# Neutrino constraints on inelastic dark matter captured in the Sun

Bhavesh Chauhan ID , Mary Hall Reno ID ,
Carsten Rott ID , Ina Sarcevic ID

###### Abstract

The flux of neutrinos from annihilation of gravitationally captured dark
matter in the Sun has significant constraints from direct-detection
experiments. However, these constraints are relaxed for inelastic dark matter
as inelastic dark matter interactions generate less energetic nuclear recoils
compared to elastic dark matter interactions. In this paper, we explore the
possibility for large volume underground neutrino experiments to detect the
neutrino flux from captured inelastic dark matter in the Sun. The neutrino
spectrum has two components: a mono-energetic “spike” from pion and kaon
decays at rest and a broad-spectrum “shoulder” from prompt primary meson
decays. We focus on detecting the shoulder neutrinos from annihilation of
hadrophilic inelastic dark matter with masses in the range 4–100 GeV and the
mass splittings in up to 300 keV. We determine the event selection criterion
for DUNE to identify GeV-scale muon neutrinos and anti-neutrinos originating
from hadrophilic dark matter annihilation in the Sun, and forecast the
sensitivity from contained events. We also map the current bounds from Super-
Kamiokande and IceCube on elastic dark matter, as well as the projected limits
from Hyper-Kamiokande, to the parameter space of inelastic dark matter. We
find that there is a region of parameter space that these neutrino experiments
are more sensitive to than the direct-detection experiments. For dark matter
annihilation to heavy-quarks, the projected sensitivity of DUNE is weaker than
current (future) Super (Hyper) Kamiokande experiments. However, for the light-
quark channel, only the spike is observable and DUNE will be the most
sensitive experiment.

##  1 Introduction

There is overwhelming evidence for the existence of dark matter (DM) in our
universe. The most widely accepted model of cosmology, $\Lambda$CDM, points
towards a non-baryonic and non-relativistic species that constitutes nearly
25% of the energy budget of our universe [1]. However, the particle nature of
DM, i.e., its mass, spin, and interactions, is yet unknown. As the Earth and
Sun move through the halo of DM trapped in our galaxy, feeble interactions
between DM and visible matter are expected. Direct detection experiments on
Earth aim to detect the nuclear and electronic recoils produced in these
scatterings. On the other hand, DM scattering off nuclei in Sun can lead to
gravitational capture and DM is gradually accumulated in the Sun [2, 3, 4, 5].
The captured DM can annihilate or decay into Standard Model particles,
producing a flux of neutrinos that can be detected by large volume neutrino
detection experiments on Earth such as Super-Kamiokande [6, 7, 8], ANTARES [9,
10], and IceCube [11, 12].

The main goal of this paper is to take a detailed look at the sensitivity of
the Deep Underground Neutrino Experiment (DUNE) [13, 14] to the flux of
neutrinos from DM annihilation in the Sun, and to compare with limits from
other neutrino and direct-detection experiments. This topic has been
extensively studied in the context of elastic scattering of DM with the
conclusion that most of the parameter space that can be probed by neutrino
experiments is already ruled out by direct-detection experiments, except
perhaps Hyper-Kamiokande [15] can do marginally better in some channels [16].
As a result, we focus our attention on inelastic dark matter proposed in Ref.
[17] to address the discrepancy between the observed annual modulation in DAMA
[18, 19, 20] and the null results in other direct-detection experiments. We
assume a simplified model where the dark sector comprises two species,
$\chi_{1}$ and $\chi_{2}$, with an endothermic mass splitting,
$\delta=M_{2}-M_{1}>0$, and contact interaction with the quarks. In this work,
we only study hadrophilic interactions and the results for leptophilic
interactions can be analogously obtained [21].

As we discuss below, the large volume underground neutrino experiments can
probe novel regions of the parameter space for inelastic dark matter that are
inaccessible to direct-detection experiments. This is owning to the fact that
the dark matter capture in the Sun is less affected by the mass splitting than
direct detection experiments (see Section 2). The subsequent neutrino flux
from inelastic DM annihilation in the Sun can be detected by the neutrino
experiments [22]. The fraction of atmospheric neutrinos that is coincident
with the position of the Sun constitute the main background. The main
challenge to the sensitivity of the neutrino experiments is the poor source-
pointing resolution at these energies, which can be attributed to the lack of
one-to-one correspondence between the direction of incident neutrino and
detected charged lepton.

It has been pointed out in Refs. [23, 24] that dark matter annihilation to
quarks in the Sun results in a large flux of mono-energetic neutrinos from
pions and kaons that decay-at-rest after thermalization (called $\pi$DAR and
KDAR respectively). The neutrinos from $\pi$DAR have energies $\sim 30$ MeV
and are extremely difficult to detect due to large atmospheric neutrino
background. As a result, the community has focused on detection of the
neutrinos from KDAR that have energies $\sim 236$ MeV that allow for
approximate reconstruction of the incident neutrino direction [25, 26, 27,
28]. This mono-energetic feature in the neutrino spectrum is often called a
“hump” or “spike”. While a positive detection of these spike neutrinos will be
a smoking-gun signal of DM capture in the Sun, the lack of any spectral
information will result in a degeneracy between the reconstructed dark matter
parameters such as its mass, cross section, and the annihilation channel. The
observation of spike neutrinos in DUNE gives a poor understanding of the
particle nature of the dark matter, and a complementary probe would be
beneficial. With this motivation, we focus on the other part of the emitted
neutrino spectrum called the “shoulder” which comprises neutrinos from the
decay of primary mesons prior to thermalization in the Sun. The shoulder is a
broad spectrum feature that extends up to the mass of the annihilating dark
matter, and strongly depends on the annihilation channel. We refer the reader
to Figure 3 for an example of the neutrino spectrum from annihilation of 25
GeV dark matter in the Sun.

| $\nu_{e}$ | $\nu_{\mu}$
---|---|---
| $E_{\nu}$ | $\theta_{\nu}$ | Ref. | $E_{\nu}$ | $\theta_{\nu}$ | Ref.
Spike ( QEL ) | SuperK | ✓ | ✗ | [25, 26] | ✓ | ✗ | –
IceCube | ✗ | ✗ | – | ✗ | ✗ | –
DUNE | ✓ | ✓ | [25, 26] | ✓ | ✓ |  [26],[28]*
Shoulder ( DIS ) | SuperK | ✓ | ✓ |  [8]*$\,{}^{\\#}$ | ✓ | ✓ |  [6, 7, 8]*
IceCube | ✓ | ✗ | – | ✓ | ✓ |  [11, 12]*
DUNE | ✓ | ✓ | [29] | ✓ | ✓ | This Work
Table 1:  We provide a bird’s eye view of the capabilities of the neutrino
experiments to detect the spike and shoulder neutrinos from the dark matter
annihilation in the Sun. The spike neutrinos undergo quasi-elastic like (QEL)
scattering and the shoulder neutrinos mostly undergo deep-inelastic scattering
(DIS). We show the ability (inability) of the detectors to _reasonably_
reconstruct the incident neutrino energy and direction with a  ✓ (  ✗ ). Note
that for the shoulder, sensitivity is same for neutrinos and anti-neutrinos.
We also provide references (Ref.) for the corresponding study, and mention
that this work discusses detection of the shoulder muon neutrinos in DUNE. The
references marked with * are by the respective collaboration.
$\,{}^{\\#}$Events from all neutrino flavors are included in Ref. [8]. See
text for details.

Observation of shoulder neutrinos will help to understand the flavor structure
of dark matter interactions with quarks. If dark matter annihilates to light
quarks ($u\bar{u}$, $d\bar{d}$, $s\bar{s}$), then the KDAR neutrinos are the
only detectable component. The KDAR neutrinos produced in the Sun are muon
neutrinos but the flavor conversion between the production point and the
detector results in all flavors of these mono-energetic neutrinos in the
detector. In a water Cherenkov detector like Super-Kamiokande, only the
$\nu_{e}$ component is detectable as $\nu_{\mu}$ is very close to the detector
threshold. DUNE can utilize the excellent low-energy resolution of Liquid
Argon Time Projection Chamber (LArTPC) and detect both $\nu_{e}$ and
$\nu_{\mu}$ components of the spike. In Ref. [25], the sensitivity of Super-
Kamiokande and DUNE to the $\nu_{e}$-spike has been studied. However, the
source-pointing resolution of a water Cherenkov detector from quasi-elastic
scattering needs to be examined carefully, and we look forward to the detailed
study by the Super-Kamiokande collaboration in future. The sensitivity of DUNE
to the $\nu_{\mu}$-spike events is studied in Ref. [28], and limits are
provided for inelastic dark matter assuming a mass splitting of 50 keV. In
this work, we include a wider parameter space, and we discuss comparisons with
direct-detection experiments.

If dark matter annihilates to heavy quarks (such as $b\bar{b}$ and/or
$c\bar{c}$), then the shoulder neutrinos can be detected by Super-Kamiokande
and IceCube which results in strong limits on spin-independent and spin-
dependent interactions of dark matter [8, 12]111The limits from ANTARES [10]
are weaker than IceCube for the dark matter mass range considered in this
paper. In Ref. [29], the sensitivity of DUNE to DM annihilation to heavy
quarks has been evaluated using electron neutrinos. As muons offer arguably
better source pointing resolution, we estimate for the first time, the
sensitivity of DUNE utilising the flux of muon neutrinos. We calculate the
energy and angular cuts on the predicted atmospheric neutrino events that will
favor the signal from dark matter annihilation in the Sun. For this, we have
used the Monte Carlo program NuWro [30] to simulate the interactions of GeV-
neutrinos with argon nuclei. We find that for the heavy-quark channel, the
shoulder neutrinos provide better sensitivity than KDAR neutrinos in DUNE.
However, the existing limits from upward-going muons in Super-Kamiokande are
stronger than our projected sensitivity from contained events in DUNE. It
would be interesting to study the sensitivity of DUNE from upward muons, but
it is beyond the scope of this work. In Table 1, we provide a bird’s eye view
of the detection capability of Super-Kamiokande, IceCube, and DUNE to the
spike and shoulder neutrinos. We tabulate if the detector can reasonably
reconstruct the incident neutrino energy and direction using spike or shoulder
neutrinos. The spike neutrinos scatter via the quasi-elastic like (QEL)
interactions, and the final state lepton is isotropic. As a result, the
sensitivity of Super-Kamiokande and DUNE is obtained from the counting of
total events in Ref. [25, 26]. Recently, it was proposed that DUNE can
reconstruct the neutrino direction using the final-state proton, which cannot
be detected in Super-Kamiokande, and results in the reduction of atmospheric
backgrounds [28]. The shoulder neutrinos mostly undergo deep-inelastic
scattering (DIS) and the kinematics of the final-state lepton is correlated
with the incident neutrino energy and direction. In Ref. [8], Super-Kamiokande
provides limits for elastic dark matter using both upward-muon and contained
events from all-flavors. It is clear from Ref. [16] that the Super-Kamiokande
sensitivity is dominated by the limits from upward muons. In our analysis, we
map these limits on to the parameter space for inelastic dark matter. The
constraints on inelastic dark matter above 100 GeV from IceCube is studied in
Ref. [31]

Due to the limitations of the Monte Carlo generator NuWro, we limit our
analysis to dark matter mass below 100 GeV. In this range, there are competing
limits from direct-detection experiments. We use the WimPyDD package [32] to
estimate the relative event-rates for elastic and inelastic dark matter, and
use the latest sensitivity of LUX-ZEPLIN (LZ) [33], XENONnT [34], PandaX-4T
[35], and the $\rm C_{3}F_{8}$ run of PICO-60 (PICO-$\rm C_{3}F_{8}$) [36]. We
find that these experiments provide a good coverage of the range of dark
matter mass considered in this paper.

