# Long-lived particles and the Quiet Sun

R. Andrew Gustafson<EMAIL_ADDRESS>Center for Neutrino Physics, Department of
Physics, Virginia Tech, Blacksburg, VA 24061, USA  Ryan Plestid
<EMAIL_ADDRESS>Walter Burke Institute for Theoretical Physics,
California Institute of Technology, Pasadena, CA 91125, USA  Ian M. Shoemaker
<EMAIL_ADDRESS>Center for Neutrino Physics, Department of Physics, Virginia
Tech, Blacksburg, VA 24061, USA  Albert Zhou  Institut für
Astroteilchenphysik, Karlsruher Institut für Technologie (KIT), D-76021
Karlsruhe, Germany

###### Abstract

The nuclear reaction network within the interior of the Sun is an efficient
MeV physics factory, and can produce long-lived particles generic to dark
sector models. In this work we consider the sensitivity of satellite
instruments, primarily the RHESSI Spectrometer, that observe the Quiet Sun in
the MeV regime where backgrounds are low. We find that Quiet Sun observations
offer a powerful and complementary probe in regions of parameter space where
the long-lived particle decay length is longer than the radius of the Sun, and
shorter than the distance between the Sun and Earth. We comment on connections
to recent model-building work on heavy neutral leptons coupled to neutrinos
and high-quality axions from mirror symmetries.

††preprint: CALT-TH/2023-023

##  I Introduction

It has long been recognized that the solar interior can serve as an efficient
factory for keV-scale physics beyond the Standard Model (BSM), e.g. solar
axions and dark photons Sikivie (1983); Dimopoulos  _et al._ (1986); Avignone
_et al._ (1987); Raffelt (1990); Battesti  _et al._ (2008); Pospelov  _et al._
(2008); An  _et al._ (2015); Graham  _et al._ (2015). In addition to thermal
production mechanisms, nuclear reactions within the Sun may also source BSM
particles up to masses and energies of roughly $15~{}{\rm MeV}$ Raffelt and
Stodolsky (1982); Raffelt (1990); Andriamonje  _et al._ (2010); Bellini  _et
al._ (2012, 2013a); Plestid (2021a, b). If a flux of long-lived particles
(LLPs) in this energy regime emanates from the solar interior, they may
transit toward the Earth and their decay products can leave detectable
signatures. It is important to emphasize that LLPs are generic consequences of
a dark sector with relatively light particles and feeble couplings to the SM
Beacham  _et al._ (2020); Agrawal  _et al._ (2021); Curtin  _et al._ (2019);
Knapen and Lowette (2022). As decay lengths become long, LLPs become
increasingly difficult to detect and strategies to attack this “lifetime
frontier” are valuable tools in the search for BSM physics. This idea has been
previously investigated, largely considering FERMI-LAT, in the high energy,
i.e. $\gtrsim 100~{}{\rm MeV}$, regime for annihilating dark matter Schuster
_et al._ (2010a); Batell  _et al._ (2010); Leane  _et al._ (2017); Leane and
Linden (2021).

In this work we point out that existing data from the RHESSI satellite
spectrometer Smith  _et al._ (2003), which observed the Quiet Sun,111Time
periods without intense surface activity such as solar flares. can place
interesting limits on dark sectors with LLPs in the range of
$\mathcal{O}(100\,\rm keV)-\mathcal{O}(1\,\rm MeV)$. This is an old idea,
first proposed by Stodolsky and Raffelt in 1982 in the context of a 200 keV
axion Raffelt and Stodolsky (1982), however, it has remained unexplored
despite new data in the intervening decades Dennis  _et al._ (2022). We
illustrate the potential sensitivity of Quiet Sun data with a number of BSM
examples, emphasizing different production mechanisms which may operate in
this mass window. A conservative analysis of existing data from RHESSI is
capable of offering complimentary constraints on production mechanisms
involving neutrino upscattering, and can probe previously untouched regions of
parameter space for axion like particles (ALPs) with masses close to $\sim
1~{}{\rm MeV}$. Upcoming missions, such as the COSI satellite Team ; Tomsick
(2021), may be able to substantially improve on the capabilities of RHESSI by
i) taking advantage of a larger instrument surface area, ii) making use of
dead time to carefully study backgrounds, and iii) taking advantage of
distinctive spectral features.

We focus on LLPs that decay primarily to photons,222We could also consider
decays to $e^{+}e^{-}$ pairs but an analysis is complicated by the magnetic
fields that surround the Earth. and have decay lengths, $\ell_{\rm LLP}$, that
satisfy

$R_{\odot}\ll\ell_{\rm LLP}\ll d_{\odot}~{},$ (1)

where $R_{\odot}$ is the radius of the Sun and $d_{\odot}$ is the distance
from the Sun to the Earth. This allows an $O(1)$ fraction of the LLPs to decay
en-route to the satellite instrument. In this limit, the flux of LLPs will
never reach any terrestrial experiment since they will decay in flight and
their daughter photons will be absorbed in the upper atmosphere. In this
sense, Quiet Sun observations are complimentary to terrestrial searches for
LLPs from the Sun such as those that have been performed by CAST Andriamonje
_et al._ (2010) and Borexino Bellini  _et al._ (2012).

