best constraints to date on different FIP
models come from this observable [46, 47]. We improve upon these previous
studies by using updated data from the Spectrometer on INTEGRAL (SPI)
instrument [48, 49] and exploring a wider set of models for the spatial
distribution of SN, revising the systematic uncertainties in these
evaluations.

In this paper, extending the companion paper Ref. [50], we propose alternative
constraints on the FIP-induced electron-positron injection. For instance, in
recent years the Voyager-1 probe has provided measurements of the local flux
of electrons outside the heliosphere [51, 52] at energies below tens of MeV.
These measurements profer the advantage that they are not significantly
affected by the solar modulation effect [53], which highly suppresses the flux
of low-energy charged CRs at Earth. For the first time, we use the Voyager-1
electron data to constrain FIPs. Likewise, the secondary emissions from the
electron-positron pairs, especially from Inverse Compton (IC) and
bremsstrahlung processes, have been studied in the context of Weakly
Interacting Massive Particles recently [54] but has not yet been applied to
the case of FIPs. We explore these secondary emission signals and perform a
systematic analysis of the available datasets for hard X-rays and low-energy
$\gamma$ rays, to determine their full constraining power on electrophilic
FIPs. The various data explored in our multimessenger approach allow us to set
stringent and robust bounds on electrophilic FIPs, competitive with limits
from $511$ keV observations.

This manuscript is organized as follows. In Section II we introduce the
phenomenology of electrophilic FIPs and characterize their production spectrum
from SN. In Section II.1, we give details on the FIP-induced injection and
diffusion of electrons and positrons in the Galaxy. Then, their secondary
photon emissions are briefly discussed in Section II.2. The various datasets,
at different energies, used to constrain FIP properties are discussed in
Section III. To summarize, we discuss constraints from:

  * •

the $511$ keV line measured by the SPectrometer on INTEGRAL (SPI, Sec. III.1);

  * •

local electron and positron fluxes detected by Voyager-1 and Alpha Magnetic
Spectrometer (AMS-02, Sec. III.2);

  * •

high-energy $\gamma$-rays in the $20$ MeV-$30$ GeV range probed by the
Energetic Gamma Ray Experiment Telescope (EGRET);

  * •

$\gamma$-rays with energies $2$-$20$ MeV revealed by the Imaging Compton
Telescope (COMPTEL);

  * •

soft $\gamma$-rays and hard X-rays, between $20$ keV and $2$ MeV, covered by
SPI;

  * •

X-rays in the $2.5-8$ keV range, studied with the X-ray Multi-Mirror Mission
(XMM-Newton);

where the X- to $\gamma$-ray signals are discussed in Section III.3. These
constraints are applied to various FIP models in Section IV. In Section V we
summarize the main results and conclude.

##  II Electrophilic FIP phenomenology

A large class of FIPs, dubbed electrophilic, feature a coupling to electrons
and positrons. An electrophilic FIP, $X$, with a mass larger than the
electron-positron energy threshold is kinematically able to produce an
electron-positron pair in its decay. For example, ALPs ($a$) with
$m_{a}\gtrsim 1$ MeV decay as $a\to e^{+}e^{-}$ and similarly for DPs. Also
massive sterile neutrinos produce electrons and positrons in their decays, for
example channels such as $\nu_{s}\to\nu_{\mu}e^{+}e^{-}$ are open for sterile
neutrinos mixed with muon neutrinos, and other decays are possible as the
sterile neutrino mass is increased. Several other FIPs can also produce
electrons and positrons in their decay, like scalars, Kaluza-Klein gravitons
and supersymmetric particles. Thus, it is important to search for signatures
of electrophilic FIPs.

Once produced in a SN, FIPs may escape from the progenitor star subtracting
energy from the core due to their weak interactions with the stellar medium.
The energy lost from the star might be comparable with losses accounted for by
neutrinos which significantly affect the SN evolution. Hence, based on the
observation of SN 1987A, it is possible to extract stringent bounds that apply
to several FIP models since cooling due to FIPs should not significantly
affect the SN dynamics. This energy-loss criterion can be cast as a constraint
on the FIP luminosity [55]

$L_{X}^{\rm tot}\lesssim 3\times 10^{52}\,{\rm erg}~{}{\rm s}^{-1}\,,$ (1)

at the beginning of the cooling phase, at about $t_{\rm pb}=1$ s, where
$t_{\rm pb}$ is the post-bounce time.

Figure 1: Left panel: Total injected flux of electrons for different FIP
masses and $N_{e}=10^{53}$, shown as function of the total energy of the
electron. Right panel: Local (unmodulated) flux of the propagated electron
population obtained by injecting the FIP-induced fluxes in the left panel.

In the case that FIPs decay electromagnetically, a more severe constraint was
obtained in Ref. [56] (see also Refs. [57, 58]) from the observation of low-
energy SN, requiring that the energy injected by FIP decay products satisfies

$L_{X}^{\rm e.m.}\lesssim 10^{50}\,{\rm erg}~{}{\rm s}^{-1}\,.$ (2)

Here, with the superscript e.m., we specified that this constraint applies to
FIP luminosity converted into an electromagnetic channel.

This bound is valid for FIPs decaying inside the SN envelope, FIPs with longer
mean-free-paths cannot be constrained. However, in this case, the copious FIP
production in a SN is expected to inject a large flux of $\gamma$-rays,
electrons and positrons in our Galaxy. Typically the electron/positron flux
will inherit the spectral properties of the parent particle. If the decay
products have an energy $E_{e}$ which is half of the parent FIP, the injected
electron/positron flux can be written as [59]

$\begin{split}\frac{dN_{e}}{dE_{e}}&=N_{e}C_{0}\left(\frac{4E_{e}^{2}-m_{X}^{2}}{E_{0}^{2}}\right)^{\beta/2}e^{-(1+\beta)\frac{2E_{e}}{E_{0}}}\,,\\\
C_{0}&=\frac{2\sqrt{\pi}\left(\frac{1+\beta}{2m_{X}}\right)^{\frac{1+\beta}{2}}E_{0}^{\frac{\beta-1}{2}}}{K_{\frac{1+\beta}{2}}\left((1+\beta)\frac{m_{X}}{E_{0}}\right)\Gamma\left(1+\frac{\beta}{2}\right)}\,,\end{split}$
(3)

assuming that FIPs are emitted with a modified blackbody spectrum, where
$E_{0}$ is related to the FIP average energy, its mass is $m_{X}>2m_{e}$,
$\beta$ is the spectral index, $K_{\frac{1+\beta}{2}}$ is the modified Bessel
function of the second kind of order $(1+\beta)/2$, $\Gamma$ is the Euler-
Gamma function and this flux is normalized such that

$\int_{m_{X}/2}^{\infty}dE_{e}\frac{dN_{e}}{dE_{e}}=N_{e}\,.$ (4)

Throughout this paper, we use $N_{e}$ to denote the number of electrons, which
is equal to the number of positrons, produced in a SN explosion via FIP
decays, i.e. $N_{e}=N_{e^{+}}=N_{e^{-}}$. The simple prescription in Eq. (3)
does not depend on the type of FIP model. This flux is obtained by assuming a
FIP decaying into an electron-positron pair. Thus, it cannot be strictly valid
for sterile neutrinos, which has more involved decay channels. However, by
changing the parameters we expect that sterile neutrino spectra can also be
roughly modeled by Eq. (3). This simple prescription for the injected lepton
flux enables us to place constraints on the electron-coupling of different
electrophilic FIPs in a model-independent way. Unless otherwise specified, in
this work we adopt the values of $E_{0}=45$ MeV, $\beta=2.5$ and $m_{X}=10$
MeV as our benchmark model.