In the next section, we present a brief overview of inelastic dark matter:
direct detection and its gravitational capture in the Sun. The evaluation of
the neutrino flux at Earth and at detectors from DM annihilation in the Sun in
reviewed in Section 3. The atmospheric background and our approach to
determining cuts to reduce the background relative to the signal of neutrinos
from DM annihilation in the Sun appear in Section 4. In Section 5 we present
our results, while Section 6 summarizes our main results and outlook.

##  2 Overview of Inelastic Dark Matter

In typical models of inelastic DM, the dark sector comprises two fermionic
mass eigenstates $\chi_{1}$ and $\chi_{2}$. Their masses differ by an
endothermic mass splitting $\delta=M_{2}-M_{1}>0$. In these models, only off-
diagonal interactions between the two mass states are allowed at the tree-
level, i.e., $\chi_{2}$ couples to $\chi_{1}$ but the self-couplings
$\chi_{1}-\chi_{1}$ and $\chi_{2}-\chi_{2}$ are forbidden. These self-
couplings may be generated at loop-level [37], and hence are sub-dominant. The
cosmological dark matter density today is dominated by the lighter state
$\chi_{1}$, however, there are circumstances when a sizable population of
$\chi_{2}$ may also survive until today. In this work, we do not make
assumptions about the details of the particle physics model and only look at
three parameters of inelastic DM — the mass of lighter DM particle
($M_{\chi}$), mass splitting ($\delta$), and the DM-nucleon scattering cross-
section ($\sigma_{\chi N}$). We will also assume isospin invariant couplings
for simplicity. As a consequence of the mass splitting, a fraction of the
energy budget of the DM interaction is spent to create the heavier state. This
changes the kinematics of the interaction, and as a result, the available
phase space for inelastic scattering is usually smaller than for elastic
scattering. In this section, we quantify the impact of the inelasticity
parameter $\delta$ on direct-detection rates as well as on gravitational
capture of inelastic DM in Sun.

###  2.1 Direct Detection

Dark matter direct-detection experiments are designed to observe the energy
transferred by galactic DM particles as they pass through the detector volume
and interact with nuclei (or electrons) of the target material. Initially
designed with a focus on the Weakly Interacting Massive Particle (WIMP)
paradigm, most of these detectors are optimized to look for signals of a
$\sim$100 GeV DM particle scattering via weak-scale interactions [38]. The
non-observation of any significant events in these experiments has
significantly constrained the viable parameter space for WIMP-like DM [35, 33,
34]. Consequently, the scope of direct detection searches has broadened,
especially at the low DM mass frontier [39, 40, 41, 42, 43, 44, 45, 36].

The rate of nuclear recoils (counts per kilogram per day) in the direct-
detection experiments is given by,

$\mathcal{R}_{\rm DD}=\int_{E_{R}^{\rm min}}^{E_{R}^{\rm
max}}dE_{R}\,\frac{\rho_{\chi}}{M_{N}M_{\chi}}\int_{v_{\rm
min}}d^{3}v\,v\,f(\vec{v})\,\frac{d\sigma_{\chi N}}{dE_{R}}$ (2.1)

where $\rho_{\chi}\sim 0.3-0.4\,\rm GeV/cm^{3}$ is the local dark matter
density [46, 47], $M_{N}$ is the nucleon mass, $M_{\chi}$ is dark matter mass,
$f(\vec{v})$ is the velocity distribution of dark matter in the vicinity of
the Earth with $\int d^{3}vf(\vec{v})=1$, and $d\sigma_{\chi N}/dE_{R}$ is the
differential DM-nucleon interaction cross section. The interaction cross
section includes the form factors which are functions of the nuclear recoil
energy $E_{R}$. The event rates in these detectors depends on the range of
recoil energies to which the detector is sensitive and on the minimum dark
matter velocity required to generate detectable nuclear recoils ($v_{min}$).

As none of the direct-detection experiments have observed a significant excess
in nuclear recoils above their background expectations, Equation 2.1 can be
used to determine their exclusion limits on elastic DM parameters $M_{\chi}$
and $\sigma_{\chi N}$. The sensitivity of these experiments depends on the
velocity distribution of DM in the vicinity of the Earth. It is usually
assumed that the DM has a Maxwellian velocity distribution in the galactic
frame with $v_{0}$=220 km/s and cutoff at 550 km/s, which is the escape
velocity of our galaxy. As the Earth is moving through the DM halo, $f(v)$ in
Equation 2.1 is the time-averaged distribution in the _boosted_ frame.

In the case of inelastic dark matter, the minimum DM velocity required to
produce a specific recoil on a nucleus of mass $M_{N}$ is increased due to the
the mass splitting,

$v_{\rm min}(E_{R})=\frac{1}{\sqrt{2M_{N}E_{R}}}\left(\frac{M_{N}}{\mu_{\chi
N}}E_{R}+\delta\right).$ (2.2)

where $\mu_{\chi N}=M_{\chi}M_{N}/(M_{\chi}+M_{N})$ is the reduced mass of
dark matter nucleus system. As a result, the available phase space for
inelastic dark matter is typically smaller when compared to the elastic
scattering $(\delta=0)$, and the event rates are relatively smaller. Not only
that, inelastic DM scattering is also more sensitive to the high-velocity tail
of the distribution.

We use WimPyDD [32] to calculate the event rates for various direct-detection
experiments supported by the package. WimPyDD can evaluate the event rates for
elastic as well as inelastic dark matter for the Xenon-based experiments
(XENONnT, and LZ) and the bubble chamber detector PICO-60 with the $\rm
C_{3}F_{8}$ target. We have also evaluated the sensitivity of PandaX which
differs from XENON and LZ primarily due to increased sensitivity at large
recoils [35]. Although other target nuclei are supported, a complete
implementation of other detectors such as CRESST [45], DarkSide [44], and CDMS
[41] in WimPyDD is beyond the scope of this work. As the sensitivity obtained
by WimPyDD is only approximate, it is advantageous to utilize the relative
event-rate

$k_{\rm DD}(M_{\chi},\delta)=\mathcal{R}_{\rm
DD}(M_{\chi},\delta)\,/\,\mathcal{R}_{\rm DD}(M_{\chi},0)\,,$ (2.3)

which is less sensitive to inaccuracies of the package, and represents the
scaling of the event rate with the mass-splitting parameter $\delta$. Using
$k_{\rm DD}$, we obtain the limits on the inelastic DM-nucleon cross-section
from a direct-detection experiment by scaling the limits for elastic DM
scattering. The exclusion limits for inelastic DM is thus given by,

$\sigma_{\rm DD}^{\rm lim}(M_{\chi},\delta)=\sigma_{\rm DD}^{\rm
lim}(M_{\chi},0)\times k_{\rm DD}(M_{\chi},\delta)\,,$ (2.4)

where $\sigma_{\rm DD}^{\rm lim}(M_{\chi},0)$ is the current limit on the
spin-independent elastic scattering cross-section. In future, one can use
$k_{\rm DD}$ to obtain new limits when there is an update from the experiment.
In Figure 1, we show the contours in $M_{\chi}-\delta$ parameter space for
fixed $k_{\rm DD}(M_{\chi},\delta)$ for PICO-$\rm C_{3}F_{8}$ and the Xenon-
based experiments LZ and PandaX. We find that for a given DM mass $M_{\chi}$,
there exists $\delta_{\rm max}$ beyond which a direct-detection experiment is
completely insensitive to DM-nucleon scattering. Moreover, $\delta_{\rm max}$
also depends on the maximum recoil threshold as evident in the larger
sensitivity of PandaX when compared to LZ and XENON. Similar conclusions have
been obtained in Ref. [48]. At the time of this writing, XENON and PandaX
provide the leading limits for DM mass between 5–9 GeV whereas LZ is most
sensitive above 9 GeV. For DM mass between 2–5 GeV, we use the limits by
PICO-$\rm C_{3}F_{8}$ 222In this range, the liquid argon based experiment,
DarkSide [44], provides leading sensitivity today and evaluating the relative
event-rate on argon is interesting. However, DM with mass less than 4 GeV is
evaporated from the Sun and the marginal improvement does not warrant a
dedicated implementation in WimPyDD. Recently, the PICO collaboration
presented their limits on inelastic dark matter in [49], and our results
obtained with Equation 2.3 are in agreement.

Figure 1:  The contours for relative event rates $k_{\rm DD}$ (see Equation
2.3) are shown for PICO-$\rm C_{3}F_{8}$ (left) and the Xenon-based
experiments (right). Note the difference in the limits of the ordinate between
figures. The contours for LZ (dashed) and PandaX (solid) are different as
PandaX is sensitive to larger nuclear recoils. The difference is significant
for DM mass above $\sim$75 GeV and mass splitting above $\sim$150 keV.

It is worth mentioning that other experiments like CRESST have enhanced
sensitivity for larger mass splittings, and future heavy-nuclei based
experiments can also probe a wider DM parameter space [48]. Moreover, the
sensitivity of these experiments is also improved by accounting for an extra-
galactic component in the DM velocity distribution [50]. However, systematic
inclusion of this high-velocity component in WimPyDD as well as gravitational
capture in the Sun (as discussed below) is beyond the scope of this work and
left as a future effort.