We perform a straightforward (and conservative) rate-only analysis, the
details of which can be found at the end of Section II. In the body of the
paper we organize our discussion along the lines of specific BSM scenarios. We
discuss neutrino upscattering in Section II and solar axion production in
Section III. We also spend time focusing on model-independent LLP constraints
in Section IV. In Section V we discuss the physics potential for dark sector
searches using future missions such as COSI. We close by summarizing our
results in Section VI.

##  II Neutrino upscattering - Transition Dipole

Figure 1: The flux of solar HNLs at Earth (ignoring decays) as calculated
through the dipole model Monte Carlo simulation, where $m_{N}=0.75$ MeV and
$d_{\mu}=2\times 10^{-11}$ $\mathrm{MeV}^{-1}$.

We begin by considering a production mechanism involving the upscattering of
solar neutrinos transiting through the Sun, e.g. $\nu A\rightarrow{\rm LLP}A$
with a $A$ a nucleus such as hydrogen or helium (see e.g. Plestid (2021a, b);
Gustafson  _et al._ (2022) for results on neutrino upscattering in the Earth).
This mechanism leverages the large solar neutrino flux which is copious in the
few-hundred keV region, and extends up to $\sim 15~{}{\rm MeV}$. Solar
neutrinos have a small probability of being absorbed in the SM because of the
small charged current scattering cross section at $E_{\nu}\sim{\rm MeV}$
energies. It is, however, possible to have BSM cross sections that exceed the
weak interaction at low energies if neutrinos couple via a transition magnetic
dipole moment Magill  _et al._ (2018); Brdar  _et al._ (2021). This can lead
to sizable conversion probabilities into an unstable right-handed neutrino,
$N$ (also called a heavy neutral lepton or HNL), for neutrinos transiting from
the center to the surface of the Sun. As it is unstable, $N$ may decay in
flight supplying a broad flux of photons in RHESSI. Similar phenomena may
occur in the aftermath of SN 1987A Brdar  _et al._ (2023); Hostert and
Pospelov  leading to tight limits below the supernova floor derived in Magill
_et al._ (2018).

Figure 2: The flux of photons at RHESSI from $N$ decays calculated using a
Monte Carlo integration with a $90^{\circ}$ opening angle, compared with the
RHESSI background in the front segment. Since the flux from decays exceeds the
background, we consider $m_{N}=0.75$ MeV, $d_{\mu}=2\times
10^{-11}\,\mathrm{MeV}^{-1}$ to be excluded.

This “dipole portal” can dominate low energy phenomenology since it is a
dimension-five operator and competes with the dimension-six four-Fermi contact
interaction at low energies. The effective Lagrangian is given by

$\mathcal{L}_{\rm
int}\supset\sum_{\alpha}d_{\alpha}F^{\mu\nu}\bar{N}\sigma_{\mu\nu}P_{L}\nu_{\alpha}~{}.$
(2)

Here, $d_{\alpha}$ represents the coupling between $N$ and each of the 3 SM
neutrinos. In this work, we consider the cases where $N$ couples to a single
flavor. This effective interaction has been studied recently in the context of
accelerator, solar, atmospheric, and collider neutrinos as well as in the
context of early universe cosmology and constraints from SN 1987A Gninenko
(2009, 2011); McKeen and Pospelov (2010); Masip and Masjuan (2011); Masip  _et
al._ (2013); Coloma  _et al._ (2017); Magill  _et al._ (2018); Brdar  _et al._
(2021); Shoemaker and Wyenberg (2019); Argüelles  _et al._ (2019); Hostert
(2019); Fischer  _et al._ (2020); Coloma  _et al._ (2020); Schwetz  _et al._
(2020); Arina  _et al._ (2021); Shoemaker  _et al._ (2021); Abdullahi  _et
al._ (2021); Shakeri  _et al._ (2020); Atkinson  _et al._ (2021); Cho  _et
al._ (2021); Kim  _et al._ (2021); Argüelles  _et al._ (2021); Ismail  _et
al._ (2021); Miranda  _et al._ (2021); Bolton  _et al._ (2021); Jodłowski and
Trojanowski (2021); Vergani  _et al._ (2021).

Unlike the monoenergetic LLP cases discussed later in this paper, the spectrum
of $E_{\nu}$ (and hence $E_{N}$) spans several orders of magnitude. For that
reason, we implement a Monte Carlo integration to sample neutrino energy,
production location, and upscattering location. We also account for flavor
transformation during the neutrino propagation (both due to adiabatic
conversion and oscillations).

We consider the Sun to be solely comprised of 1H and 4He with densities given
by the Standard Solar Model Bahcall and Pinsonneault (2004); Bahcall  _et al._
(2006); Bahcall . All scattering is calculated to be off free nucleons,
ignoring the coherent enhancement due to helium. This only leads to an $\sim
10\%$ change in the bounds, which we will see is a much smaller effect than
uncertainty in the detector opening angle/background. The cross section for
scattering on a free proton is given by
$\differential\sigma_{\mathrm{dip}}=\differential\sigma_{1}+\differential\sigma_{2}$
with,

