# High-Energy Neutrinos From Millicharged Dark Matter Annihilation in the Sun

Asher Berlina,b<EMAIL_ADDRESS>Dan Hoopera,c,d<EMAIL_ADDRESS>aTheory
Division, Fermi National Accelerator Laboratory  bSuperconducting Quantum
Materials and Systems Center (SQMS), Fermi National Accelerator Laboratory
cUniversity of Chicago, Department of Astronomy and Astrophysics  dUniversity
of Chicago, Kavli Institute for Cosmological Physics

###### Abstract

Millicharged dark matter particles can be efficiently captured by the Sun,
where they annihilate into tau leptons, leading to the production of high-
energy neutrinos. In contrast to the Earth, the high temperature of the Sun
suppresses the fraction of millicharged particles that are bound to nuclei,
allowing for potentially high annihilation rates. We recast existing
constraints from the IceCube Neutrino Observatory and use this information to
place new limits on the fraction of the dark matter that is millicharged. This
analysis excludes previously unexplored parameter space for masses of
$m_{\chi}\sim(5-100)\ \text{GeV}$, charges of $q_{\chi}\sim 10^{-3}-10^{-2}$,
and fractional abundances as small as $f_{{}_{\text{DM}}}\sim 10^{-5}$.

††preprint: FERMILAB-PUB-24-0353-T

##  I Introduction

Although all known particles carry values of electric charge that are of order
unity (or are electrically neutral), models featuring elementary particles
with much smaller quantities of charge have received a significant amount of
interest in recent years Haas  _et al._ (2015); Izaguirre and Yavin (2015);
Magill  _et al._ (2019); Kelly and Tsai (2019); Acciarri  _et al._ (2020);
Harnik  _et al._ (2019); Acciarri  _et al._ (2020); Perez  _et al._ (2024);
Plestid  _et al._ (2020); Bloch  _et al._ (2021); Foroughi-Abari  _et al._
(2021); Harnik  _et al._ (2021); Marocco and Sarkar (2021); Berlin  _et al._
(2023a); Chen  _et al._ (2023); Budker  _et al._ (2022); Argüelles Delgado
_et al._ (2021); Afek  _et al._ (2021). Such millicharged particles can arise,
for example, if an effective charge is generated through the kinetic mixing of
the photon with a new light dark photon,
$\mathscr{L}\supset(\epsilon/2)F^{\mu\nu}F^{\prime}_{\mu\nu}$, where
$F^{\prime}_{\mu\nu}$ is the dark photon field strength and $\epsilon$ is a
small dimensionless parameter. This mixing causes particles, $\chi^{\pm}$,
charged under this new $U(1)^{\prime}$, to acquire an effective electric
charge, $q_{\chi}=\epsilon\,e^{\prime}/e$, where $e^{\prime}$ is the
$U(1)^{\prime}$ gauge coupling and $e$ is the standard electromagnetic
coupling Holdom (1986).

Within the context of effective field theory, any value of $\epsilon\ll 1$ is
technically natural. If the Standard Model is embedded within a Grand Unified
Theory, however, this mixing is generated only through loops of particles that
carry both hypercharge and $U(1)^{\prime}$ charge. At the one-loop level, the
expected size of this mixing is given by

$\epsilon\sim\frac{e^{\prime}e}{16\pi^{2}}\ln\bigg{(}\frac{M^{\prime}}{M}\bigg{)}\simeq
6\times
10^{-4}\times\bigg{(}\frac{e^{\prime}}{e}\bigg{)}\,\ln\bigg{(}\frac{M^{\prime}}{M}\bigg{)}\,,$
(1)

where $M^{\prime}/M$ is the ratio of the masses in the loop Holdom (1986);
Cohen  _et al._ (2010); Baumgart  _et al._ (2009). From this perspective,
particles with $q_{\chi}\sim 10^{-3}$ are particularly well motivated.

In this work, we consider the possibility that a fraction of the cosmological
dark matter density consists of millicharged particles. Although observations
of the cosmic microwave background and measurements of the primordial element
abundances have been used to place strong constraints on charged dark matter,
these observations remain consistent with the possibility that a small
fraction, $f_{{}_{\text{DM}}}\lesssim 10^{-2}$, of the dark matter could carry
significant electric charge Dubovsky  _et al._ (2004); de Putter  _et al._
(2019); Kovetz  _et al._ (2018); Buen-Abad  _et al._ (2022); Stebbins and
Krnjaic (2019). In particular, a millicharged dark matter subcomponent could
arise as a thermal relic of the early universe. Assuming standard thermal
freeze-out, the requirement of perturbative unitarity Griest and Kamionkowski
(1990) can be used to place the following lower limit on the fractional
abundance of this component:

$f_{{}_{\text{DM}}}\gtrsim 10^{-10}\times(m_{\chi}/\text{GeV})^{2}~{},$ (2)

where $m_{\chi}$ is the mass of the millicharged particle, $\chi^{\pm}$.

Even for very small fractional abundances, underground searches for dark
matter scattering have been used to place extremely strong constraints on
millicharged subcomponents, testing particles with charges as small as
$q_{\chi}\sim 10^{-10}/f_{{}_{\text{DM}}}^{1/2}$ Emken  _et al._ (2019). These
experiments are almost completely insensitive to particles with sufficiently
large values of $q_{\chi}$, however, since such particles scatter many times
in the terrestrial overburden, shedding their kinetic energy and thermalizing
down to room temperature, $300\ \text{K}\simeq 26\ \text{meV}$, well below the
threshold of typical large-scale experiments. As a result, millicharged
particles with $q_{\chi}\gtrsim 10^{-4}\times(m_{\chi}/\text{GeV})^{1/2}$
reside within a blind spot of existing direct detection efforts Emken  _et
al._ (2019).