###  2.2 Gravitational Capture in Sun

Figure 2:  The contours for relative capture rate $k_{\rm cap}$ (see Equation
2.8) are compared with the sensitivity of direct-detection experiments. The
figure on the left focuses on the small-mass parameter space. The shaded
regions show non-zero relative event rate in direct-detection experiments,
$k_{\rm DD}\geq 0$, for PICO-C3F8 (purple), XENON and LZ (cream), and PandaX
(blue). The gray shaded region represents the evaporation mass of dark matter,
which we fix at 4 GeV for all $\delta\geq 0$. One can see that there is a
significant parameter space where direct-detection experiments are not
sensitive, but where indirect detection from a significant gravitational
capture and annihilation of DM in the Sun is possible.

When a DM particle from the galactic halo falls into the gravitational
potential of the Sun, it accelerates to a velocity
$w(r)=\sqrt{u^{2}+v_{esc}^{2}(r)}$, where $u$ is the DM velocity outside the
gravitational potential and $v_{esc}$ is the local escape velocity, both in
units of $c$ (natural units). The inelastic interaction between incident DM
and a target nucleus in the Sun is kinematically allowed only if
$w^{2}(r)>2\delta/\mu$ where $\mu$ is the reduced mass of the DM–nucleus
system [51, 52]. As a result, for typical values of the mass splitting, only
the high-velocity tail of DM distribution results in a gravitational capture.
The differential capture rate by the $i$-th isotope in the Sun is given by,

$\frac{dC_{i}}{dV}=n_{\chi}n_{i}(r)\int du\frac{f(u)}{u}w(r)\int
dE_{R}\frac{d\sigma}{dE_{R}}\,,$ (2.5)

where $f(u)$ is the DM velocity distribution boosted to the frame of the Sun,
$n_{\chi}=\rho_{\chi}/M_{\chi}$, and $n_{i}(r)$ is the local number density of
the target isotope. The limits for integration over the recoil energy and the
velocity are non-trivial as the phase-space for capture is significantly
reduced due to the mass splitting $\delta$. We follow the method outlined in
Ref. [53] for evaluating the integrals in Equation 2.5. The _total_ rate of
capture of DM in Sun is given by,

$\Gamma_{\rm cap}=\sum_{i}\int_{0}^{R_{\rm sun}}4\pi
r^{2}dr\frac{dC_{i}}{dV}\,,$ (2.6)

where we sum over all 29 isotopes given by the B16-AGSS09 solar model [54]. As
the capture rate for inelastic dark matter with spin-dependent interactions is
very small [52, 53], we only look at spin-independent scatterings of dark
matter. Moreover, we assume that the number density of captured dark matter in
the Sun has reached equilibrium and the rate of annihilation is given by the
steady state solution,

$\Gamma_{\rm A}=\frac{\Gamma_{\rm cap}}{2}.$ (2.7)

Lastly, the evaporation of low-mass dark matter will be affected by a non-zero
mass splitting [55]. We assume that any change in reduced evaporation would be
dominated by the reduction in capture rate and do not consider dark matter
mass below 4 GeV [4, 56]. In detailed models, a small elastic-scattering cross
section is always generated at one-loop level which aids in attaining
equilibrium and contributes to evaporation.

Similar to the application of direct-detection limits on elastic DM to
inelastic DM, we are interested in the relative reduction in the capture rate
due to the inelastic DM mass splitting, so we determine

$k_{\rm cap}=\rm\Gamma_{cap}(M_{\chi},\delta)\,/\,\Gamma_{cap}(M_{\chi},0)\,.$
(2.8)

In Figure 2, we show the contours of constant $k_{\rm cap}$ in the
$M_{\chi}-\delta$ parameter space. As expected, the inclusion of the mass
splitting yields a smaller capture rate as compared to the elastic DM
interactions (i.e., $\delta=0$ line in the figure). We also show the regions
of parameter space where complementary limits from direct-detection
experiments exist from Xenon-based experiments as well as from PICO-$\rm
C_{3}F_{8}$. It is interesting to note that there exists a region of parameter
space that is inaccessible to direct-detection experiments but does not suffer
from an exponential reduction in capture rate. In this region, even though the
capture rate is reduced by factor of $0.1-0.01$, there are no competing limits
from direct-detection experiments. The only way to probe this space is through
neutrino detection experiments. In Ref. [57] discusses the complementarity
between direct-detection experiments and capture in the Sun of inelastic dark
matter to obtain halo-independent bounds.

##  3 Neutrino Flux from DM annihilation in Sun

Figure 3: _Left:_ The energy-scaled spectrum of muon neutrinos and anti-
neutrinos per DM annihilation ($E_{\nu}\,d\varphi_{\beta}/dE_{\nu}$) in the
Sun is shown for a benchmark DM mass of 25 GeV and for two annihilation
channels - ‘$b\bar{b}$’ (blue) and ‘$u\bar{u}$’ (light blue). The neutrino
(anti-neutrino) spectrum is shown with solid (dashed) curve. The _spike_ in
the $\nu_{\mu}$ spectrum is due to primary and secondary $K^{+}$ that decay-
at-rest (KDAR) in the Sun. For illustration, we represent the mono-energetic
feature with a Gaussian of width 10 MeV which is the typical energy resolution
of a LArTPC detector at these energies. _Right:_ The energy-scaled spectrum of
non-muon flavors is shown. The spectrum of anti-neutrinos is same as
neutrinos. _Both:_ The broad feature from $E_{\nu}=500$ MeV to
$E_{\nu}=M_{\chi}$ in all neutrino spectra, called the _shoulder_ , is
significant only for the heavy-quark channel. There are additional features in
the neutrino spectra at below 250 MeV, notably from three-body decays of
kaons, which are not shown here.

In a simplified and minimal model of inelastic DM, it is reasonable to assume
that the gravitationally captured DM in the Sun annihilates to standard model
particles. In hadrophilic models, the DM annihilates to a quark-antiquark pair
which hadronize in the solar media. The subsequent decays of the mesons result
in neutrinos that escape the Sun. In detailed models of DM, it is also
possible that the dominant annihilation channel is to leptons or beyond
standard model particles [58], however, we do not consider these models here.
We also ignore the details of the mediator and only assume scalar contact
interactions.

The flux of neutrinos of flavor $\alpha$ at a detector on Earth is given by,

$\frac{d\Phi_{\alpha}^{\rm D}}{dE_{\nu}}=\frac{\Gamma_{\rm A}}{4\pi
d^{2}}\sum_{\beta}P_{\alpha\beta}\,\frac{d\varphi_{\beta}}{dE_{\nu}}$ (3.1)

where, $d\varphi_{\beta}/dE_{\nu}$ is the energy-spectrum of neutrinos
produced per annihilation and $P_{\alpha\beta}$ is the average conversion
probability which includes propagation inside the Sun, through vacuum to the
Earth, and propagation inside the Earth to the detector location. In this
work, we assume that $d=1$ A.U. is the average distance between the Sun and
Earth, and we ignore the effects of the eccentricity of the Earth’s orbit. We
use the publicly available package PPPC 4 DM$\nu$ [59] to estimate the
neutrino flux as well as the DM annihilation rate, $\Gamma_{\rm A}$. The
spectrum of neutrinos of various flavors at the source, without any flavor
conversion effects, is shown in Figure 3 for a benchmark DM mass of 25 GeV for
two different annihilation channels, to ‘$u\bar{u}$’ and to ‘$b\bar{b}$’. Note
that there are two distinct features: a mono-energetic _spike_ in the
$\nu_{\mu}$ spectrum, and a broad _shoulder_ in all six components ($\nu_{e}$,
$\bar{\nu}_{e}$, $\nu_{\mu}$, $\bar{\nu}_{\mu}$, $\nu_{\tau}$, and
$\bar{\nu}_{\tau}$). The spike at 236 MeV arises from primary and secondary
$K^{+}$ that decay at rest [23, 24], whereas the shoulder is mostly due to
prompt decays of primary mesons produced after hadronization. For the spike,
the integrated form of Equation 3.1 can be obtained as,

$\Phi^{\rm spike}=\frac{\Gamma_{\rm A}}{4\pi d^{2}}\times
P_{\mu\mu}\times\frac{2M_{\chi}r_{K}}{2m_{K}}\mathcal{B}$ (3.2)

where $r_{K}$ is the energy fraction that yields kaons at rest [25],
$m_{K}\sim 494$ MeV is the mass of charged kaon, and $\mathcal{B}=0.68$ is the
charged kaon branching fraction to mono-energetic $\nu_{\mu}$. In Ref. [28],
the DUNE collaboration reported the projected sensitivity of the flux of these
236 MeV neutrinos in a DM model independent form as,

$\Phi^{\rm spike}\,(400\,\text{kton-yr})\leq 10^{2}\,\rm m^{-2}\,s^{-1}$ (3.3)

translates to limits on the DM capture rate in the Sun. For an honest
comparison, we use the sensitivity from generator-level analysis without
accounting for the detector effects in reconstruction, which reduces the
sensitivity by an order or magnitude. The sensitivity of DUNE to shoulder
neutrinos from DM annihilation is the Sun is the main subject of this paper.

One should note that the shoulder is significant only if DM annihilates to
heavy quarks, whereas the spike is present in both light-quark and heavy-quark
channels [25]. As we do not discuss the detailed model of inelastic DM, we
consider these two channels, $u\bar{u}$ and $b\bar{b}$, independently as
benchmark annihilation channels with unit branching fraction. Moreover, we do
not consider channels such as $W^{+}W^{-}$ and $t\bar{t}$ as the main focus of
the paper is on DM masses below 100 GeV. Lastly, we do not explicitly consider
leptonic channels such as $\tau\bar{\tau}$ and $\nu\bar{\nu}$ as the results
for these channels can be analogously obtained.

Figure 4:  The flavor conversion probabilities, averaged over the position of
the Sun over a period of one year at the proposed location of DUNE far-
detector, are shown for neutrinos (left) and antineutrinos (right) as
functions of energy.