$\begin{split}\frac{\differential\sigma_{1}}{dE_{r}}=&\alpha(2d)^{2}F_{1}^{2}\bigg{(}\frac{1}{E_{r}}-\frac{1}{E_{\nu}}\\\
&+\frac{m_{N}^{2}(E_{r}-2E_{\nu}-m_{p})}{4E_{\nu}^{2}E_{r}m_{p}}+\frac{m_{N}^{4}(E_{r}-m_{p})}{8E_{\nu}^{2}E_{r}^{2}m_{p}^{2}}\bigg{)}~{},\end{split}$
(3)

and

$\begin{split}\frac{\differential\sigma_{2}}{dE_{r}}=&\alpha
d^{2}\mu_{n}^{2}F_{2}^{2}\bigg{[}\frac{2m_{p}}{E_{\nu}^{2}}\big{(}(2E_{\nu}-E_{r})^{2}-2E_{r}m_{p}\big{)}\\\
&+m_{N}^{2}\frac{E_{r}-4E_{\nu}}{E_{\nu}^{2}}+\frac{m_{N}^{4}}{E_{\nu}E_{r}}\bigg{]}~{}.\end{split}$
(4)

Here, $F_{1}$ and $F_{2}$ are electromagnetic form factors, $\mu_{n}$ is the
magnetic moment of the nucleon in question, $E_{r}$ is the recoil energy, and
$m_{p}$ is the proton mass Borah  _et al._ (2020); Arrington (2004). Since the
neutrino energy is much less than the proton mass, the HNL energy $E_{N}$ is
nearly identical to the neutrino energy $E_{\nu}$. Thus, the flux of HNLs has
similar features to the solar neutrino flux (see Fig. 1).

The HNL has decay channels $N\rightarrow\nu_{\alpha}\gamma$. We consider the
$\nu$ to be massless, and the decays to be isotropic in the rest frame of the
HNL.333In complete generality the HNL may have some angular correlation with
its polarization, but this depends on the details of the model e.g. Dirac vs.
Majorana neutrinos Balantekin  _et al._ (2019) and we neglect this in what
follows. The decay length is calculated as

$\lambda=\frac{4\pi}{d_{\alpha}^{2}m_{N}^{3}}\gamma\beta.$ (5)

The Monte Carlo simulation samples locations for $N$ decays along with the
energy and direction of the decay photon. This is used to calculate the
resulting photon flux with respect to energy and angle observed by RHESSI. We
consider opening angles of $1^{\circ}$ and $90^{\circ}$, where we reject all
photons arriving at larger angles. The background flux observed by RHESSI is
calculated by using the reported number of counts and effective area of the
front segment (ignoring narrow peaks) Smith  _et al._ (2003). We reject a
parameter point if the flux from $N$ decays exceeds the observed flux at any
energy (see Fig. 2).

Figure 3: Excluded parameter space for a muon neutrino transition dipole
moment. Along with our bounds, we show 90% CL exclusions from Borexino $e-\nu$
scattering Brdar  _et al._ (2021); bor (2018), terrestrial solar neutrino
upscattering Plestid (2021b), Supernova 1987A Magill  _et al._ (2018), big-
bang nucleosynthesis and the cosmic microwave background Brdar  _et al._
(2021). For RHESSI excluded parameter space, we include exclusions from taking
a $1^{\circ}$ opening angle and a $90^{\circ}$ opening angle. The star
represents the parameter point show in Fig. 1 and Fig. 2.

Our resulting exclusion curves from the RHESSI data are shown in Fig. 3 for a
muon neutrino dipole coupling. We find that RHESSI data can offer a
complimentary (and direct) probe of regions of parameter space that are
already probed by SN-1987A. Constraints are strongest in the low mass region
(sub-MeV), and this may also be probed using coherent elastic neutrino nucleus
scattering. We see that the exclusions for the three neutrino flavors all have
similar values in Fig. 4.

Figure 4: Excluded parameter space of a transition dipole moment for each of
the three active neutrinos. We see that the constraints all take a similar
form, only varying by $\mathcal{O}(1)$ factors.

##  III Heavy solar axions

Another production mechanism is solar axions with energies in excess of
$E_{a}\gtrsim 500$ keV. These energies are too high to allow for thermal
production (except for in exponentially suppressed tails), and so the
background photon fluxes are much smaller than for typically considered keV
solar axion searches. The study of MeV-scale solar axions has a long history,
and they have been searched for in terrestrial experiments such as Borexino
and CAST Andriamonje  _et al._ (2010); Bellini  _et al._ (2012). As we discuss
below, satellite measurements of the Quiet Sun provide a complimentary probe
that excels for decay lengths that are short relative to the Earth-Sun
distance.

It is worth highlighting recent work on model building for axions with an
extended matter content Fukuda  _et al._ (2015); Agrawal and Howe (2018); Hook
_et al._ (2020); Dunsky  _et al._ (2023). These models are motivated by the
axion quality problem and seek to protect the axion against Planck suppressed
corrections. The simplest mechanism to achieve this is to simply break the
canonical relation $f_{a}m_{a}\approx f_{\pi}m_{\pi}$ and to allow for $m_{a}$
to be “heavy” relative to predictions of conventional (i.e. DSVZ Dine  _et
al._ (1981); Zhitnitsky (1980) or KSVZ Kim (1979); Shifman  _et al._ (1980))
axion models. It is interesting to note that these independent model building
considerations often push the mass and couplings of the axion into regions of
parameter space that are well suited for solar axion detection; we will
comment on this in great detail below. For instance, following the benchmark
scenarios presented in Dunsky  _et al._ (2023) one finds that masses in the
$\sim 10$ MeV regime with axion decay constants $f_{a}\sim 10^{-5}~{}{\rm
GeV}^{-1}$ fall squarely within the “natural” window of parameter space whilst
simultaneously predicting a sizeable coupling to nucleons and a decay length
that is a few times longer than the radius of the Sun. For slightly lighter
axions, solar production and detection is a useful complimentary probe.