We will now discuss in detail the non-standard injection of electrons and
positrons in the Galaxy, caused by FIP decays, and its relevant
phenomenological consequences.

Figure 2: Left panel: Electron spectrum at the Earth location for different
injection parameters, see Eq. (3). In both panels we use $N_{e}=10^{53}$ and a
FIP mass $m_{X}=10$ MeV. Right panel: Galactic map representing the density of
propagated electrons generated from FIPs, at an energy of $1$ MeV, and in
galactic coordinates.

###  II.1 Galactic electron-positron flux induced by electrophilic FIPs

In order to calculate the spectrum and galactic distribution of electrons and
positrons produced in FIP decays, we solve the diffusion equation of these
particles using the DRAGON2 code [60,
61].111https://github.com/cosmicrays/DRAGON2-Beta_version DRAGON2 is an
advanced CR propagation code designed to self-consistently solve the
diffusion-advection-loss equation describing CR transport for all species
involved in the CR network, including CRs of both astrophysical and exotic
origin (e.g., from dark matter annihilations/decays).

Norm. Energy | $E_{0}$ | $4~{}\operatorname{GeV}$
---|---|---
Diffusion coeff. | $D_{0}$ | $1.02\times 10^{29}\operatorname{cm}^{2}\operatorname{s}^{-1}$
Diffusion index | $\delta$ | $0.49$
Break energy | $E_{b}$ | $312~{}\operatorname{GeV}$
Index break | $\Delta\delta$ | $0.20$
Smooth. param. | $s$ | $0.04$
$\beta$ exponent | $\eta$ | $-0.75$
Halo height | $H$ | $8~{}\operatorname{kpc}$
Alfvèn velocity | $v_{A}$ | $13.4\operatorname{km}\operatorname{s}^{-1}$
Table 1: Main propagation parameters used in our analysis [42].

We work under the hypothesis that FIPs decay relatively close, at a distance
smaller than $\sim\mathcal{O}(10)~{}\operatorname{kpc}$ [46], to the SN where
they are produced. Thus, we simulate the injection of electrons and positrons
as happening directly from SN distributed following the spatial distribution
determined by the Ferriere distribution [62], ‘Ferr’, convolved with the
Steiman-Cameron distribution [63] of the spiral arms (four-arm model), and the
injection spectrum given by Eq. (3). Alternatively, we use the Lorimer
distribution [64], ‘Lor’, in order to estimate the uncertainties related to
the SN distribution. The electron/positron injection flux is an input of the
code. We define it as

$\frac{d\Phi_{e}}{dE_{e}}=\frac{\Gamma_{cc}}{4\pi
d_{SN}^{2}}\frac{dN_{e}}{dE_{e}}\,,$ (5)

where $\Gamma_{cc}=2$ SN per century is the rate of core-collapse SN
explosions in the Galaxy [65] and $d_{SN}=10.2$ kpc is an effective length-
scale necessary to calculate the total area of emission from the SN. It is
roughly the average distance between galactic SN calculated from the source
distribution employed in this work. This parameter is obtained by imposing
that the injected number of particles per unit of energy in Eq. (3) is equal
to the integral of the total flux density of particles (obtained convolving
Eq. (3) with the spatial distribution of sources) over the volume of the
Galaxy. Furthermore, we remark that electrons and positrons propagate and
interact in the Galaxy on a timescale of $10^{3}$-$10^{6}$ years, extremely
long compared to the SN explosion rate. Therefore, we can model the lepton
injection as time-independent and smoothly following the SN distribution.

In this setup, we employ the spatially-constant diffusion coefficient derived
from the analyses of secondary-to-primary flux ratios in Ref. [66] and adapted
in Ref. [42] for the 3D source, gas and magnetic field distribution of the
Galaxy (B/C best-fit model). In particular, for the diffusion coefficient
defined in Eq. (3.3) of Ref. [66], the parameters are shown in Tab. 1. We
adopt the magnetic field model derived by Ref. [67], with a normalization of
the disk, halo and turbulent magnetic field intensities set to the values
found in Ref. [68] from the study of synchrotron radiation. The energy density
distribution of the radiation fields has been taken from Ref. [69].

In the left panel of Fig. 1 we show the FIP-induced electron/positron flux
injected for a single SN, assuming a fixed total number of injected positrons
$N_{e}=10^{53}$, obtained by Eq. (5) for various FIP masses. These injections
give rise to a galactic electron flux that, after propagation, is shown in the
right panel of Fig. 1. Here, the flux is evaluated at the Earth location and
we neglect the solar modulation effect, which is negligible outside the
heliosphere. From this comparison we note that different masses in the
$2$-$20$ MeV range lead to very similar spectra at their peak. Moreover, their
propagated spectra are very similar below $\sim 20$ MeV, as we see from the
right panel of Fig. 1. Another peculiar feature is that the propagated
electron/positron flux peaks at an energy slightly below the peak one of the
injected flux, because of the energy lost by leptons during their propagation.
Moreover, there is another peak at higher energies, around $\sim 100$ MeV, due
to reacceleration of electrons/positrons by plasma waves [70, 71]. We limit
this analysis to $m_{X}\lesssim 20$ MeV because in this range our conclusions
are quite mass-independent and above this threshold the FIP production starts
to be Boltzmann suppressed. The discussion until now is only focused on the
mass dependence of the FIP-induced galactic lepton flux. However, this flux
depends also on the FIP spectral properties, i.e. the choice of $E_{0}$ and
$\beta$ in Eq. (3). This is precisely what is shown in the left panel of Fig.
2. Here, we illustrate how the choice of parameters affect the predicted
electron/positron flux at a fixed FIP mass $m_{X}=10$ MeV and total number of
injected positrons $N_{e}=10^{53}$, for values of $30~{}\operatorname{MeV}\leq
E_{0}\leq 60~{}\operatorname{MeV}$ and $2\leq\beta\leq 3$. The black, red and
blue lines share the same $E_{0}$ and different spectral indices, pinching the
spectrum as $\beta$ increases. The cyan and purple lines embrace the
variability of $E_{0}$ in the interval $30$-$60$ MeV, showing a suppression of
the flux for lower injection energies. From this comparison we conclude that
the results of our analysis are not strongly affected by the precise shape of
the spectrum in Eq. (3), for parameters varying in the discussed range. This
makes it possible to apply our results to a large number of FIP models,
without expecting major quantitative differences. For a detailed study of
these constraints in realistic FIP models we defer to an upcoming publication
[72]. The spatial distribution of the lepton population is also relevant for
the upcoming discussion. As expected for charged particles, this population
follows the spiral structure of the Galaxy, as shown in the right panel of
Fig. 2 for an electron energy of $1$ MeV.