We quantify the effect of flavor conversion in the Earth using the latest
neutrino oscillation parameters from NuFIT[60]. We use the nuCraft software
[61] to evaluate the flavor conversion probabilities due to vacuum-mixing as
well as Earth-matter effects [62, 63]. The parametric resonance [64, 65],
which is important for Earth-crossing neutrinos in the energy range 3–10 GeV
[66], is included in nuCraft. We evaluate the neutrino flux at 1 A.U. which
includes flavor conversion in the Sun and in vacuum using PPPC 4 DM$\nu$. This
flux without Earth-crossing, $d\Phi_{\mu}^{\rm E}/dE_{\nu}$, is multiplied by
flavor conversion probabilities depending on the zenith angle of the Sun at a
given time and location on Earth. As we are interested in event-rates in
detectors over a span of several years of runtime, we average over the
position of the Sun over the period of one year and obtain an annual averaged
flavor conversion probability. Subsequently, the (annually-averaged) flux of
$\nu_{\alpha}$ at a detector location is obtained by,

$\frac{d\Phi_{\alpha}^{\rm D}}{dE_{\nu}}=\frac{d\Phi_{e}^{\rm
E}}{dE_{\nu}}\times\langle P^{ED}_{e\alpha}\rangle+\frac{d\Phi_{\mu}^{\rm
E}}{dE_{\nu}}\times\langle P^{ED}_{\mu\alpha}\rangle+\frac{d\Phi_{\tau}^{\rm
E}}{dE_{\nu}}\times\langle P^{ED}_{\tau\alpha}\rangle$ (3.4)

where $\langle P_{\alpha\beta}\rangle$ represents the conversion probabilities
averaged over the zenith angles of the Sun. The antineutrino flux at the
detector can be obtained analogously. We use AstroPy [67] to calculate the
zenith-angles of the Sun ($\psi$) at the proposed location of the DUNE far-
detector, and we calculate $P_{\alpha\beta}^{ED}(E_{\nu},\psi)$ using nuCraft
for $100\times 365$ samples of $\psi$ over a period of one year, before
averaging. Our primary interests are the $\nu_{\mu}$ and $\bar{\nu}_{\mu}$
fluxes from the shoulder since the final-state charged leptons in the charged-
current interactions are best candidates to identify the direction of the
source, and $\mu^{\pm}$ offer the best angular resolution in DUNE. We evaluate
$\langle P_{\alpha\beta}\rangle$ for neutrinos and antineutrinos and present
them in Figure 4.

##  4 Atmospheric Neutrino Background

Figure 5:  We show the product of the angle-integrated fluxes of atmospheric
$\nu_{\mu}$ (solid) and $\bar{\nu}_{\mu}$ (dashed) and charged-current cross
sections at DUNE. We have explicitly shown the effect of flavor-conversion in
Earth: the lighter blue curves show the same $\nu_{\mu}$ and $\bar{\nu}_{\mu}$
fluxes times cross sections without neutrino oscillations in the Earth. Even
though the fluxes are similar, $\bar{\nu}_{\mu}$ contribute less to the event
rate due to relatively smaller cross section.

As discussed in the previous section, our focus is on low-mass DM particles
which produce neutrinos with energies ranging from 500 MeV to 100 GeV. At
these energy scales, one of the key challenges is to efficiently identify and
reject the significant background arising from atmospheric neutrinos.
Atmospheric neutrinos, originating from interactions between cosmic rays and
the Earth’s atmosphere, reach the detector from all directions [68]. A
fraction of them can appear to come from the direction of the Sun. For a
typical detector exposure that we consider, these neutrinos cannot be
distinguished from the neutrinos originating from DM annihilation in the Sun,
and thus limit the sensitivity of any terrestrial detector. In this section,
we take a close look at the expected distributions of atmospheric neutrino
events in DUNE and propose event selection criteria that would significantly
reduce the atmospheric neutrino background.

As detailed estimates of the atmospheric neutrino flux at the location of the
DUNE far-detector are unavailable, we adopt the flux predictions for Super-
Kamiokande provided by Honda et al. [68]. This is a reasonable assumption as
the proposed site for the DUNE is at a similar latitude to Super-Kamiokande
[66]. The effect of atmospheric neutrino flavor conversion due to propagation
in the Earth is evaluated using nuCraft [61] using the best-fit values of
neutrino oscillation parameters and assuming the Normal Hierarchy of neutrino
masses. As noted above, to obtain the energy-dependent cross sections as well
as differential distributions, we use the neutrino Monte Carlo generator NuWro
that simulates neutrino and antineutrino interactions with argon nuclei.
Although Ref. [68] provides zenith and azimuth dependent flux, we use the all-
sky-averaged flux as input to NuWro, and we assume that the distribution is
isotropic at the energy scales in consideration. Similar to Ref. [26], we use
the spectral function approach to model the nuclei and set nucleus_target=2
and sf_method=1 in the NuWro parameter initialization file. Other parameters
are set to their default values. We only look at charged-current interactions.
All neutral-current interactions channels are switched off. ITo obtain
generalized results that are utilized later for the neutrino flux from DM
annihilation in Sun, we also separately simulate interactions from mono-
energetic $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ with energies between 100 MeV and
100 GeV. As mentioned earlier, we only focus on the muon neutrinos and
antineutrinos as we expect better source-pointing resolution for tracks as
opposed to cascade-like events from electron and tau flavors.

Our focus is directed towards interactions that result in muon-like events,
where the interaction point is situated within the volume of the detector.
This is different from Super-Kamiokande, which primarily utilizes upward-going
muons that mostly originate in the rock outside the detector [7]. The upward
muons are advantageous as they have much larger target volume, whereas the
_contained_ events in DUNE carry more information about the incident
neutrinos. Due to low thresholds and excellent particle identification
capabilities of LArTPCs, the events will have additional detectable final-
state particles (like protons and pions) that can help with reconstruction,
however, we do not utilise them. The differential event rate for contained
events from charged-current interactions of atmospheric muon neutrinos can be
expressed as,

$\frac{dN^{\rm atm}}{dE_{\nu}}=\frac{M_{\rm
target}\,T}{\text{kton}}\frac{N_{A}\times 10^{9}}{A}\times\sigma_{\rm
tot}\frac{d\Phi^{\rm atm}}{dE_{\nu}}$ (4.1)

where $\sigma_{\rm tot}$ is the total charged-current cross section from all
channels, $M_{\rm target}$ is the fiducial mass of the target nuclei (atomic
mass $A$), and $T$ is the exposure time. In Figure 5, we show the flux of
atmospheric $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ folded with their energy-
dependent interaction cross section, with and without accounting for flavor
conversions in the Earth. As DUNE is not expected to be magnetized, it cannot
differentiate between $\mu^{-}$ and $\mu^{+}$, and the detector is only
sensitive to the aggregate flux. We expect around 2300 events per 34 kton-yr
in DUNE from atmospheric $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ from all angles
with energies above 100 MeV, with most of the events arising from neutrinos
with energies below 5 GeV. Even though their flux is similar,
$\bar{\nu}_{\mu}$ contribute less to the event rate due to relatively smaller
cross-section. However, not all of these events would be considered as
background in the search for DM annihilation in the Sun. The irreducible
background comes from the atmospheric neutrinos that ‘appear’ to be coming
from the direction of the Sun.

The LArTPC technology that will be employed by the DUNE detector is expected
to have excellent energy and angular resolution for charged particles,
especially at the energies of interest [14]. Even if the $\mu^{\pm}$-tracks
are perfectly reconstructed, the energy and direction of the incident neutrino
can only be imperfectly determined. Since our aim is to determine the
direction of the flight of the neutrino in DUNE, we need to determine event-
selection criteria that minimize this uncertainty. We follow Ref. [26] for
this purpose, albeit for high-energy neutrinos. The event selection criteria
for $\mu^{\pm}$-tracks in DUNE has two components: a muon energy threshold
($E_{\mu}^{\rm th}$), and the other, on maximum value of the relative
direction of the muon track to the Sun ($\theta_{\mu}^{\rm c}$). The remainder
of this section is dedicated to determining appropriate values for these
parameters.

As evident from Figure 5, a large fraction of atmospheric neutrino events can
be reduced by selecting high-energy tracks above a threshold ($E_{\mu}^{\rm
th}$). These tracks that come from the tail of the $\mu^{\pm}$ energy
distribution. In this work, we consider three benchmark cases for
$E_{\mu}^{\rm th}$: 500 MeV, 1 GeV, and 5 GeV. As similar cuts will be
eventually imposed on the DM signal as well, one has to be conservative
because very strict selection criteria will also reduce event-rate from DM
annihilation, thus resulting in low sensitivity. While there may exist
alternative choices that optimize the signal-to-background ratio, we do not
perform this exercise. Our choices represent a low-, moderate-, and high-
energy cutoff, and we shall explore their implications.

Figure 6:  The fractions of muons produced above threshold, $\eta_{\rm
th}^{\rm atm}$ (see Equation 4.3) as a function of incident energy, for
$\nu_{\mu}$ (solid) and $\bar{\nu}_{\mu}$ (dashed) are shown for three
benchmark choices of the track energy threshold $E^{\rm th}_{\mu}$ as 500 MeV,
1 GeV, and 5 GeV.

The number of muon-like tracks from atmospheric neutrinos in DUNE with energy
above the threshold $E_{\mu}^{\rm th}$ is obtained by

$N_{\rm th}^{\rm atm}=\sum_{\mu^{\pm}}\int dE_{\nu}\frac{dN^{\rm
atm}}{dE_{\nu}}\times\eta_{\rm th}^{\rm atm}(E_{\nu},E_{\mu}^{\rm th})$ (4.2)

where $\eta_{\rm th}^{\rm atm}$ is the fraction of final-state muons that are
produced above threshold for a given neutrino energy, and given by,

$\eta_{\rm th}^{\rm atm}(E_{\nu},E^{\rm
th}_{\mu})=\frac{1}{\sigma(E_{\nu})}\int\frac{d\sigma(E_{\nu},E_{\mu})}{dE_{\mu}}\Theta(E_{\mu}-E^{\rm
th}_{\mu})\,dE_{\mu}\equiv\frac{{\rm N_{MC}}(E_{\mu}>E^{\rm th}_{\mu})}{{\rm
N_{MC,\rm total}}}.$ (4.3)

In the equation above, ${\rm N_{MC}}(E_{\mu}>E^{\rm th}_{\mu})$ is the number
of simulated events in NuWro that pass the energy selection criterion, and
${\rm N_{MC,\rm total}}=5\times 10^{6}$ is the total number of simulated
events in NuWro for a given neutrino/antineutrino energy. In Figure 6, we show
the energy dependence of $\eta_{\rm th}^{\rm atm}$ for both neutrinos and
antineutrinos for the three benchmark choices of $E^{\rm th}_{\mu}$. As
expected, the selected fraction of events is identically zero at low energies
($E_{\nu}^{\rm min}\sim m_{\mu}+E^{\rm th}_{\mu}$), and it monotonically
increases with incident neutrino energy. We also note that the selection
efficiency for anti-neutrinos is typically higher than that of neutrinos. The
resulting numbers of events for muon-like tracks in DUNE from atmospheric
neutrinos and anti-neutrinos are tabulated in Table 2 for the three benchmark
energy threshold and assuming an exposure of 34 kton-yr.