The primary production mechanism for heavy solar axions is the
$p~{}d\rightarrow\\!~{}^{3}{\rm He}~{}\gamma$ reaction which takes place in
the solar $pp$ chain. Other mechanisms are energetically allowed, such as $M1$
transitions in the CNO chain Massarczyk  _et al._ (2022), and $e^{+}e^{-}$
annihilation from ${}^{8}{\rm B}$ neutrinos in the solar interior, however, we
find that the production rates are sufficiently small so as to be
uninteresting.

The flux of axions (prior to decay) can be related to the flux of $pp$
neutrinos, and depends on the isovector coupling of axions to nucleons
$g_{3aN}$ Donnelly  _et al._ (1978). The axions must first escape the Sun and
then decay before reaching Earth. The escape probability depends both on axion
absorption and decay processes. Putting all of this together and setting ${\rm
BR}_{a\gamma\gamma}=1$, we arrive at the flux of axions arriving at a detector
orbiting the Earth,

$\frac{\Phi_{\gamma}}{\Phi_{\nu}^{(pp)}}=0.54|g_{3aN}|^{2}\quantity[\frac{p_{a}}{p_{\gamma}}]^{3}\quantity[\mathrm{e}^{-R_{\odot}/\ell_{\rm
abs}}-\mathrm{e}^{-d_{\odot}/\ell_{\rm dec}}]~{},$ (6)

where $\ell_{\rm abs}^{-1}=\ell_{\rm MFP}^{-1}+\ell_{\rm dec}^{-1}$ with
$\ell_{\rm MFP}^{-1}$ the averaged mean free path in the Sun and $\ell_{\rm
dec}$ the axion decay length. The coupling $g_{3aN}$ is the isovector coupling
strength of the axion to nucleons, and $p_{a}/p_{\gamma}$ is the ratio of
three-momenta between an axion and photon emitted with $E=5.49~{}{\rm MeV}$.
The $pp$ neutrino flux is given by $\Phi_{\nu}^{(pp)}=6\times 10^{10}~{}{\rm
cm}^{-2}{\rm s}^{-1}$. We account for axion-absorption, Primakoff scattering,
and axion electron scattering in our calculation of $\ell_{\rm MFP}^{-1}$.

Our results are shown in Fig. 5. We note that our exclusions depend on the
axion nucleon coupling, captured by $g_{3aN}$, and the decay constant
$g_{a\gamma\gamma}$. If $g_{a\gamma\gamma}$ vanishes at some scale
$\mu=\mu_{0}$, but $g_{aee}\neq 0$ then an effective
$g_{a\gamma\gamma}\sim(\alpha/4\pi)g_{aee}/m_{e}$ will be generated via a
1-loop triangle diagram, and in this way one can re-cast our limits444This
requires accounting for the branching ratio to photons, as well as adjusting
the decay length. in terms of those on $g_{aee}$. We do not include exclusions
from SN1987 typically plotted in the $m_{a}-g_{a\gamma\gamma}$ plane because
the values of $g_{3aN}$ that are required to produce a sufficient axion flux
in the Sun lead to axion trapping within a core-collapse supernova Chang  _et
al._ (2018).555This occurs because axion-nucleon scattering leads to mean free
paths much shorter than the typical size of a supernova, trapping the axions.
This is an important distinction between the hadronically coupled axion models
we considered here vs. an axion like particle which couples exclusively to
photons (see e.g. Jaeckel and Spannowsky (2016)). The solar axion constraints
we discuss here are therefore complimentary to supernova cooling ones. If the
axion nucleon coupling, $g_{aN}$, is large enough to evade SN-1987 bounds via
self trapping then it is also large enough to be probed with RHESSI data. Low
energy supernovae observations have been used to place constraints on axions
which decay in-flight and deposit energy to the ejecta Caputo  _et al._
(2022). Additionally, axions produced in neutron star mergers have been
constrained using X-ray observation Diamond  _et al._ (2023). These
constraints also disappear in the strong coupling regime, and are
complimentary to ours. Constraints from NA62 Cortina Gil  _et al._ (2021),
E787 Adler  _et al._ (2004), and E949 Artamonov  _et al._ (2009) are subject
to $O(m_{K}^{4}/m_{\rho}^{4})$ hadronic uncertainties in the prediction of
$K\rightarrow a\pi$ Bauer  _et al._ (2021); Dunsky  _et al._ (2023). Finally,
our constraints on $g_{a\gamma\gamma}$ lie above the ceiling of searches
performed with the Borexino collaboration Bellini  _et al._ (2012) because we
are sensitive to decay lengths much shorter than $d_{\odot}$. This is
demonstrative of the way in which constraints from solar axion may compliment
existing search techniques using accelerator based experiments, underground
detectors, and astrophysical constraints.