###  II.2 Secondary emissions from the FIP-induced electron-positron
population

During their propagation in the Galaxy, electrons/positrons injected by FIPs
interact with the interstellar gas and radiation fields (ISRFs), i. e. mainly
optical and ultraviolet (UV) light from stars, infrared (IR) light from dust
and the Cosmic Microwave Background (CMB) as well as the galactic magnetic
field. Together, the IC emission, from the boost of the low energy ISRF
photons, and the bremsstrahlung emission, from the interaction of
electrons/positrons with the interstellar gas, produce a continuous emission
from X-rays at the keV scale to $\gamma$ rays at the MeV scale. Therefore, one
can benefit from MeV $\gamma$-ray observations as well as the rich data in
X-ray observations in order to probe the injected lepton population.

Figure 3: X-to-$\gamma$-ray emission from a FIP with mass $m_{X}=10$ MeV in a
region around the galactic center ($|l|<40^{\circ}$ and $|b|<40^{\circ}$).
Here, we show the bremsstrahlung emission from the interaction with atomic
(red line, labeled as HI) and molecular (magenta line, labeled as H2) gas as
well as the IC emission from the interaction with the CMB (blue line, labeled
as CMB) and all the ISRFs, including CMB, IR, optical and UV fields (green
line, labeled as ISRF). Figure 4: Comparison of the predicted FIP signals, for
$m_{X}=10$ MeV and $N_{e}=10^{54}$, (black lines) with the different datasets
employed in this work. Since each dataset is extracted from a different region
of the sky, the theoretical predictions are calculated for the spatial region
and energy range of the corresponding dataset. The electron flux is indicated
with a dashed line and the secondary X-to-$\gamma$-ray emission with solid
lines. In addition, the calculated emission corresponding to the $511$ keV
line is also shown.

In this work, we model the secondary diffuse emission due to the interactions
of this electron/positron population using the HERMES code [73], which
performs a numerical integration of the emission along the line of sight at
each galactic position and energy using detailed gas maps and updated ISRF
models [74, 75]. In Fig. 3 we show the spectra of the different components
producing the IC and bremsstrahlung emissions from the electrons injected by a
$10$ MeV FIP for a region with $|b|<40^{\circ}$ and $|l|<40^{\circ}$ in
latitude and longitude around the galactic center. In particular, the IC
emission from the interaction with the CMB and the rest of ISRFs is shown in
blue and green, respectively, while the bremsstrahlung emission produced from
the interaction with atomic and molecular gas are shown as red and magenta
lines respectively. Bremsstrahlung emission, for a single scattering event,
produces a flat photon spectrum that drops at a cutoff energy of the same
order of magnitude of the charged particle energy. This explains the behavior
of the red and magenta lines in Fig. 3. The IC emission promotes the energy of
the ambient photons in the ISRFs proportionally to $E^{2}$. For example, a
typical ISRF photon has an energy around a few eV, and it gains an energy of
$(E/m_{e})^{2}\sim 3600$ times higher when interacting with an electron of
$E\sim 30$ MeV, thereby reaching a final energy at the keV-scale. At lower
energies, scattering on the CMB dominates. These processes explain why the IC
emission is more important at low energies while the bremsstrahlung emission
peaks above a few tens of MeV.

A distinctly different process for secondary emission is the positron
annihilation with galactic electrons, contributing to the $511$ keV photon
line, which is a subject of notable interest [76, 48, 49, 77]. The FIP-
injected positrons are slowed down through elastic scatterings with the
interstellar gas eventually forming a positronium bound state with the ambient
electrons, then annihilating almost at rest [78]. Only around $25\%$ of
positrons form (singlet) para-positronium states, which decay into two
photons, each with energy almost exactly equal to the electron mass,
contributing to the $511$ keV line. The (triplet) orthopositronium state,
formed in $75\%$ of cases, decays into three photons with a continuous
spectrum. This latter contribution will be neglected. The impact of ALPs and,
subsequently of DPs and sterile neutrinos, on the $511$ keV line was discussed
in Refs. [46, 47].

For a given number of injected positrons $N_{e}$ it is possible to calculate
the $511$ keV photon flux as

$\frac{d\phi_{\gamma}^{511}}{d\Omega}=2k_{ps}N_{\rm e}\Gamma_{cc}\int
ds\,s^{2}\frac{n_{cc}(x_{s,b,l},y_{s,b,l},z_{s,b,l})}{4\pi s^{2}}\,,$ (6)

where $k_{ps}=1/4$ is the fraction of positronium decays contributing to the
$511$ keV line signal, $\Gamma_{cc}$ is the galactic SN rate introduced in Eq.
(5), $n_{cc}$ is the SN density distribution, $d\Omega=dldb\cos b$ is the
solid angle element with $l$ longitude and $b$ latitude and the coordinates of
points at distance $s$ from the Earth contributing to the signal are
determined in spherical coordinates by

$\begin{split}x_{s,b,l}&=x_{0}+s\cos l\cos b\,,\\\ y_{s,b,l}&=y_{0}+s\sin
l\cos b\,,\\\ z_{s,b,l}&=z_{0}+s\sin b\,,\\\ \end{split}$ (7)

given that the Earth is located at $(x_{0},y_{0},z_{0})=(-8.2,0,0)$ kpc [79].
The recipes discussed in this Section to calculate secondary emissions from
the injected lepton fluxes, are applied throughout this work to set
constraints on $N_{e}$.