Component |  $E_{\mu}^{\rm th}$=500 MeV |  $E_{\mu}^{\rm th}$=1 GeV |  $E_{\mu}^{\rm th}$=5 GeV
---|---|---|---
$\nu_{\mu}$ | 884 | 552 | 120
$\overline{\nu}_{\mu}$ | 325 | 232 | 58
Total | 1209 | 784 | 178
Table 2:  The expected number of events in DUNE from atmospheric muon
neutrinos and antineutrinos assuming an exposure of 34 kton-yr. The event
rates are shown for the three benchmark values of the muon energy threshold,
$E_{\mu}^{\rm th}$. We expect $\sim$2300 events/(34 kton-yr) from the
neutrinos and anti-neutrinos above 100 MeV. Figure 7: Geometry of a charged-
current interaction is illustrated. DUNE has excellent energy and angular
resolution for tracks. However, the direction of the flight of incident
neutrino can only be determined statistically.

For a given muon-like track in DUNE, one can reconstruct the initial energy of
the charged particle based on the rate of energy deposition along the track.
Assuming perfect resolution, one can also obtain the initial direction of the
charged particle. This is, however, insufficient to infer the direction of the
initial neutrino that interacted in the detector due to the lack of one-to-one
correspondence between the neutrino and charged-lepton directions. One can
only statistically infer whether the incident neutrino path lies within a cone
of half-angle $\theta_{\mu}^{\rm c}$ with respect to the direction of the
charged particle: $\theta_{\mu}<\theta_{\mu}^{\rm c}$ where $\theta_{\mu}$ is
the true angle between the muon and neutrino trajectories. Depending on the
position of the Sun with respect to the muon track, $\theta_{\mu}^{\rm\,sun}$,
we want to determine a criterion to accept/reject the event. The geometry of
the interaction is illustrated in Figure 7.

Figure 8:  The first three figures (_top left, top right, and bottom left_)
show the angular distribution of muon-like events originating from atmospheric
neutrinos in DUNE for the three benchmark choices of the energy threshold,
$E^{\rm th}_{\mu}$, considered in this paper. We show the contributions of
$\mu^{-}$ (dashed) and $\mu^{+}$ (dotted) separately, but only their aggregate
(solid) is observable. _Bottom right:_ The fraction of events for which
$\theta_{\mu}<\theta_{\mu}^{\rm\,c}$ as a function of $\theta_{\mu}^{\rm\,c}$
is shown for the three benchmark choices of $E^{\rm th}_{\mu}$ for
$\mu^{+}+\mu^{-}$ (solid), $\mu^{-}$ (dashed) and $\mu^{+}$ (dotted). The
horizontal grey line is $f[\theta_{\mu}<\theta_{\mu}^{\rm\,c}]=0.90$ which
determines the numerical value of $\theta_{\mu}^{\rm\,90}$.

To determine this directionality criterion, we look at the angular
distribution of $\mu^{-}$ and $\mu^{+}$ originating from atmospheric
$\nu_{\mu}$ and $\bar{\nu}_{\mu}$ respectively, as well as their aggregate.
The distributions obtained for the three benchmark values of $E^{\rm
th}_{\mu}$ are shown in the first three panels of Figure 8. As expected, the
muons (anti-muons) are more collinear with neutrino (anti-neutrino) direction
for larger values of the muon energy threshold, and the total event rate is
smaller. From this angular distribution, we determine a numerical value of
$\theta_{\mu}^{\rm c}$ such that nearly 90% of the incident neutrinos lie
inside the cone of half-angle $\theta_{\mu}^{\rm c}$ with the track as axis
and the interaction point as the vertex of the cone, and call it
$\theta_{\mu}^{\rm\,90}$. The fraction of events for which
$\theta_{\mu}<\theta_{\mu}^{\rm c}$ as a function of $\theta_{\mu}^{\rm c}$ is
shown in the last panel of Figure 8. Although we have explicitly shown the
contributions of $\mu^{-}$ and $\mu^{+}$, only $\mu^{-}+\mu^{+}$ is observable
as none of the detectors in consideration can differentiate between them. It
is clear that the value of $\theta_{\mu}^{\rm\,90}$ depends on the choice of
$E^{\rm th}_{\mu}$, and subsequently we have three benchmark scenarios:

  1. 1.

Low-cutoff: $E^{\rm th}_{\mu}$ = 0.5 GeV and $\theta_{\mu}^{\rm\,90}$ =
40$\degree$

  2. 2.

Moderate-cutoff: $E^{\rm th}_{\mu}$ = 1 GeV and $\theta_{\mu}^{\rm\,90}$ =
30$\degree$

  3. 3.

High-cutoff: $E^{\rm th}_{\mu}$ = 5 GeV and $\theta_{\mu}^{\rm\,90}$ =
15$\degree$.

In principle, it should be possible to perform an event-by-event analysis
where the muon-like track is accepted only if $E_{\mu}>E^{\rm th}_{\mu}$ and
the position of the Sun in the sky relative to the muon track is inside the
conical section with half-angle $\theta_{\mu}^{\rm\,90}$, i.e.,
$\theta_{\mu}^{\rm\,sun}<\theta_{\mu}^{\rm\,90}$. The geometry of the
interaction is illustrated in Figure 7.

It is possible to improve the atmospheric neutrino background estimate by
including the full angular dependence of the atmospheric neutrino flux,
however, this is beyond the scope of the paper and left as future work which
could also include true predictions for atmospheric neutrinos at the location
of DUNE, as well as the uncertainty from reconstruction. For our estimates, we
assume that the atmospheric neutrino flux at these energies is isotropic and
subsequently, the event selection efficiency considering both the energy
threshold and direction of the track is,

$\eta_{\rm sel}^{\rm atm}=\eta_{\rm th}^{\rm
atm}\times\frac{1}{2}\left(1-\cos\theta_{\mu}^{\rm\,90}\right)$ (4.4)

where both $\eta_{\rm th}^{\rm atm}$ and $\theta_{\mu}^{\rm\,90}$ depend on
the choice of $E^{\rm th}_{\mu}$. Subsequently, the _irreducible_ number of
events from atmospheric neutrinos in DUNE is given by,

$\displaystyle N_{\rm sel}^{\rm atm}$ $\displaystyle=\sum_{\mu^{\pm}}\int
dE_{\nu}\frac{dN^{\rm atm}}{dE_{\nu}}\times\eta_{\rm sel}^{\rm
atm}(E_{\nu},E_{\mu}^{\rm th},\theta_{\mu}^{\rm\,90})$ (4.5)
$\displaystyle\equiv N_{\rm th}^{\rm
atm}\times\frac{1}{2}\left(1-\cos\theta_{\mu}^{\rm\,90}\right),$ (4.6)

where $N_{\rm th}^{\rm atm}$ are given in Table 2. Any event selection
criterion on the background must also be imposed on the signal, and is
explored in the next section.

Another potential source of background comes from solar atmospheric neutrinos
[69, 70, 71]. These are produced in cosmic ray interactions with the Sun’s
atmosphere. These neutrinos are typically sub-dominant as compared to Earth’s
atmospheric neutrinos, even when isolating events in the direction of the Sun.
For DUNE, we estimate the event rate from solar atmospheric neutrinos to be
$\leq 1$/(400 kton-yr). We have evaluated the neutrino “floor” for spin-
independent interactions and find that it is approximately two orders of
magnitude below the projected sensitivity of shoulder neutrinos. We do not
explicitly show the neutrino floor for brevity.

##  5 Results and Discussion

In the previous section, we identified three benchmark event-selection
criteria that can minimize the background from atmospheric neutrinos, and we
obtained the source pointing resolution of DUNE. In this section, we evaluate
the impact of these criteria on the signal from dark matter annihilation in
the Sun to obtain the projected sensitivity for DUNE from shoulder neutrinos.
For the spike neutrinos, we use Equation 3.2 and the recent DUNE results
summarised in Equation 3.3, together with Equation 2.8 for our new constraints
inelastic dark matter.

Due to the lack of one-to-one correspondence between incident-neutrino energy
and the outgoing-lepton energy and direction, a fraction of events from
shoulder neutrinos from DM annihilation in the Sun will not satisfy the event
selection criteria. A signal-event is missed if either the outgoing-lepton has
kinetic energy below threshold, or if it is produced outside the cone of half-
angle $\theta_{\mu}^{c}$ with axis along the direction of the Sun at the time
of event. The fraction of the events from DM annihilation in the Sun that pass
the selection criteria are given by,

$N_{\rm sel}^{\rm sig}=\frac{M_{\rm target}\,T}{\text{kton}}\frac{N_{A}\times
10^{9}}{A}\times\sum_{\mu^{\pm}}\int dE_{\nu}\,\sigma_{\rm
tot}(E_{\nu})\,\frac{d\Phi^{\rm D}}{dE_{\nu}}\times\eta_{\rm sel}^{\rm
sig}(E_{\nu},E_{\mu}^{\rm th},\theta_{\mu}^{\rm\,90})$ (5.1)

where $\eta_{\rm sel}^{\rm sig}$ is the _signal-selection efficiency_ , and
determined by,

$\displaystyle\eta_{\rm sel}^{\rm sig}(E_{\nu},E^{\rm
th}_{\mu},\theta_{\mu}^{\rm\,90})$ $\displaystyle=\frac{1}{\sigma_{\rm
tot}(E_{\nu})}\int{dE_{\mu}\,d\cos\theta_{\mu}}\frac{d\sigma}{dE_{\mu}d\cos\theta_{\mu}}\,\Theta(E_{\mu}-E^{\rm
th}_{\mu})\,\Theta(\theta_{\mu}^{\rm\,90}-\theta_{\mu})\,$
$\displaystyle\equiv\frac{{\rm N_{MC}}(E_{\mu}>E^{\rm
th}_{\mu},\theta_{\mu}<\theta_{\mu}^{\rm\,90})}{{\rm N_{MC,\rm total}}}.$
(5.2)

We have evaluated $\eta_{\rm sel}^{\rm sig}$ for $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ using NuWro for the three benchmark scenarios considered in
this paper, and the results are shown in Figure 9 (left).