Constraints from big bang nucleosynthesis (BBN) will generically apply both
because the axions we consider have lifetimes in the vicinity of a few
seconds, and because the same reaction, $p~{}d\rightarrow\\!~{}^{3}{\rm
He}~{}\gamma$, is a key driver of BBN. In the absence of any additional dark
sector decay modes, measurements of $N_{\rm eff}$ will generically exclude
axions with masses below 5 MeV or so. These constrains can be alleviated if
the dark sector contains additional degrees of freedom see e.g. Dunsky  _et
al._ (2023). Searches for gamma rays from the Quiet Sun offer a complimentary
direct probe of axion (or other light particle) production that is independent
of early universe cosmology.

Figure 5: Contours of $g_{3aN}$ for which the solar axion flux of photons
would overwhelm the the RHESSI background measurements for the front segments.
Sensitivity is exhausted for $g_{3aN}\sim 1\times 10^{-5}$ however further
reach can be obtained with better data and/or a more sophisticated analysis.

We consider a $90^{\circ}$ opening angle for our signal, meaning all decays
between the Sun’s surface and Earth’s orbit contribute. The monoenergetic
nature of the axion means the photon flux is constant in energy (see Section
IV for more details on monoenergetic production). We demand that this flux
exceed $1.8\times 10^{-3}\mathrm{s}^{-1}\mathrm{cm}^{-2}\mathrm{keV}^{-1}$ for
photon energies above 1 MeV, so that this flux is above the observed RHESSI
background flux in the front segments.

##  IV Model-Independent Searches

Let us now consider a model-independent production of LLPs (here called
$\phi$) which decay via $\phi\rightarrow\gamma\gamma$. In this simplified
model, we consider all production to occur at the solar center, and $\phi$
only interacts with SM physics through its decay. We also assume there is no
preferential direction for decay in the rest frame of $\phi$, so the flux of
photons is a uniform distribution between $E_{\gamma,\mathrm{min}}$ and
$E_{\gamma,\mathrm{max}}$ where
$E_{\gamma,\mathrm{max/min}}=1/2\times(E_{\phi}\pm\sqrt{E_{\phi}^{2}-m_{\phi}^{2}})$.
Inverting this equation, we find $E_{\phi}\geq
E_{\gamma}+m_{\phi}^{2}/(4E_{\gamma})$ (we will call this lowest energy
$E_{\phi,\mathrm{min}}$). Therefore, if we know the rate of production
$R_{\phi}$ and decay length $\lambda$ as a function of $E_{\phi}$, then we can
determine the BSM flux of photons at Earth.

$\begin{split}\frac{\differential\Phi_{\gamma}}{\differential
E_{\gamma}}=&\frac{2}{4\pi d_{\odot}^{2}}\times\\\
&\int^{\infty}_{E_{\phi,\mathrm{min}}}\differential
E_{\phi}\frac{e^{-R_{\odot}/\lambda(E_{\phi})}-\mathrm{e}^{-d_{\odot}/\lambda(E_{\phi})}}{\sqrt{E_{\phi}^{2}-m_{\phi}^{2}}}\frac{\differential
R_{\phi}}{\differential E_{\phi}}\end{split}$ (7)

One particularly well motivated morphology is where $\phi$ has a mono-
energetic production spectrum. This would occur if $\phi$ is produced via a
2-body decay $\chi\rightarrow\phi X$ or via annihilation
$\chi\chi\rightarrow\phi X$ for $v_{\chi}\ll 1$. Performing the integral in
Eq. 7 with a delta-function distribution leads to a flux of photons that is
constant in energy between $E_{\gamma,\mathrm{min}}$ and
$E_{\gamma,\mathrm{max}}$.

Figure 6: Comparison of photon fluxes for different $\phi$ production
scenarios. The fluxes are normalized so that the total production rate is
$N_{\phi}=10^{28}s^{-1}$, and the decay length is $\lambda=10R_{\odot}$ at
$E_{\phi}=1\mathrm{MeV}$. We consider $m_{\phi}$ to be negligibly small.

Remaining more agnostic to the source of $\phi$ production, we may consider a
power-law distribution with respect to energy for $E_{b}\leq E_{\phi}\leq
E_{u}$

$\frac{\differential R_{\phi}}{\differential
E_{\phi}}\bigg{|}_{\mathrm{power}}=R_{c}\times
E_{\phi}^{c}~{}\Theta(E_{u}-E_{\phi})\Theta(E_{\phi}-E_{b})~{}.$ (8)

For $m_{\phi}\ll E_{\gamma},E_{\phi}$ the photon flux is calculable in closed
form,

$\begin{split}&\frac{\differential\Phi_{\gamma}}{\differential
E_{\gamma}}\bigg{|}_{\mathrm{power}}=\frac{2R_{c}}{4\pi d_{\odot}^{2}}\\\
&\times\bigg{[}\bigg{(}\frac{R_{\odot}\tilde{E}}{\tilde{\lambda}}\bigg{)}^{c}\bigg{(}\Gamma\big{(}-c,\frac{R_{\odot}\tilde{E}}{\tilde{\lambda}E_{u}}\big{)}-\Gamma\big{(}-c,\frac{R_{\odot}\tilde{E}}{\tilde{\lambda}E_{l}}\big{)}\bigg{)}\\\
&-\bigg{(}\frac{d_{\odot}\tilde{E}}{\tilde{\lambda}}\bigg{)}^{c}\bigg{(}\Gamma\big{(}-c,\frac{d_{\odot}\tilde{E}}{\tilde{\lambda}E_{u}}\big{)}-\Gamma\big{(}-c,\frac{d_{\odot}\tilde{E}}{\tilde{\lambda}E_{l}}\big{)}\bigg{)}\bigg{]}~{},\end{split}$
(9)

with $\Gamma(a,x)$ the incomplete gamma function,
$E_{l}=\max\\{E_{b},E_{\phi,\mathrm{min}}\\}$, and $\tilde{\lambda}$ is the
decay length at characteristic energy $\tilde{E}$. We normalize to the total
rate of $\phi$ produced, $N_{\phi}$. For the mono-energetic case, we have
$N_{\phi}=R_{\phi}$, while for the power-law production, we have