##  III Constraints on electron/positron injection by FIPs

Before discussing the specific details for the analyses of each of the
datasets employed in this work, we provide a general view of the different
observations and expected FIP-induced signals over a large energy range (see
Fig. 4). In particular, the black lines are the theoretical predictions for
the multimessenger signals induced by an exotic electron/positron injection of
$N_{e}=10^{54}$ positrons per SN explosion. The benchmark values for the
energy spectrum of the FIP decay products are assumed and $m_{X}=10$ MeV. Our
theoretical predictions are compared with various datasets, shown with
different colors. Note that the black dashed line indicates the FIP-induced
electron signal (scaled by a factor $5\times 10^{-5}$ for sake of clarity)
that can be probed by measurements of the unmodulated electron flux by
Voyager-1. The black solid lines indicate the predicted X-to-$\gamma$-ray
signal, covering the energy range from $1$ keV to $200$ MeV, including the
$511$ keV line. The various measurements considered are obtained in different
observation regions. Thus, we compare data and FIP signals calculated in the
same portion of the sky as indicated in Fig. 4 for each dataset. In
particular, we show data from the MOS detector (XMM-Newton, blue points) [80],
SPI detector on INTEGRAL (red points) [81], COMPTEL (orange points) [82] and
EGRET telescope (green points) [83], representing the X-to-$\gamma$-ray
emission and Voyager-1 (purple points) [52], representing the direct electron
signal. Fig. 4 helps to illustrate the behavior of the FIP-induced signals as
a function of energy and how they compare with data. For instance, the
$\gamma$-ray signal above $\sim 20$ MeV is extremely low compared to EGRET
observations because of the strong suppression of the bremsstrahlung emission
at high energies. Therefore, we do not expect to place competitive constraints
with this instrument. At lower energies, like in the $2$-$20$ MeV range, the
bremsstrahlung emission produces a sizable photon flux, peaked at $\sim 10$
MeV and iscomparable with the COMPTEL measurements. As the energy is lowered,
between $20$ keV and $2$ MeV, the IC on ISRFs becomes responsible for a large
fraction of the X-ray signal. Here, SPI data for the broad emission do not
constitute a strong constraint. By contrast, SPI measurements of the $511$ keV
line set the leading constraints. Finally, in the $1$-$10$ keV region where
MOS data are available, the X-ray emission is dominated by IC emission on CMB
and IR photon background. These low-energy observations are very constraining,
as evident from Fig. 4. For comparison, in Fig. 4 we also show the primary
electron flux at MeV energies and the Voyager-1 measurements, competitive to
X-ray searches in probing exotic electron injections.

In the following, we describe our analysis and results for every energy range
and dataset considered.

###  III.1 $511$ keV signal

Figure 5: Left panel: Latitude profile for the FIP-induced $511$ keV line
signal for our benchmark choice of parameters, $m_{X}=10$ MeV and
$N_{e}=10^{54}$. Observations of SPI are shown in black, with the associated
$1\sigma$ experimental error (calculated as the quadrature sum of systematic
and statistical errors). Results obtained with the different SN distributions
are shown with different colors, green for Lor and blue for Ferr. The profiles
for the 2D distributions, i.e. not considering the spiral arms structure of
the Galaxy, are shown as dashed lines. Right panel: As described in the left
panel, but for the longitude profile and $N_{e}=10^{53}$.

Although obtaining a precise measurement of the galactic diffuse $\gamma$-ray
emission at $511$ keV is challenging given the backgrounds involved and the
technological limitations, the SPI instrument has provided data collected for
more than $20$ years [48, 49]. In particular, the latitude and longitude
profiles of this emission allow us to constrain the injection of positrons in
the Galaxy from different mechanisms. Here, we employ these datasets to derive
bounds on the amount of positrons $N_{e}$ injected from electrophilic FIPs,
calculated with the approach explained in Sec. II.2. We also show how these
limits are sensitive to important systematic uncertainties and we associate an
uncertainty range to the value of $N_{e}$ that can be constrained. We remark
that the energy-dependence of this signal is almost independent on the FIP
mass, which mildly affects the intensity of the signal for $1~{}{\rm
MeV}\lesssim m_{X}\lesssim 20~{}{\rm MeV}$. Above this mass, the FIP
production in SN starts to be Boltzmann suppressed and it is more difficult to
achieve a sizable $N_{e}$ in an unconstrained region of the parameter space,
depending on the FIP model. We mention that in our calculations we are also
including the continuous emission generated by the same lepton population at
$511$ keV, which does not account for more than a $6\%$ of the signal in the
considered parameter range.

In Fig. 5 we show the expected $511$ keV FIP-induced signal compared to the
measured SPI latitude, $N_{e}=10^{54}$, (left panel) and longitude,
$N_{e}=10^{53}$, (right panel) profiles. We show the predicted profiles for
different source distributions, modeling the SN distribution as 2D or 3D, when
neglecting or including the spiral arms structure of the Galaxy. As we show in
the left panel, SPI data of the latitude profile roughly follows the behavior
of the FIP-induced signal. A radically different situation is shown in the
right panel, where the FIP-related signal is peaked at $\sim\pm 20^{\circ}$ in
longitude, in contrast with measurements peaked towards the galactic center.
This is due to the SN distribution, that is not peaked at the galactic center.
This different morphology makes it evident that the $511$ keV signal cannot be
fitted by the proposed exotic injection of positrons. Hence, this kind of
signal is unable to explain the high emission at central longitudes, which is
not fully understood in terms of conventional astrophysical mechanisms either
[84]. However, we can set a $95\%$ Confidence Level (CL) upper limit on
$N_{e}$, the amount of electrons/positrons produced per SN explosion, as
reported in Tab. 2. The comparison between limits obtained for different SN
distributions allow us to see the impact of this important systematic
uncertainty, up to a factor $\sim 2$, in our calculations. As we see from Tab.
2 the latitude profile does not give a strict constraint on $N_{e}$, a factor
$\sim 15$ weaker than the bound from the longitude distribution, reaching
$N_{e}\lesssim 0.86\times 10^{53}$.

$N_{e}(\times 10^{53})$ $95\%$ CL | Lor | Ferr
---|---|---
Latitude profile | $12.9$ | $19.7$
Longitude profile | $0.86$ | $1.31$
Long. profile ($|l|<20^{\circ}$) | $1.52$ | $2.50$
Smeared Lat. profile | $2.60$ | $4.00$
Smeared Long. profile | $0.17$ | $0.20$

Table 2: Upper limits at $95\%$ CL on $N_{e}$ for the Lorimer (Lor) and
Ferriere (Ferr) SN spatial distributions for the latitude and longitude
profiles of the $511$ keV line [49]. More details are provided in the text.

The obtained bounds are about one order of magnitude weaker than the upper
limit $N_{e}\lesssim 1.4\times 10^{52}$, obtained in Refs. [46, 47]. This
discrepancy can be explained by the more realistic SN distribution used in
this work, where we account for the spiral arms structure of the Galaxy.
Moreover, up to this point we neglected the smearing effect due to positron
propagation before annihilation [46, 47]. We also derive limits on $N_{e}$
taking this effect into account. We employ the same procedure as used in Eq.
(18) of Ref. [46] imposing a smearing scale of $1$ kpc. In Fig. 6 we show the
effect of smearing in the predicted $511$ keV longitude profile for the Ferr
SN distribution. As we see, smearing the signal leads to change in the
morphology of the profile and allows us to set the most optimistic
constraints, as displayed in Tab. 2. These limits are a factor of a few better
than using our main analysis and are in very good agreement with the results
of Refs. [46, 47].