$E^{\rm th}_{\mu}$ | $\theta_{\mu}^{\rm\,90}$ | $N^{\rm atm}_{\rm sel}$ | $N^{\rm excl}_{90}$
---|---|---|---
0.5 GeV | 40$\degree$ | 1664 | 57.5
1.0 GeV | 30$\degree$ | 618 | 34.4
5.0 GeV | 15$\degree$ | 36 | 9.4
Table 3:  The number of expected atmospheric background events that pass the
selection criterion ($N^{\rm atm}_{\rm sel}$) and the number of signal events
needed for 90% C.L. exclusion ($N^{\rm excl}_{90}$) are tabulated. We have
assumed an exposure of 400 kton-yr for DUNE.

To obtain the projected 90% C.L. exclusion limits, we follow Ref. [26] and
assume a Poisson distribution for the event-rate from the atmospheric
background. We assume a detector exposure of 400 kton-yr, and the observed
number of events is set equal to the expected number of background events,
rounded to the nearest integer. A signal is considered excluded at 90% C.L.,
if the total number of events, i.e., the sum of background and signal, exceeds
the observed number of events in at least 90% of the cases. For the three
scenarios considered in this paper, the number of expected atmospheric
background events that pass the selection criterion ($N^{\rm atm}_{\rm sel}$),
and the number of signal events needed for 90% C.L. exclusion ($N^{\rm
excl}_{90}$) are presented in Table 3.

Figure 9:  Left: The signal-selection efficiency, $\eta_{\rm sel}^{\rm sig}$,
for the three benchmark choices of muon energy thresholds ($E^{\rm th}_{\mu}$)
are shown for $\nu_{\mu}$ (solid) and $\overline{\nu}_{\mu}$ (dashed). The
cone half-angle ($\theta_{\mu}^{\rm c}$) is determined by the choice of
$E^{\rm th}_{\mu}$ (see Table 3). Right: The sensitivities of DUNE to the
shoulder neutrinos from DM annihilation to $b\bar{b}$ for the three benchmark
choices are shown along with the sensitivity to spike neutrinos (blue) from
Ref. [28], for elastic DM ($\delta=0)$. The exclusion limits from the direct-
detection experiment PICO60-C3F8 [36] are shown using a gray curve.

The projected sensitivity in the $M_{\chi}-\sigma_{\chi p}^{\rm SI}$ plane for
elastic DM (i.e., $\delta=0$) is obtained by considering $N_{\rm sel}^{\rm
sig}\geq N_{90}^{\rm
excl{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}g}}$.
The results are shown in Figure 9 (right). It is interesting to note that the
low-energy-cutoff scenario has larger sensitivity at small DM mass ($\leq$ 7
GeV) and the high-energy-cutoff scenario has larger sensitivity at larger DM
mass ($\geq$ 20 GeV). This can be attributed to two different factors. First,
a larger threshold on the muon energy implies that event-rate from DM with
small mass is suppressed. Second, the muons with larger energy are more
collinear with the incident neutrino. As a result, not only the background is
suppressed due to small $\theta_{\mu}^{\rm c}$, but also the fraction of
events that satisfy the directionality criterion is larger, and subsequently
the signal-to-background ratio is enhanced.

Unsurprisingly, the entire parameter space for elastic DM that can lead to
detectable signal in DUNE is already ruled out by direct-detection
experiments. In the Figure 9 (right), we only show the limits by PICO60-$\rm
C_{3}F_{8}$ from Ref. [36]. Other direct-detection experiments such as XENON,
LZ, and Panda-X are even more constraining. Their upper bounds on the spin-
independent cross section $\sigma_{\chi p}^{\rm SI}$ are below the scale shown
on this plot. This is one of the primary reasons for investigating inelastic
DM, for which the direct-detection experiments are less sensitive. Moving
forward, we choose the moderate-cutoff scenario (i.e., $E^{\rm th}_{\mu}$ = 1
GeV and $\theta_{\mu}^{\rm c}$ = 30$\degree$) to forecast our projected
sensitivity for DUNE to detect the shoulder neutrinos from annihilation of
inelastic DM captured in the Sun.

Figure 10:  The exclusion limits with neutrino experiments and direct-
detection experiments on the parameter space of mass-splitting ($\delta$) and
spin-independent scattering cross-section with nucleon ($\sigma_{\chi p}^{\rm
SI}$) are shown for two choices of DM mass ($M_{\chi}$). The limits for DM
annihilation to light-quark channel (left) and heavy-quark channel (right) are
shown separately. The projected 90% C.L. exclusion limits from DUNE with 400
kton-yr exposure are shown for spike and shoulder neutrinos independently. The
existing 90% C.L. exclusion limits from Super-Kamiokande with 3903 days
exposure are shown using the red solid curve (using Equation 2.8 and limits on
elastic interactions from Ref. [8]). The projected sensitivity of Hyper-
Kamiokande with 11 years exposure is shown with the red dashed curve (using
Equation 2.8 and Ref. [16]). The existing limits from IceCube-DeepCore with 7
years of exposure is shown in green (using Equation 2.8 and Ref. [12]), but
note that they are weaker than Super-K at these DM masses. The limits from
Super-Kamiokande and Hyper-Kamiokande are dominated by upward-muons and only
available for DM annihilation to $b\bar{b}$. The shaded region is ruled out
from the direct detection experiments LZ [33] PICO-60 [36].

For inelastic DM with non-zero mass splitting $\delta$, the capture rate in
the Sun is reduced by a factor $k_{\rm cap}(M_{\chi},\delta)\leq 1$ (see
Equation 2.8) whereas the spectrum of neutrinos is not affected333If the
heavier state is long-lived and constitutes a minor component of the
cosmological DM density, only the capture rate in the Sun needs to be suitably
modified and the spectrum of neutrinos remains the same.. As a result, the
limiting sensitivity for DM-nucleon cross-section for inelastic DM can be
obtained from the limits obtained for elastic DM by a simple translation,

$\sigma_{\rm cap}^{\rm lim}(M_{\chi},\delta)=\sigma_{\rm cap}^{\rm
lim}(M_{\chi},\delta=0)\times k_{\rm cap}^{-1}(M_{\chi},\delta)$ (5.3)

where $\sigma_{\rm cap}^{\rm lim}(M_{\chi},\delta=0)$ are the limits on the
DM-nucleon cross-section obtained from the capture of elastic DM (shown in
Figure 9). In Ref. [8], Super-Kamiokande utilized their upward-going muons
data to look for neutrinos from DM annihilation in the Sun. Their limits are
presented for elastic DM. We translate their $\delta=0$ limits to limits on
inelastic DM using Equation 5.3. In Ref. [16], the sensitivity of Hyper-
Kamiokande using upward-going muons was presented for SD interactions. We
translate these limits for inelastic dark matter with an additional factor of
$\Gamma_{\rm cap}^{\rm SD}(M_{\chi})/\Gamma_{\rm cap}^{\rm SI}(M_{\chi})$ in
Equation 5.3 to account for the difference in capture rate. Analogously, the
limiting sensitivity of direct-detection experiments for inelastic DM can be
obtained from the limits on elastic DM,

$\sigma_{\rm DD}^{\rm lim}(M_{\chi},\delta)=\sigma_{\rm DD}^{\rm
lim}(M_{\chi},\delta=0)\times k_{\rm DD}^{-1}(M_{\chi},\delta)$ (5.4)

where ‘DD’ in the subscript refers to the specific direct-detection
experiment. As described earlier, we limit our attention to PICO-C3F8, LZ, and
PandaX for a good coverage of the range of DM mass. Moreover, the capture of
inelastic DM in the Sun with spin-dependent (SD) interactions is highly
suppressed [52, 53], and we only present our sensitivity forecasts for spin-
independent (SI) scattering cross-section of DM with nucleons.

With the translated limits from other neutrino experiments and direct-
detection experiments, we present our results for DUNE in Figure 10 and Figure
11. The sensitivity is projected for muon-like _contained_ tracks with
moderate cutoff (i.e., $E_{\mu}^{\rm th}=0.5$ GeV and
$\theta_{\mu}^{\,90}=30$°) and assuming a fiducial exposure of 400 kton-yr.

In Figure 10, we show our results for the light-quark channel and the heavy-
quark channel separately. This is to highlight the fact that other neutrino
experiments (such as Super-Kamiokande and IceCube) can only detect shoulder
muon neutrinos and antineutrinos, which is prominent for heavy-quark channel
only. In case of dark matter annihilation to light-quarks, the shoulder is
suppressed. The $\nu_{\mu}$-spike is only detectable at DUNE. In principle,
Super-Kamiokande and DUNE can also detect the $\nu_{e}$-spike (originating due
to flavor conversion) for the light-quark channel, and the projected
sensitivity has been obtained in [25]. We do not show these limits as our
discussion is focused on detection of muon-like tracks. One also expects some
events from tau neutrinos and anti-neutrinos from DM annihilation without any
background from atmospheric neutrinos. However, their source-pointing
resolution is difficult to determine without dedicated detector simulations,
and we encourage DUNE collaboration to consider this. In Figure 11, we show
the limits and projected sensitivities of neutrino experiments and direct
detection experiments for four benchmark values of the mass splitting
$\delta$. For brevity, we show the limits from light-quark channel and the
heavy-quark channel in the same figure, even though they can be mutually
exclusive.

Figure 11:  The complementarity of neutrino experiments and direct-detection
experiments on the parameter space of DM mass ($M_{\chi}$) and spin-
independent scattering cross-section with nucleon ($\sigma_{\chi p}^{\rm SI}$)
is shown for four choices of mass-splitting ($\delta$). The projected 90% C.L.
exclusion limits from DUNE with 400 kton-yr exposure are shown for spike and
shoulder neutrinos separately and assuming light- and heavy-quark annihilation
channels independently. The existing limits Super-Kamiokande and the projected
sensitivity of Hyper-Kamiokande are obtained using Equation 2.8 and Refs. [8]
and [16] respectively. Note that the limits from Super-K and Hyper-K assumes
annihilation to $b\bar{b}$ channel. The gray-shaded region is ruled out from
the direct detection experiments. The existing limits from IceCube-DeepCore
[12]are not shown because they are weaker than Super-Kamiokande at lower DM
mass and weaker than direct-detection at larger DM mass.