$R_{c}=\begin{cases}\frac{N_{\phi}(c+1)}{E_{u}^{c+1}-E_{b}^{c+1}}&\mathrm{for}~{}~{}~{}c\neq-1~{},\\\
\frac{N_{\phi}}{\log(E_{u}/E_{b})}&\mathrm{for}~{}~{}~{}c=-1~{}.\end{cases}$
(10)

Constraints on the number of $\phi$ produced per second in the Sun are shown
in Fig. 7. Constraints are set as described at the end of Sections II and III.

##  V Future prospects

(a) Mono-energetic $\phi$ production  (b) $\phi$ production uniform in energy
(c = 0)  (c) $\phi$ production with a linear dependence on energy (c = 1)  (d)
$\phi$ production inversely proportional to energy (c = -1).

Figure 7: Exclusion of $\phi$ production for some of the special cases
considered. Production rates above the lines are excluded. In all cases, the
mass is considered negligible, and there is no production below 10 keV
($E_{b}=10\,\mathrm{keV}$).

In the above discussion, we have found that re-purposing existing RHESSI data
is able to provide interesting constraints on light dark sectors with MeV
scale LLPs. Our analysis should be viewed as a proof of principle and
certainly underestimates the sensitivity of experiments like RHESSI to new
physics models. The major limitations in our analysis are a lack of reliable
peak-subtracted spectra and the ability to suppress backgrounds (see DeRocco
_et al._ (2022); Frerick  _et al._ (2023) for recent work in the keV regime
for more sophisticated statistical analyses). For example, much of the
background for RHESSI comes not from solar activity but rather from cosmic ray
interactions with the Earth’s atmosphere i.e. the radiation comes from the
rear rather than forward field of view. Much of this background can presumably
be suppressed (or perhaps eliminated) with a future instrument, especially if
a dedicated search is performed. In what follows we sketch potential
improvements using a near-term MeV telescope. For concreteness we will anchor
our discussion around the COSI satellite.666We thank Albert Shih for pointing
out the COSI mission to us.

RHESSI operated with minimal shielding to minimize weight. This made the
instrument an effectively “all sky” observation with a high level of cosmic
ray background activity. In contrast COSI will operate with active shielding,
and its further use of Compton kinematic discrimination offers further
background reduction Boggs and Jean (2000). Moreover, ongoing work to better
understand gamma ray emission from the Quiet Sun will further improve on
irreducible backgrounds Orlando and Strong (2021); Orlando  _et al._ (2023).

Other strategies that could be pursued with a future instrument are to go
beyond the rate-only analysis presented above. For example, COSI will have 25%
sky coverage and excellent angular resolution. One could image the MeV photon
flux differential in both energy and angular position. Depending on the
lifetime of LLPs a “halo” of photons could be searched for outside the solar
corona. The shape of the photon distribution will be model dependent, but can
be computed using the Monte Carlo simulations outline above. Similarly, taking
advantage of COSI’s large field of view, other local planetary systems could
be used to search for LLPs. This was suggested recently in the context of
Jupiter where the capture of light dark matter is better motivated Leane and
Linden (2021); Acevedo  _et al._ (2023).

Finally, let us comment on a second channel of interest: ${\rm LLP}\rightarrow
e^{+}e^{-}$. This may occur for a dark vector which dominantly decays via
$V\rightarrow e^{+}e^{-}$, and has recently been considered (in the context of
large volume underground detectors) for the same
$p~{}d\rightarrow\\!~{}^{3}{\rm He}~{}\gamma$ reaction considered here D’Eramo
_et al._ (2023). A search for electrons and/or positron would require accurate
modeling for propagation through magnetic fields in the vicinity of the Earth.

##  VI Conclusions and Outlook

We have discussed simple particle physics models that predict an MeV flux of
photons produced by the Sun. The generic requirement is the existence of some
LLP which can efficiently transport energy from the interior (fueled by
nuclear reactions) to beyond the Sun’s surface. Provided the LLP has a
sizeable branching ratio to final states including at least one photon e.g.
$\gamma\gamma$, $\nu\gamma$, and/or $e^{+}e^{-}\gamma$ final states, one can
search for energetic gamma rays emanating from the Quiet Sun.

We find that constraints from existing data from RHESSI, with a very
conservative analysis strategy, can probe small pockets of untouched parameter
space for both MeV scale axions and a neutrino dipole portal. In both cases,
the RHESSI analysis provides complimentary coverage to existing search
strategies (including cosmological probes such as BBN).