Figure 6: Comparison of the predicted $511$ keV longitude profile for the Ferr
3D SN distribution accounting (green line) and not accounting (blue line) for
smearing of the signal. The predicted profile without smearing is multiplied
by a factor of $5$ to facilitate the comparison. Here, $N_{e}=2\times
10^{52}$.

However, in contrast with Refs. [46, 47] we find that the smearing effect acts
to strengthen the constraint instead of making it weaker. This finding can be
understood by examining the right panel of Fig. 5. Here, we highlight how the
3D spiral structures suppress the X-ray flux in the most constraining bin at a
longitude $l\sim 30^{\circ}$ compared to smooth 2D distributions employed in
previous analyses. Thus, roughly speaking, smearing our 3D distributions make
the profiles more similar to the 2D ones and the bound is subsequently
strengthened. This conclusion is confirmed by the observation that a smearing
of the 2D distribution makes the bound weaker, as in Refs. [46, 47].
Therefore, 3D distributions introduce inhomogeneities on very small angular
scales, that might weaken the bound, especially at large longitudes, where the
most constraining data points are. We do not expect that these structures are
observable because of the smearing effect, leading to a stronger constraint.
It is important to mention that the approximate smearing of the distribution
is a very simple prescription to roughly approximate realistic particle
diffusion.

At this point we comment on why constraints obtained with high longitude data
might lead to artificially stronger bounds. In Fig. 5 we observe that some of
the data points for $|l|>20^{\circ}$ show negative values of the diffuse flux
at $511$ keV. This is due to the fact that there are important systematic
uncertainties in the derivation of these measurements and it is not easy to
estimate them for these high-longitude points. Indeed, even the $1\sigma$
upper limit for the most constraining data point at $l\sim 30^{\circ}$ is
negative and the bound is set by the $2\sigma$ upper limit. This feature is
responsible for making this constraint so stringent. Issues with the
background subtraction at high longitudes can be easily understood if one
considers a roughly constant background noise emission: the measurement where
the signal is lower would have a lower associated signal-to-noise ratio,
leading to more significant uncertainties. Therefore, in order to avoid the
measurements most affected by these systematic uncertainties, we derive
conservative limits from the longitude profile, removing large longitude data
points in the region $|l|>20^{\circ}$. These constraints, for the different SN
distributions, are shown in the third line of Tab. 2, and constitute the most
conservative ones, being around a factor $\sim 2$ weaker than the non-smeared
constraints previously discussed. This analysis allows us to associate
uncertainty bands on the $511$ keV bound and subsequently $N_{e}$. More
precisely, the most stringent bound is obtained by high longitude data points
including smearing effects and the 3D Lor SN profile

$N_{e}\lesssim 0.17\times 10^{53}\,.$ (8)

The weakest constraint is obtained by conservatively neglecting smearing
effects, considering only low-longitude data points and the 3D Ferr SN
distributions

$N_{e}\lesssim 2.50\times 10^{53}\,.$ (9)

Therefore we can conservatively associate more than one order of magnitude of
uncertainty to this constraint.

###  III.2 Local electron/positron measurements

Electrons generated in electrophilic FIP decays would contribute to the
galactic diffuse electron population at MeV energies. This allows us to use
Voyager-1 measurements of the local electron flux to constrain their
injection. For a simple order of magnitude estimate, we can evaluate the flux
of injected electrons in the Galaxy as

$\frac{d\Phi_{e}}{dE_{e}}\simeq\frac{\Gamma_{cc}}{4\pi
d_{SN}^{2}}\frac{N_{e}}{\Delta E_{e}}\,,$ (10)

where $\Delta E_{e}\simeq 50$ MeV is the energy range over which electrons are
injected. Comparing this number with the electron flux measured by Voyager-1
in Fig. 7 (black points), we realize that the FIP-induced flux leaves an
imprint on this observable if $N_{e}\simeq 10^{54}$. This expectation is
confirmed by the full numerical evaluation of the electron flux (blue line),
shown in Fig. 7. This is compared to the Voyager-1 measurements in the
$4$-$10$ MeV range. Here, we have added a line obtained from a power-law fit
of the Voyager-1 data that allows us to illustrate how our limit on $N_{e}$
could improve (up to a factor of $3$) if data at slightly higher energies
would be available. We notice that the uncertainties related to determining
the propagation parameters play a minor role in this energy range. However, we
observe that the main uncertainty could be the halo height employed, for which
these signals scale as the square root (the larger the halo height, the larger
the signal). Hence the limits derived from the Voyager-1 data have an
associated systematic uncertainty of less than $50\%$, corresponding to
extreme values of the halo height in the range $3$-$20$ kpc.

Figure 7: Comparison of the electron flux measured by Voyager-1 with the FIP-
induced signal for $N_{e}=1.3\times 10^{54}$ and $m_{X}=10$ MeV. We include,
as an orange dashed line, the result of a power-law fit to this data for
completeness (see the text for more details).

Figure 8: Left panel: Effect of reacceleration on the predicted FIP-induced
electron signal for $N_{e}=10^{53}$ and $m_{X}=10$ MeV. Comparison of the
electron Voyager-1 data with the FIP-induced electron spectrum for different
values of the Alfvèn velocity (VA). Right panel: Similar to the left panel,
but for positron Solar-modulated spectra compared to the AMS-02 positron data.

The left panel of Fig. 8 aids in illustrating the weak dependence of the
Voyager-1 bound to systematic uncertainties. Precisely, this comparison shows
the effect of reacceleration. At energies above $\sim 100$ MeV the effect of
reacceleration in the spectrum of the FIP-induced electron population becomes
dramatic. It basically implies a gain of energy for low-energy CR particles
due to the interaction with magnetic plasma waves. The effect of
reacceleration is proportional to the square of the speed of the plasma waves,
i.e. the Alfvèn velocity and inversely proportional to the spatial diffusion
coefficient. In conclusion, different Alfvèn velocities do not affect the
limits from Voyager-1, since the electron spectrum remains roughly unchanged
below $\sim 20$ MeV, while these fluxes are enhanced by orders of magnitude
above $100$ MeV depending on the exact value of this parameter.