The main result of our new analysis of the current neutrino experiments,
Super-Kamiokande and IceCube, is that they already rule out a large portion of
the inelastic dark matter parameter space that is usually inaccessible to
direct-detection experiments. We find that for the heavy-quark channel, the
projected sensitivities from both spike and shoulder muon neutrinos at DUNE
are weaker than the Super-Kamiokande limits we have determined here. Note that
the limits from Super-Kamiokande are dominated by upward going muons which
originate outside the detector volume, and thus have a much larger target
mass, resulting in better sensitivity. In the heavy-quark channel, only Hyper-
Kamiokande is expected to improve the limits in future. For the light-quark
channel, DUNE can detect the $\nu_{\mu}$-spike efficiently. In the light-quark
channel, the projected limits from shoulder neutrinos is weaker than the
spike, as expected. As Figure 10 and Figure 11 show, DUNE can constrain a
region of $(M_{\chi},\delta)$ parameter space for inelastic dark matter
annihilation to light quarks where Super-Kamiokande, Hyper-Kamiokande and DD
experiments do not have sensitivity.

##  6 Summary and Outlook

In models of dark matter with sizable interactions with standard model
particles, the dark matter in the galactic halo can be gravitationally
captured by the Sun. The subsequent annihilation of the captured dark matter
results in a novel neutrino flux on Earth. The detectable neutrino spectrum
has two components: a spike at $\sim$236 MeV from kaon decays-at-rest, and a
broad-spectrum shoulder from prompt decays of mesons. DUNE is a good candidate
to study the spike neutrinos, and has been widely discussed in the literature
[25, 26, 72, 28], whereas the sensitivity to shoulder neutrinos is only
discussed for electron neutrinos [29]. We evaluate, for the first time, the
sensitivity of DUNE from _contained_ GeV-scale tracks originating from the
shoulder $\nu_{\mu}$ and $\bar{\nu}_{\mu}$. To reduce the atmospheric neutrino
background, we determine event selection criteria in DUNE that can identify
neutrinos coming from the direction of the Sun. Unsurprisingly, the parameter
space probed by DUNE for dark matter with elastic interactions with nucleons
is already ruled out by direct detection experiments. For inelastic dark
matter [17], we find that there is interesting parameter space which cannot be
probed by direct-detection experiments but where neutrino experiments can
provide complementary sensitivity.

We find that the sensitivity of neutrino experiments depends on the flavor
structure of inelastic dark matter interactions. We have considered separately
couplings to both heavy-quarks and light-quarks in dark matter annihilation in
the Sun. For coupling to heavy-quarks, both the spike and shoulder neutrinos
are detectable, and we find that current limits from Super-Kamiokande [8] and
IceCube [12] already rule out a large part of the inelastic dark matter
parameter space that is inaccessible to direct-detection experiments (see
Figure 10 and Figure 11). In this channel, our projected sensitivity for
contained events in DUNE is weaker than the current limits from upward-going
muons from Super-Kamiokande, and IceCube. We encourage Super-Kamiokande to
perform a dedicated search for inelastic dark matter using their new
atmospheric neutrino data [73]. We also encourage IceCube and Hyper-Kamiokande
to search for inelastic dark matter in the Sun. For coupling to light-quarks,
only the spike is detectable as the flux of shoulder neutrinos is relatively
small. In this scenario, DUNE excels due to its better performance at low-
energy, and will probe currently unexplored parts of the inelastic dark matter
parameter space. In this paper, we have extended the analysis of Ref. [28] to
include a wider range of mass splittings. In the future, a positive signal
from spike in DUNE, and null results from shoulder neutrinos in other neutrino
experiments will strongly hint towards inelastic dark matter coupling to
light-quarks only.

The DUNE collaboration could improve our analysis by including the effects of
detector resolution on the reconstruction of neutrino energy and direction.
For the spike neutrinos, the DUNE collaboration finds that these effects
reduce the sensitivity by a factor of $\sim$10 [28]. The shoulder neutrinos
are higher in energy, and we expect the detector resolution to be better, so
these effects should be smaller. For shoulder neutrinos, it is reasonable to
assume that the angular distribution of the background atmospheric neutrinos
is isotropic. Including the location-specific zenith-angle dependence of the
atmospheric neutrinos in the Monte Carlo generators will result in a small
changes in the projected sensitivity presented in this paper. We encourage the
DUNE collaboration to consider a dedicated study of GeV-scale atmospheric
neutrinos in the future.

## Acknowledgements

We would like to thank Sergio Palomares-Ruiz, Kate Scholberg, Raj Gandhi,
Sandra Robles, Kajetan Niewczas, Gaurav Tomar, Ivan Martinez-Soler, and
Juergen Reichenbacher for helpful discussions and comments. BC and MHR are
supported by US Department of Energy Grant DE-SC-0010113. IS is supported by
the U. S. Department of Energy Grant DE-FG02-13ER41976/sc-0009913. C. Rott
acknowledges support from NSF Grant No. PHY-2309967 and from the National
Research Foundation of Korea (NRF) for the Basic Science Research Program
NRF-2020R1A2C3008356. For facilitating portions of this research, BC and MHR
wish to acknowledge the Center for Theoretical Underground Physics and Related
Areas (CETUP*), The Institute for Underground Science at Sanford Underground
Research Facility (SURF), and the South Dakota Science and Technology
Authority for hospitality and financial support, as well as for providing a
stimulating environment. It must be noted that this analysis has been
performed independently by the authors and does not reflect the opinions of
the DUNE collaboration.