Our major motivation is a simple proof of principle that MeV-scale LLPs with
decay lengths larger than the radius of the Sun can be efficiently searched
for using solar telescopes. The analysis presented here is conservative and
fairly crude; we define exclusions by the condition that the BSM signal
prediction exceeds the total signal observed in any energy window by RHESSI.
Constraints and/or discovery potential could be substantially improved with a
better understanding of instrument backgrounds and more sophisticated analysis
techniques. For example, one could make use of angular profiles of incident
photons to search for new physics, as an LLP flux will produce a photon flux
outside the stellar corona with a predictable angular shape/morphology. We
encourage future missions with MeV scale instrumentation below the cut-off of
Fermi-Lat, such as COSI Team ; Tomsick (2021), to consider searches for BSM
particles, with the Sun being a well-motivated engine for MeV-scale LLPs.

## Acknowledgments

This work benefited from feedback at the Simon’s Center for Geometry and
Physics, and RP would like to specifically thank Simon Knapen, Rebecca Leanne,
and Jessie Shelton for useful discussions. We thank Albert Shih for helpful
discussions regarding the RHESSI instrument. We benefited from feedback on
this manuscript from Rebccea Leanne and Elena Pinetti.

This work is supported by the U.S. Department of Energy, Office of Science,
Office of High Energy Physics, under Award Number DE-SC0011632 and by the
Walter Burke Institute for Theoretical Physics. RP acknowledges support from
the U.S. Department of Energy, Office of Science, Office of High Energy
Physics, under Award Number DE-SC0011632 and the Neutrino Theory Network
Program Grant under Award Number DEAC02-07CHI11359 and the US DOE under Award
Number DE-SC0020250.

##  Appendix A Inefficient production mechanisms

In this section we discuss production mechanisms which we have found to be too
inefficient to allow for detection prospects with our RHESSI analysis.

###  A.1 Mass-Mixing portal for HNLs

Another BSM model involving HNLs has $N$ couple directly to active neutrinos
through added elements in the PMNS matrixAtkinson  _et al._ (2021); Plestid
(2021a); Friedrich  _et al._ (2021); Bryman and Shrock (2019); Bellini  _et
al._ (2013b); Argüelles  _et al._ (2020); Liventsev  _et al._ (2013); Asaka
and Shaposhnikov (2005); Asaka  _et al._ (2005); Atre  _et al._ (2009);
Johnson  _et al._ (1997); Levy (2018); Formaggio  _et al._ (1998); Gorbunov
and Shaposhnikov (2007); Drewes (2013); Bondarenko  _et al._ (2018); Boyarsky
_et al._ (2009); Berryman  _et al._ (2020); Ballett  _et al._ (2020); Gelmini
_et al._ (2021); Orloff  _et al._ (2002); Boiarska  _et al._ (2021); Sabti
_et al._ (2020); Acciarri  _et al._ (2021). Active neutrinos contain a small
admixture of the HNLs along with the three known mass states,

$\nu_{\alpha}=U_{\alpha N}N+\sum_{i=1}^{3}U_{\alpha i}\nu_{i}~{},$ (11)

where $U_{\alpha N}$ represents the coupling of HNLs to active neutrinos.
Since the Sun only has nuclear reactions that produce electron neutrinos, our
constraint is on $U_{eN}$. The $N$ flux from upscattering is subdominant by
orders of magnitude to that from direct production. Therefore, the flux is
given by rescaling the neutrino flux

$\Phi_{N}=|U_{Ne}|^{2}\Phi_{\nu}\sqrt{1-m_{N}^{2}/E_{N}^{2}}.$ (12)

For the masses considered here, there are only three decay channels; i)
$N\rightarrow 3\nu$, ii) $N\rightarrow\nu\gamma$, and iii) $N\rightarrow\nu
e^{+}e^{-}$. As with other production mechanisms, we only consider signals
from photons. The geometry of this decay (into a massless neutrino and photon)
is identical to the case of the dipole portal. The decay rate for each of the
processes follows the general form

$\Gamma_{N\rightarrow SM}\propto G_{F}^{2}|U_{eN}|^{2}m_{N}^{5}~{},$ (13)

which has the steep power-law dependence on mass typical of weak decays. We
find that since decay lengths are always long enough to fall outside the range
given in Eq. 1 that sensitivity from RHESSI is subdominant to searches at
Borexino (which benefits from a large detector volume) and from direct
laboratory searches (see Fig. 8).

Figure 8: Excluded parameter space for HNLs in the mass mixing model. The
dashed line shows the RHESSI exclusion, while shaded regions come from BeEST
Friedrich  _et al._ (2021), PIENU Bryman and Shrock (2019), and Borexino
Bellini  _et al._ (2013b).

###  A.2 Captured dark matter in the Sun

If heavy dark matter, $\chi$, has interactions beyond gravity, it may scatter
within large celestial bodies and become gravitationally captured. The Sun,
being by far the most massive object in the solar system, is a strong
candidate in searching for the signals from captured $\chi$ Press and Spergel
(1985); Gould (1987a, b); Garani and Palomares-Ruiz (2017); Frandsen and
Sarkar (2010); Schuster  _et al._ (2010b); Bell and Petraki (2011); Feng  _et
al._ (2016); Allahverdi  _et al._ (2017); Leane  _et al._ (2017); Arina  _et
al._ (2017); Albert  _et al._ (2018); Nisa  _et al._ (2019); Niblaeus  _et
al._ (2019); Cuoco  _et al._ (2020); Serini  _et al._ (2019); Mazziotta  _et
al._ (2020); Bell  _et al._ (2021).