In principle, for high reacceleration, we could derive constraints on $N_{e}$
from positron AMS-02 data at GeV energies (see the right panel of Fig. 8). The
solar-modulated spectrum of the FIP signals for different values of the Alfvèn
velocity, from $5$ km/s to $40$ km/s is shown. Because of the lower background
production of positrons, the positron channel would provide stronger
constraints than the electron flux at those energies. Remarkably, we observe
that for large enough values of the Alfvèn velocity ($V_{A}\sim
30$-$40\operatorname{km}/\operatorname{s}$) we would be able to obtain strong
constraints from AMS-02. As opposed to the Voyager-1 data, that are obtained
from outside the heliosphere, AMS-02 is located near Earth, meaning that the
particles that it measures are affected by the heliosphere and the solar
magnetic fields. Unfortunately, the interaction with the heliosphere
introduces a strong suppression in the flux of these particles for small
reacceleration. Analyses of secondary CRs usually favor values of the Alfvèn
velocity between a few km/s and $\sim 30$ km/s [85], so values above this
would be in conflict with CR observations. As a matter of fact, our benchmark
value is $13$ km/s, which yields a FIP-induced signal many orders of magnitude
below AMS-02 data. The main reason that Voyager-1 data can provide stronger
constraints than AMS-02, is the suppression of the flux of CR particles from
their interaction with the heliosphere. Regarding positron fluxes, we cannot
set constraints on FIPs because of the strong variability of the signal with
the Alfvèn velocity uncertainties. In the case of maximal reacceleration the
constraint would be comparable with the Voyager-1 bound.

In conclusion, we observe that Voyager-1 provides very valuable observations
to constrain electrophilic FIPs at MeV masses. In particular, the limits
derived from Voyager-1 data are of the order of $N_{e}\simeq 1$-$2\times
10^{54}$, which are compatible with those found from the analysis of the SPI
profiles. Additionally, we remark that these bounds are much less affected by
the different sources of systematic uncertainties present in the modeling of
the $511$ keV signal.

###  III.3 X-to-$\gamma$-ray data

If electrophilic FIPs produce electrons and positrons with energies above tens
of MeV, these can emit high-energy photons via IC and bremsstrahlung. This
flux is probed by diffuse $\gamma$-ray observations from keV to MeV energies
in various galactic regions. The quality of data in this energy range is not
as high as it is at the GeV scale, mainly due to the high instrumental
backgrounds. We have collected a variety of valuable measurements and show
that by using different observational data, referring to different energy
ranges and angular positions in the sky, we can set bounds on FIPs competitive
with ones arising from the $511$ keV line and direct observations of
electrons. In particular, we use the observations from the following
experiments:

  * •

EGRET [83] measurements at energies higher than $20$ MeV and above the
galactic plane,

  * •

COMPTEL [82] data in a region around the center of the Galaxy in the range
$2$-$20$ MeV,

  * •

SPI [86, 81] data between $20$ keV and $2$ MeV for two sectors of the sky
around the galactic center,

  * •

MOS [80] (in the XMM-Newton mission) observations from $2.5$ to $8$ keV in
several angular rings around the center of the Galaxy.

With the combination of these datasets we cover a wide energy range, from keV
to tens of MeV, for different regions of the sky, making a comprehensive
evaluation of the FIP constraints possible. In all the cases we consider the
full uncertainties (statistical and systematic errors) reported by the
experiments and obtain limits at $95\%$ CL from a $\chi^{2}$ analysis. The
obtained limits are shown in Tab. 3. In the following, we discuss the results
from each dataset.

|  $N_{e}(\times 10^{55})$ $95\%$ CL
---|---
EGRET | $2.2$
COMPTEL | $0.18$
SPI ($|l|=|b|<47.5^{\circ}$) | $6.2$
SPI ($|b|<15^{\circ}$, $|l|<30^{\circ}$) | $2.1$
MOS/XMM (ring 1) | $0.35$
MOS/XMM (ring 3) | $0.047$
MOS/XMM (ring 4) | $0.06$
MOS/XMM (ring 30) | $10.1$
MOS/XMM (Combined) | $0.027$

Table 3: Limits on $N_{e}$ at the $95\%$ CL derived from the fit of the
predicted secondary X-to-$\gamma$-ray emissions to the datasets discussed in
the text.

Constraints from EGRET data: In the upper left panel of Fig. 9, we show the
comparison of the FIP-induced signals expected for $N_{e}=2\times 10^{55}$ and
$m_{X}=10$ MeV with the data extracted from Ref. [83], for the region of the
Galaxy at $20^{\circ}<|b|<60^{\circ}$ and $0^{\circ}<l<360^{\circ}$. In this
case, the peak of the bremsstrahlung emission, at $E_{\gamma}\sim 10$ MeV,
lies at energies below the available EGRET data and, therefore, the signal’s
flux falls down rapidly in the EGRET measurements energy range, leading to
weak constraints in comparison to those derived from the $511$ keV line or
Voyager-1 data. However, the lowest-energy data points allow us to still
derive relevant constraints from this dataset.

Figure 9: Comparison of different measurements with the predicted secondary
signal for $m_{X}=10$ MeV and evaluated in the region of the sky where the the
data is extracted in each case. We show the total $1\sigma$ error bars
reported by the experiments. Top-left panel: Comparison with EGRET (CGRO)
data, extracted from Ref. [83]. Top-right panel: COMPTEL (CGRO) [82]. Middle-
left panel: SPI (Integral) [86]. Middle-right panel: SPI (Integral) [81].
Bottom-left panel: MOS data at the Ring 3 [80]. We shade the region not used
in the fits to this data since we consider only the range $2.5$-$8$ keV.
Bottom-right panel: MOS data at Ring 30 [80].

Constraints from COMPTEL data: In the upper right panel of Fig. 9, we show how
the expected FIP signals compare to COMPTEL data [82] for the region
$|b|<20^{\circ}$ and $|l|<60^{\circ}$. Here, we only use the data points above
the region of the emission lines at $2.23$ MeV, produced from the formation of
deuterium in the interstellar medium, and $\simeq 1.8$ MeV where lines are
caused by the decay of 26Al into 26Mg and its subsequent de-excitation. We are
not including these sources of background, whose modeling is far beyond the
scope of this work. Since the bremsstrahlung peak emission coincides with the
position of these data points, strong constraints can be derived using this
dataset, competitive to those from the $511$ keV SPI latitude profiles and
those from Voyager-1 data. In particular, we obtain an upper limit in the
number of electrons/positrons injected of $N_{e}\sim 1.8\times 10^{54}$ at
$95\%$ CL (see Tab. 3).

Constraints from SPI data: In the middle left panel of Fig. 9 we show a
comparison of the FIP-induced signal with the data of the diffuse $\gamma$-ray
emission derived in Ref. [86] for a region with latitude and longitude smaller
than $47.5^{\circ}$ around the galactic center. This dataset lies slightly
below the bremsstrahlung peak emission and leads to very weak constraints on
the injected number of electrons and positrons from electrophilic FIPs when
compared to the other limits derived in this work.

In the middle right panel of Fig. 9, we show the SPI data obtained by Ref.
[81] at $|b|<15^{\circ}$ and $|l|<30^{\circ}$. Note that this is a region very
concentrated around the galactic center, leading to slightly stronger
constraints than in the previous case. This dataset covers the energy region
from $\sim 20$ keV to $\sim 2$ MeV, where the FIP emission transitions from
being dominated by the bremsstrahlung emission to being dominated by the IC
emission (below $\sim 1$ MeV). In this figure, for illustrative purposes we
show the associated line emission at $511$ keV from positronium annihilation.
Although we remark that the constraint on $N_{e}$ from this dataset is derived
using the continuous emission. As we see, the line emission exceeds by far the
measurements at $511$ keV for electron injection values which are not ruled
out by SPI continuous emission. These limits are shown in Tab. 3.