## References

  * [1] Planck collaboration, N. Aghanim et al., _Planck 2018 results. VI. Cosmological parameters_ , _Astron. Astrophys._ 641 (2020) A6 [1807.06209].
  * [2] W. H. Press and D. N. Spergel, _Capture by the sun of a galactic population of weakly interacting massive particles_ , _Astrophys. J._ 296 (1985) 679.
  * [3] K. Griest and D. Seckel, _Cosmic Asymmetry, Neutrinos and the Sun_ , _Nuclear Physics B_ B283 (1987) 681.
  * [4] A. Gould, _WIMP Distribution in and Evaporation From the Sun_ , _Astrophys.J._ 321 (1987) 560.
  * [5] A. Gould, _Cosmological density of WIMPs from solar and terrestrial annihilations_ , _Astrophys. J._ 388 (1991) 338.
  * [6] Super-Kamiokande collaboration, S. Desai et al., _Search for dark matter WIMPs using upward through-going muons in Super-Kamiokande_ , _Phys. Rev. D_ 70 (2004) 083523 [hep-ex/0404025].
  * [7] Super-Kamiokande collaboration, T. Tanaka et al., _An Indirect Search for WIMPs in the Sun using 3109.6 days of upward-going muons in Super-Kamiokande_ , _Astrophys. J._ 742 (2011) 78 [1108.3384].
  * [8] Super-Kamiokande collaboration, K. Choi et al., _Search for neutrinos from annihilation of captured low-mass dark matter particles in the Sun by Super-Kamiokande_ , _Phys. Rev. Lett._ 114 (2015) 141301 [1503.04858].
  * [9] ANTARES collaboration, S. Adrian-Martinez et al., _First results on dark matter annihilation in the Sun using the ANTARES neutrino telescope_ , _JCAP_ 11 (2013) 032 [1302.6516].
  * [10] ANTARES collaboration, S. Adrian-Martinez et al., _Limits on Dark Matter Annihilation in the Sun using the ANTARES Neutrino Telescope_ , _Phys. Lett. B_ 759 (2016) 69 [1603.02228].
  * [11] IceCube collaboration, M. G. Aartsen et al., _Search for dark matter annihilations in the Sun with the 79-string IceCube detector_ , _Phys. Rev. Lett._ 110 (2013) 131302 [1212.4097].
  * [12] IceCube collaboration, R. Abbasi et al., _Search for GeV-scale dark matter annihilation in the Sun with IceCube DeepCore_ , _Phys. Rev. D_ 105 (2022) 062004 [2111.09970].
  * [13] DUNE collaboration, B. Abi et al., _Deep Underground Neutrino Experiment (DUNE), Far Detector Technical Design Report, Volume I Introduction to DUNE_ , _JINST_ 15 (2020) T08008 [2002.02967].
  * [14] DUNE collaboration, B. Abi et al., _Deep Underground Neutrino Experiment (DUNE), Far Detector Technical Design Report, Volume II: DUNE Physics_ , 2002.03005.
  * [15] Hyper-Kamiokande collaboration, K. Abe et al., _Hyper-Kamiokande Design Report_ , 1805.04163.
  * [16] N. F. Bell, M. J. Dolan and S. Robles, _Searching for dark matter in the Sun using Hyper-Kamiokande_ , _JCAP_ 11 (2021) 004 [2107.04216].
  * [17] D. Tucker-Smith and N. Weiner, _Inelastic dark matter_ , _Phys. Rev. D_ 64 (2001) 043502 [hep-ph/0101138].
  * [18] DAMA collaboration, R. Bernabei et al., _Search for WIMP annual modulation signature: Results from DAMA / NaI-3 and DAMA / NaI-4 and the global combined analysis_ , _Phys. Lett. B_ 480 (2000) 23.
  * [19] DAMA collaboration, R. Bernabei et al., _First results from DAMA/LIBRA and the combined results with DAMA/NaI_ , _Eur. Phys. J. C_ 56 (2008) 333 [0804.2741].
  * [20] DAMA, LIBRA collaboration, R. Bernabei et al., _New results from DAMA/LIBRA_ , _Eur. Phys. J. C_ 67 (2010) 39 [1002.1028].
  * [21] R. Garani and S. Palomares-Ruiz, _Dark matter in the Sun: scattering off electrons vs nucleons_ , _JCAP_ 05 (2017) 007 [1702.02768].
  * [22] J. Shu, P.-f. Yin and S.-h. Zhu, _Neutrino Constraints on Inelastic Dark Matter after CDMS II_ , _Phys. Rev. D_ 81 (2010) 123519 [1001.1076].
  * [23] C. Rott, J. Siegal-Gaskins and J. F. Beacom, _New Sensitivity to Solar WIMP Annihilation using Low-Energy Neutrinos_ , _Phys. Rev. D_ 88 (2013) 055005 [1208.0827].
  * [24] N. Bernal, J. Martín-Albo and S. Palomares-Ruiz, _A novel way of constraining WIMPs annihilations in the Sun: MeV neutrinos_ , _JCAP_ 08 (2013) 011 [1208.0834].
  * [25] C. Rott, S. In, J. Kumar and D. Yaylali, _Dark Matter Searches for Monoenergetic Neutrinos Arising from Stopped Meson Decay in the Sun_ , _JCAP_ 11 (2015) 039 [1510.00170].
  * [26] C. Rott, S. In, J. Kumar and D. Yaylali, _Directional Searches at DUNE for Sub-GeV Monoenergetic Neutrinos Arising from Dark Matter Annihilation in the Sun_ , _JCAP_ 01 (2017) 016 [1609.04876].
  * [27] C. Rott, D. Jeong, J. Kumar and D. Yaylali, _Neutrino Topology Reconstruction at DUNE and Applications to Searches for Dark Matter Annihilation in the Sun_ , _JCAP_ 07 (2019) 006 [1903.04175].
  * [28] DUNE collaboration, A. A. Abud et al., _Searching for solar KDAR with DUNE_ , _JCAP_ 10 (2021) 065 [2107.09109].
  * [29] A. Bueno, R. Cid, S. Navas, D. Hooper and T. J. Weiler, _Indirect detection of dark matter WIMPs in a liquid argon TPC_ , _JCAP_ 01 (2005) 001 [hep-ph/0410206].
  * [30] T. Golan, C. Juszczak and J. T. Sobczyk, _Final State Interactions Effects in Neutrino-Nucleus Interactions_ , _Phys. Rev. C_ 86 (2012) 015505 [1202.4197].
  * [31] R. Catena and F. Hellström, _New constraints on inelastic dark matter from IceCube_ , _JCAP_ 10 (2018) 039 [1808.08082].
  * [32] I. Jeong, S. Kang, S. Scopel and G. Tomar, _WimPyDD: An object–oriented Python code for the calculation of WIMP direct detection signals_ , _Comput. Phys. Commun._ 276 (2022) 108342 [2106.06207].
  * [33] LUX-ZEPLIN collaboration, J. Aalbers et al., _First Dark Matter Search Results from the LUX-ZEPLIN (LZ) Experiment_ , _Phys. Rev. Lett._ 131 (2023) 041002 [2207.03764].
  * [34] (XENON Collaboration)††, XENON collaboration, E. Aprile et al., _First Dark Matter Search with Nuclear Recoils from the XENONnT Experiment_ , _Phys. Rev. Lett._ 131 (2023) 041003 [2303.14729].
  * [35] PandaX-4T collaboration, Y. Meng et al., _Dark Matter Search Results from the PandaX-4T Commissioning Run_ , _Phys. Rev. Lett._ 127 (2021) 261802 [2107.13438].
  * [36] PICO collaboration, C. Amole et al., _Dark Matter Search Results from the Complete Exposure of the PICO-60 C 3F8 Bubble Chamber_, _Phys. Rev. D_ 100 (2019) 022001 [1902.04031].
  * [37] B. Chauhan, _Sub-MeV Self Interacting Dark Matter_ , _Phys. Rev. D_ 97 (2018) 123017 [1711.02970].
  * [38] M. W. Goodman and E. Witten, _Detectability of Certain Dark Matter Candidates_ , _Phys. Rev. D_ 31 (1985) 3059.
  * [39] SuperCDMS collaboration, R. Agnese et al., _Search for Low-Mass Weakly Interacting Massive Particles with SuperCDMS_ , _Phys. Rev. Lett._ 112 (2014) 241302 [1402.7137].
  * [40] SuperCDMS collaboration, R. Agnese et al., _New Results from the Search for Low-Mass Weakly Interacting Massive Particles with the CDMS Low Ionization Threshold Experiment_ , _Phys. Rev. Lett._ 116 (2016) 071301 [1509.02448].
  * [41] SuperCDMS collaboration, R. Agnese et al., _Improved WIMP-search reach of the CDMS II germanium data_ , _Phys. Rev. D_ 92 (2015) 072003 [1504.05871].
  * [42] CRESST collaboration, G. Angloher et al., _Results on light dark matter particles with a low-threshold CRESST-II detector_ , _Eur. Phys. J. C_ 76 (2016) 25 [1509.01515].
  * [43] EDELWEISS collaboration, L. Hehn et al., _Improved EDELWEISS-III sensitivity for low-mass WIMPs using a profile likelihood approach_ , _Eur. Phys. J. C_ 76 (2016) 548 [1607.03367].
  * [44] DarkSide collaboration, P. Agnes et al., _Low-Mass Dark Matter Search with the DarkSide-50 Experiment_ , _Phys. Rev. Lett._ 121 (2018) 081307 [1802.06994].
  * [45] CRESST collaboration, A. H. Abdelhameed et al., _First results from the CRESST-III low-mass dark matter program_ , _Phys. Rev. D_ 100 (2019) 102002 [1904.00498].
  * [46] R. Catena and P. Ullio, _A novel determination of the local dark matter density_ , _JCAP_ 08 (2010) 004 [0907.0018].
  * [47] J. Bovy and S. Tremaine, _On the local dark matter density_ , _Astrophys. J._ 756 (2012) 89 [1205.4033].
  * [48] N. Song, S. Nagorny and A. C. Vincent, _Pushing the frontier of WIMPy inelastic dark matter: Journey to the end of the periodic table_ , _Phys. Rev. D_ 104 (2021) 103032 [2104.09517].
  * [49] PICO collaboration, E. Adams et al., _Search for inelastic dark matter-nucleus scattering with the PICO-60 CF 3I and C3F8 bubble chambers_, 2301.08993.
  * [50] G. Herrera, A. Ibarra and S. Shirai, _Enhanced prospects for direct detection of inelastic dark matter from a non-galactic diffuse component_ , _JCAP_ 04 (2023) 026 [2301.00870].
  * [51] S. Nussinov, L.-T. Wang and I. Yavin, _Capture of Inelastic Dark Matter in the Sun_ , _JCAP_ 08 (2009) 037 [0905.1333].
  * [52] A. Menon, R. Morris, A. Pierce and N. Weiner, _Capture and Indirect Detection of Inelastic Dark Matter_ , _Phys. Rev. D_ 82 (2010) 015011 [0905.1847].
  * [53] M. Blennow, S. Clementz and J. Herrero-Garcia, _Pinning down inelastic dark matter in the Sun and in direct detection_ , _JCAP_ 04 (2016) 004 [1512.03317].
  * [54] N. Vinyoles, A. M. Serenelli, F. L. Villante, S. Basu, J. Bergström, M. C. Gonzalez-Garcia et al., _A new Generation of Standard Solar Models_ , _Astrophys. J._ 835 (2017) 202 [1611.09867].
  * [55] M. Blennow, S. Clementz and J. Herrero-Garcia, _The distribution of inelastic dark matter in the Sun_ , _Eur. Phys. J. C_ 78 (2018) 386 [1802.06880].
  * [56] R. Garani and S. Palomares-Ruiz, _Evaporation of dark matter from celestial bodies_ , _JCAP_ 05 (2022) 042 [2104.12757].
  * [57] S. Kang, A. Kar and S. Scopel, _Halo-independent bounds on Inelastic Dark Matter_ , 2308.13203.
  * [58] J. Smolinsky and P. Tanedo, _Dark Photons from Captured Inelastic Dark Matter Annihilation: Charged Particle Signatures_ , _Phys. Rev. D_ 95 (2017) 075015 [1701.03168].
  * [59] P. Baratella, M. Cirelli, A. Hektor, J. Pata, M. Piibeleht and A. Strumia, _PPPC 4 DM $\nu$: a Poor Particle Physicist Cookbook for Neutrinos from Dark Matter annihilations in the Sun_, _JCAP_ 03 (2014) 053 [1312.6408].
  * [60] I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz and A. Zhou, _The fate of hints: updated global analysis of three-flavor neutrino oscillations_ , _JHEP_ 09 (2020) 178 [2007.14792].
  * [61] M. Wallraff and C. Wiebusch, _Calculation of oscillation probabilities of atmospheric neutrinos using nuCraft_ , _Comput. Phys. Commun._ 197 (2015) 185 [1409.1387].
  * [62] L. Wolfenstein, _Neutrino Oscillations in Matter_ , _Phys. Rev. D_ 17 (1978) 2369.
  * [63] S. P. Mikheyev and A. Y. Smirnov, _Resonance Amplification of Oscillations in Matter and Spectroscopy of Solar Neutrinos_ , _Sov. J. Nucl. Phys._ 42 (1985) 913.
  * [64] E. K. Akhmedov, _Neutrino oscillations in inhomogeneous matter. (In Russian)_ , _Sov. J. Nucl. Phys._ 47 (1988) 301.
  * [65] P. I. Krastev and A. Y. Smirnov, _Parametric Effects in Neutrino Oscillations_ , _Phys. Lett. B_ 226 (1989) 341.
  * [66] K. J. Kelly, P. A. N. Machado, I. Martinez-Soler and Y. F. Perez-Gonzalez, _DUNE atmospheric neutrinos: Earth tomography_ , _JHEP_ 05 (2022) 187 [2110.00003].
  * [67] Astropy collaboration, T. P. Robitaille et al., _Astropy: A Community Python Package for Astronomy_ , _Astron. Astrophys._ 558 (2013) A33 [1307.6212].
  * [68] M. Honda, M. Sajjad Athar, T. Kajita, K. Kasahara and S. Midorikawa, _Atmospheric neutrino flux calculation using the NRLMSISE-00 atmospheric model_ , _Phys. Rev. D_ 92 (2015) 023004 [1502.03916].
  * [69] K. C. Y. Ng, J. F. Beacom, A. H. G. Peter and C. Rott, _Solar Atmospheric Neutrinos: A New Neutrino Floor for Dark Matter Searches_ , _Phys. Rev. D_ 96 (2017) 103006 [1703.10280].
  * [70] C. A. Argüelles, G. de Wasseige, A. Fedynitch and B. J. P. Jones, _Solar Atmospheric Neutrinos and the Sensitivity Floor for Solar Dark Matter Annihilation Searches_ , _JCAP_ 07 (2017) 024 [1703.07798].
  * [71] J. Edsjo, J. Elevant, R. Enberg and C. Niblaeus, _Neutrinos from cosmic ray interactions in the Sun_ , _JCAP_ 06 (2017) 033 [1704.02892].
  * [72] C. Rott, S. In, J. Kumar and D. Yaylali, _New Dark Matter Search Strategies at DUNE_ , _J. Phys. Conf. Ser._ 1342 (2020) 012004 [1710.03822].
  * [73] Super-Kamiokande collaboration, K. Abe et al., _Atmospheric neutrino oscillation analysis with external constraints in Super-Kamiokande I-IV_ , _Phys. Rev. D_ 97 (2018) 072001 [1710.09126].