For the case of symmetric dark matter with a long-lived particle mediator,
there is the interaction $\chi\bar{\chi}\rightarrow\mathrm{LLPs}$. The
energies of these final observable particles are $\mathcal{O}(m_{\chi})$.
However, as discussed in Garani and Palomares-Ruiz (2017), for thermal relic
annihilation cross sections, short range interactions with SM, and $m_{\chi}$
below a few GeV, most of the $\chi$ evaporates from the Sun before
annihilating. Even Jupiter, which has a cooler core than the Sun, would have
evaporation be the dominant effect for $m_{\chi}\leq 0.7\mathrm{GeV}$ Garani
and Palomares-Ruiz (2022), far above the energy sensitivity of RHESSI. We note
that in the presence of long-range $\chi-\mathrm{SM}$ interactions,
evaporation may be suppressed Leane and Linden (2021); Acevedo  _et al._
(2023). However, this is a model-dependent scenario, and is not considered in
this work.

We also considered the case of asymmetric dark matter with self-interactions
via a scalar $\phi$ with a Yukawa like interaction
$\mathcal{L}\subset\bar{\chi}\chi\phi$ . As there is no annihilation, in the
absence of evaporation, the $\chi$ population grows indefinitely. Virialized
dark matter passing through the Sun can scatter on the trapped overdensity and
produce LLPs via the bremsstrahlung-like reaction
$\chi\chi\rightarrow\chi\chi\phi$. In order to produce MeV gamma rays, we
require heavy dark matter, $m_{\chi}\gtrsim 1~{}{\rm TeV}$, such that there is
sufficient available kinetic energy $m_{\chi}v_{\chi}^{2}\gtrsim 1~{}{\rm
MeV}$.777Dark matter nucleon scattering cannot induce MeV bremsstrahlung (i.e.
via $\chi N\rightarrow\chi N\phi$), because the available kinetic energy is
set by $m_{N}v_{\chi}^{2}\sim 1~{}{\rm keV}\times(v_{\chi}/10^{-3})^{2}$. This
is most easily seen in the rest frame of the heavy dark matter. In order to
produce a sufficiently large flux of LLPs, we require a sizeable
$\chi\chi\rightarrow\chi\chi$ cross section. This can only be achieved with a
light mediator for TeV scale (or heavier) dark matter. The cross section
relies on a small momentum transfers. Non-relativistic kinematics, however,
demand a parametrically larger momentum transfer in the bremsstrahlung like
reaction than for elastic scattering. For example, demanding
$E_{\phi}\sim\mathcal{O}(\mathrm{MeV})$ bremsstrahlung, requires a momentum
transfer on the order of $\Delta p^{2}\sim m_{\chi}E_{\phi}\sim(1~{}{\rm
GeV})^{2}$. Due to this kinematic supression, we find that RHESSI is incapable
of setting competitive limits even with the most generous/optimistic model
building choices to maximize the bremsstrahlung like cross section.

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available in every region, especially outside of Europe and North America.

##### Electricity Trading.

Our approach, while perhaps more sophisticated than the one used by the UBA
(see Section 6.1.2), is based on the SFOT approach, not on the more complex
(and probably more accurate) MRIO approach (see Section S2.2). This means that
our study only considers electricity exchange with direct neighbors, but
ignores additional, network-wide interactions.

##### Storage Cycling.

The method to calculate virtual emissions for the electricity flows going into
and coming out of pumped hydro storage units in this study is a simplified
one. It assumes a constant grid EF for these electricity flows, instead of
calculating a grid EF at a higher temporal resolution. Should the inflows and
outflows occur at times when the grid EF is systematically higher or lower
than the annual average grid EF, this approach will introduce an error.

##### Transformation & Distribution.

Similarly, our calculation of T&D losses does not temporally (or spatially)
disaggregate losses in the grid. Also, the losses at different voltage levels
of the grid are not analyzed separately.

###  S5.4 Future Work

Besides the obvious opportunities for future work, i.e. addressing the
limitations mentioned in Section S5.3 (to the extent possible), we want to
briefly highlight some additional promising avenues for research and
applications.

To find out if and how it applies to a different context, we would welcome it
if researchers were to use the methodology demonstrated in this study for
calculating grid EF for other countries and time periods. Some data sources
may have to be adjusted (e.g. for the primary-energy referenced EF 17). The
results could be compared to the grid EF calculated by official institutions
(IEA, EEA, national institutions).

Another interesting approach would be to engage in a type of global
sensitivity analysis (GSA). While in this study, we only investigated single
effects in isolation, it may be worth mapping the entire solution space more
systematically. Existing approaches from mathematics 76, 77, some of which
have been already applied to life cycle assessment 78, 79, 80, may be a good
starting point. The findings may e.g. provide additional insight into how
individual aspects (effects) interact with one another.

Moving away from research, and towards application, it may be worth developing
a user interface (UI) for the methodology. This UI could e.g. be an
interactive, browser based solution which requires no installations by the
user. This UI would lower the barriers to using the methodology and applying
it to Scope 2 emission accounting.

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