Figure 10: Representation of the limits on $N_{e}$ at the $95\%$ CL derived
from the fit of the predicted secondary X-to-$\gamma$-ray emissions to the
datasets discussed in the text. The discussed uncertainties in the evaluation
of the limits from the profiles of the $511$ keV emission are shown as hatched
bars. These limits are valid for FIP masses below $\sim 20$ MeV.

Constraints from XMM data: We use the data from the MOS detector in the energy
range from $1$ keV to above $10$ keV, which was provided by Ref. [80] divided
in galactocentric rings that are $6^{\circ}$ wide. In this case, the most
central rings lead to a higher FIP-induced signal since the density of SN
tends to increase towards the galactic center. We observe that this is the
trend that our constraints follow, being stronger for the inner rings and
weaker in the external rings. However, the limits derived from the two
innermost rings (Rings 1 and 2) are weaker than the limit derived from Ring 3
(the strongest limit), as it was observed in Ref. [87], which is likely due to
the very high astrophysical backgrounds and lower FIP-induced signals that are
present in the very central regions. Here, we show the comparison of the
calculated signals with MOS data at the most constraining ring (Ring 3) and
the most external ring (Ring 30) in the lowest row of Fig. 9. We observe that,
while an injection of electrons of $N_{e}=5\times 10^{53}$ is already above
MOS data in the $2$-to-$4$ keV range for Ring 3 (left panel), the most
external ring only allow us to constrain $N_{e}$ to be smaller than $\sim
1\times 10^{56}$. In particular, the $95\%$ CL limits that we show in Tab. 3
are derived with the data from $2.5$ to $8$ keV, as performed by Refs. [80,
87], due to the background noise in the detector at lower and higher energies
respectively. In addition to this, the combination of all the rings allows us
to set a constraint that is a factor 2 stronger. Remarkably, these are the
strongest limits derived from the continuous X-to-$\gamma$-ray emission. They
are notably even stronger than those from the Voyager-1 electrons, and
competitive with those of the longitude profile of the $511$ keV line, which
were the strongest imposed so far. This, added to the fact that MOS
measurements and its uncertainties are much more robust than those evaluated
for the $511$ keV line, makes the MOS instrument exceptionally valuable for
constraining FIPs. In addition, we remark that higher reacceleration values,
improve these limits by a factor of a few, making the MOS data even more
valuable for constraining electrophilic FIP signals.

Summary of the constraints: In Fig. 10, we summarize the limits obtained on
the number of electrons/positrons injected by FIPs with $m_{X}=10$ MeV. We
expect these constraints to be valid for the entire $1$-$20$ MeV range of FIP
masses. We also show hatched regions for the limits extracted from the
emission profile of the $511$ keV line, representing the uncertainties that we
have estimated and propagated. Notably, the data from the MOS detector leads
to constraints compatible with the lower limit obtained from the longitude
profile of the $511$ keV line. We remark that, as we commented above, SPI data
seems to be largely affected by systematic uncertainties not well accounted
for, while the measurements by the MOS detector seem to be very robust,
leading to more reliable constraints. Similar reasoning applies for
theoretical uncertainties on the calculation of these fluxes. Additionally, we
also notice that Voyager-1 data provides constraints that are compatible with
those from the latitude profile of the $511$ keV line. Among the $\gamma$-ray
datasets used here, the measurements from COMPTEL provide the strongest
constraints.

| ALPs |  DPs |  $\nu_{s}-\mu s$ |  $\nu_{s}-\tau s$
---|---|---|---|---
$\alpha$ | 0 | 2 | 1.64 | 1.52
$N_{0}$ | $5.2\times 10^{56}$ | $1.5\times 10^{57}$ | $1.3\times 10^{59}$ | $1.3\times 10^{59}$
$g_{0}$ | $g_{ae}=2.7\times 10^{-13}$ | $\epsilon=7.9\times 10^{-11}$ | $|U_{\mu s}|=1.8\times 10^{-2}$ | $|U_{\tau s}|=0.11$
$\tau(g_{0})$ | 0.14 | 3.06 | 0.004 | 0.37
Table 4: Parameters of Eq. (11) to fit $N_{e}$ as function of the FIP coupling
for ALPs, DPs and sterile neutrinos mixed with muon or tau flavors.

##  IV Application to FIP constraints

The upper limits on the electron/positron injection discussed in Sec. III can
be applied to several FIP models. In the literature, some interest was devoted
towards constraining sterile neutrino properties [43, 47], ALPs [46] and DPs
[47]. The same models can be studied in the light of the complementary
constraints introduced in this work. If these particles exist in nature, we
expect them to be produced in SN and possibly have a coupling with
electrons/positrons. Even though accurately revisiting all these constraints
is beyond the scope of this work, we will briefly sketch how the existing
constraints improve and become more robust.

In a very general fashion, we can write down the number of positrons produced
by FIPs in a single SN explosion as function of the FIP-electron coupling $g$

$\begin{split}N_{e}=N_{0}\left(\frac{g}{g_{0}}\right)^{\alpha}e^{-\left(\frac{g}{g_{0}}\right)^{2}\tau(g_{0})}\,,\end{split}$
(11)

where $N_{0}$, $g_{0}$ and $\tau(g_{0})$ are parameters fitting various
physical situations. The parameter $\alpha=\\{0,2\\}$ represents the
possibility of having production of FIPs and their decay controlled by the
same coupling $g$ or not. Namely, as discussed in Ref. [46] a copious amount
of positrons is produced by ALPs emitted by a SN via their coupling to protons
and decaying via their electron-interaction $g=g_{ae}$. In this case, since
production and decay are caused by different couplings, $\alpha=0$. Otherwise,
in the most common case, both decay and production are proportional to the
same coupling and $\alpha=2$, like for DPs, mixed with photons with a coupling
$g=\epsilon$ and sterile neutrinos mixed with active ones through
$g=|U_{\alpha s}|$ for $\alpha=\mu,\tau$ [47]. In the case of sterile
neutrinos $\alpha$ can assume a value different from $2$ in order to
schematically take into account the contribution of type Ib/c SN to the
positron production. The behavior of the fitting function in Eq. (11) can be
understood by looking at the extreme regimes of free-streaming,
$(g/g_{0})^{2}\tau(g_{0})\ll 1$, and trapping $(g/g_{0})^{2}\tau(g_{0})\gg 1$.
In the first case, for $\alpha=2$, the positron number increases proportional
to $g^{2}$, since the FIP production increases. In the opposite limit, the
exponential suppression is severe because of FIPs decaying inside the SN
envelope and no positrons being able to escape the star. In these terms, the
requirement that FIPs decay close to the SN translates into a lower limit on
the FIP opacity

$\left(\frac{g}{g_{0}}\right)^{2}\tau(g_{0})\gtrsim\frac{R_{*}}{R_{\rm
dec}}\simeq 3\times 10^{-11}\,,$ (12)

where $R_{*}\simeq 10^{7}$ km is the SN (Ib/c) envelope radius and $R_{\rm
dec}\lesssim 10$ kpc is the average distance between SN.

We use Eq. (11) to fit the positron production from ALPs (with $g_{ae}\gtrsim
10^{-14}$), DPs and sterile neutrinos, all the particles having a mass of 10
MeV. We summarize the fitting parameters in Tab. 4, based on the numerical
calculations presented in Refs. [46, 47, 88]. It is easy to verify that, with
the reported parameters, the condition in Eq. (12) is always satisfied for the
parameter range considered. Therefore, for most of the considered FIP models,
except for ALPs with weak coupling to electrons, our assumption that the
positron injection happens close to the SN is confirmed. Now we are in a
position to estimate the exclusion regions for the various FIP parameters,
given the phenomenology discussed in this work. From Fig. 10 we notice that
the SPI constraint on the 511 keV line longitudinal profile is the strongest
constraint on the electron-positron injection from SN, reaching

$N_{e}\lesssim 1.7\times 10^{52}\,,$ (13)

in agreement with Refs. [47, 46]. In this work we showed that this constraint,
even though it is the strongest one, is affected by several theoretical
uncertainties related to the SN distribution, choice of the most constraining
dataset and modeling of the positron propagation in the Galaxy (see Sec.
III.1). These uncertainties may relax the bound by up to one order of
magnitude. We propose a more robust bound set by MOS observations of the IC
signal at a few keV caused by the leptons injected by FIP decays. With these
observations we are able of setting a very robust constraint of

$N_{e}\lesssim 2.7\times 10^{53}\,.$ (14)

A summary of the various constraints translated to FIP couplings are shown in
Fig. 11. Note that these are only rough estimates of the constraints because
we used the approximate fitting function in Eq. (11) and not the full
numerical approach, approximating the non-trivial effect of type Ib/c SN [47].
For a first estimate, we expect this approach to be reasonably accurate and
especially useful to give an idea of how the different constraints reflect on
FIP properties. As expected, the MOS bound is comparable with the 511 keV one,
giving the most stringent and robust constraint on $N_{e}$. In addition, the
Voyager-1 and COMPTEL constraints help to set a slightly weaker constraint.
This combination of data allows us to robustly exclude a large portion of the
various FIPs parameter space. We leave a more accurate evaluation of the
various constains for FIPs of different masses for a future work [72].

Figure 11: Limits on the considered FIP models with $m_{X}=10$ MeV at the
$95\%$ CL: ALPs (upper left panel), DPs (upper right panel), sterile neutrinos
mixed with muon (lower left panel) and tau neutrinos (lower right panel). The
most conservative constraints are shown for the SPI bound on the 511 keV line.
The color code is in agreement with Fig. 10. Note that the ALP constraint
extends to small couplings $g_{ae}\sim\mathcal{O}(10^{-14})$ until the
condition in Eq. (12) is satisfied. The sterile neutrino mixing angle in the
lower left panel, in the upper limit, is assumed to saturate to one.

##  V Conclusions

Supernovae (SNe) provide valuable information about the existence and
properties of electrophilic Feebly Interacting Particles (FIPs), which are
essential for investigating exotic physics. The objective of this research is
to analyze and constrain the production of FIPs in SNe using different
astrophysical messengers, thereby playing a vital role in exploring the
interplay between new physics and astrophysical probes.

Our study focuses on exploring the multimessenger signals generated by FIPs
that decay into electron-positron pairs. We aim to demonstrate the
effectiveness of this phenomenology in constraining the production and
properties of various FIPs, such as axion-like particles, sterile neutrinos,
and dark photons. The production of electrons and positrons by FIPs
contributes to the overall diffuse background of these particles in our
Galaxy. Additionally, when these particles interact with the interstellar gas,
they give rise to secondary emissions of photons across a wide spectrum,
ranging from X-rays to $\gamma$ rays. By accurately modeling these emissions
and considering the $511$ keV signal produced by positron annihilation, we can
establish constraints on the total flux of injected electrons and positrons,
regardless of the specific FIP model being considered.

We have significantly improved constraints on FIP models when comparing with
previous studies. This includes refining the assessment of uncertainties in
the SPI analysis, in order to establish more reliable constraints. Using
measurements from Voyager-1, accurately determining how the local electron
flux beyond the heliosphere allows us to constrain FIPs, for the first time.
Furthermore, we have systematically examined secondary emissions from
electron-positron pairs, specifically the inverse Compton and bremsstrahlung
processes, by analyzing available datasets from X-ray to $\gamma$-ray
energies. These additional data provide stringent limits on electrophilic
FIPs. In addition, after obtaining limits on the production of these particles
and their injection of electrons/positrons, we show how these limits can be
translated to concrete parameters defining the properties of axion-like
particles, sterile neutrinos, and dark photons.

In summary, our study emphasizes the importance of galactic signals produced
by SNe in the exploration of FIPs. Our journey into the phenomenology of
electrophilic FIPs from SN started in Ref. [50], where we proved the
effectiveness of the XMM-Newton data in constraining exotic physics. Moving
one step further, in this work we employ a multimessenger approach
incorporating observations from diverse cosmic messengers to enhance our
comprehension of the FIP phenomenology. The exhaustive analysis that we
carried out sets a new standard for the methodologies employed in this type of
investigation. Our research establishes a solid groundwork for future
inquiries to shed light on the characteristics of FIPs and their significance
in astroparticle physics.

###### Acknowledgements.

SB would like to thank Jordan Koechler and Marco Cirelli for useful
discussions regarding XMM-Newton data. This article is based upon work from
COST Action COSMIC WISPers CA21106, supported by COST (European Cooperation in
Science and Technology). SB is supported by funding from the European Union’s
Horizon 2020 research and innovation programme under grant agreement No.
101002846 (ERC CoG “CosmoChart”) as well as support from the Initiative
Physique des Infinis (IPI), a research training program of the Idex SUPER at
Sorbonne Université. The work of PC is supported by the European Research
Council under Grant No. 742104 and by the Swedish Research Council (VR) under
grants 2018-03641 and 2019-02337. PDL is supported by the European Research
Council under grant 742104 and the Swedish National Space Agency under
contract 117/19 This project used computing resources from the Swedish
National Infrastructure for Computing (SNIC) under project Nos. 2021/3-42,
2021/6-326, 2021-1-24 and 2022/3-27 partially funded by the Swedish Research
Council through grant no. 2018-05973.

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