The efficient scattering of millicharged particles with astrophysical objects
within our Solar System has a drastic effect on their local distribution and
density. For $m_{\chi}\gg 1\ \text{GeV}$, the characteristic thermal velocity
of these particles will be well below the gravitational escape velocity of the
Sun or Earth. As a result, such particles remain gravitationally bound to
these objects after scattering, accumulating as a dense hydrostatic gas over
the age of the Solar System.

The possible accumulation of dark matter particles within stellar bodies has
been studied extensively, dating back to the 1980s Krauss  _et al._ (1985);
Press and Spergel (1985); Griest and Seckel (1987); Gould (1987a, b); Gould
and Raffelt (1990). It has also long been appreciated that such particles
could give rise to interesting signatures, arising from their annihilation
into Standard Model states. Examples include the prompt annihilation into
neutrinos Silk  _et al._ (1985); Srednicki  _et al._ (1987); Kamionkowski
(1991); Barger  _et al._ (2002); Flacke  _et al._ (2009) or long-lived
mediators Schuster  _et al._ (2010); Batell  _et al._ (2010) in the Sun,
annihilations in the local volume of terrestrial neutrino detectors Pospelov
and Ramani (2021); McKeen  _et al._ (2023), and annihilations in Earth’s core
Mack  _et al._ (2007); Pospelov and Ray (2024).

Signals arising from the annihilation of millicharged particles
gravitationally bound to Earth was studied in Ref. Pospelov and Ramani (2021).
However, in this case, an additional complication arises from the fact that
$\chi^{-}$ can bind with atomic nuclei, $N$. In turn, the resulting Coulomb
barrier between $(\chi^{-}N)$ and $\chi^{+}$ drastically suppresses the local
annihilation rate of millicharged particles. For $q_{\chi}\gg 10^{-5}$, the
binding energy of $(\chi^{-}N)$ greatly exceeds the temperature of Earth’s
core, exponentially suppressing the rate of terrestrial-based annihilation
signals.

In this paper, we instead focus on the capture and annihilation of
millicharged particles in the Sun, leading to the production of high-energy
neutrinos through $\chi^{+}\chi^{-}\to\tau^{+}\tau^{-}$ followed by the decay
of the tau leptons into neutrinos. In contrast to millicharged particles in
the Earth, the high temperature of the solar interior suppresses the fraction
of these particles that will be bound to nuclei, making this an ideal
environment to produce detectable fluxes of annihilation products. We recast
the results of existing searches for high-energy neutrinos from the Sun, as
recently performed by the IceCube Collaboration Aartsen  _et al._ (2017);
Abbasi  _et al._ (2022). Using this data, we place new limits on millicharged
dark matter, excluding previously unexplored parameter space corresponding to
$m_{\chi}\sim(5-100)\ \text{GeV}$ and $q_{\chi}\sim 10^{-3}-10^{-2}$ for
fractional abundances as small as $f_{{}_{\text{DM}}}\sim 10^{-5}$.

The remainder of this paper is organized as follows. In Secs. II and III, we
discuss the dynamics associated with the capture and annihilation of
millicharged dark matter in the Sun. Sec. IV addresses modifications that can
arise if $\chi^{-}$ can efficiently form bound states with solar nuclei. This
formalism is applied in Sec. V to derive constraints, using existing data from
the high-energy neutrino telescope, IceCube. We then compare these limits to
other bounds on millicharged dark matter. In Sec. VI, we summarize our results
and discuss directions for future work.

##  II Capture and Annihilation of Dark Matter in the Sun

The total number millicharged particles that are gravitationally bound to the
Sun, $N_{\chi}$, evolves in time according to

$\frac{dN_{\chi}}{dt}\simeq\Gamma_{\text{cap}}-N_{\chi}^{2}\,K_{\text{ann}}~{},$
(3)

where the first and second terms on the right-hand side correspond to the
capture and annihilation rates, respectively. Here, we have neglected the
effects of millicharged particle evaporation, since for $m_{\chi}\gtrsim 1\
\text{GeV}$ their thermal velocities are smaller than the gravitational escape
velocity near the Sun’s surface,
$v_{\text{esc}}=(2GM_{\odot}/R_{\odot})^{1/2}\simeq 2\times 10^{-3}$, where
$M_{\odot}$ and $R_{\odot}$ are the solar mass and radius, respectively.

Within the parameter space that we will consider in this work, nearly every
millicharged particle that passes through the Sun will scatter, become
gravitationally bound, and reach kinetic equilibrium Pospelov and Ramani
(2021); Berlin  _et al._ (2024). To determine the capture rate, we therefore
simply calculate the rate at which such particles enter the Sun. The
gravitational pull of the Sun significantly increases the flux of dark matter
particles that reach its surface. The distance of closest approach to the Sun
for a dark matter particle entering the Solar System with an impact parameter,
$b$, and a velocity, $v_{{}_{\text{DM}}}$, is given by

$R_{\text{min}}=\frac{G\,M_{\odot}}{v^{2}_{{}_{\text{DM}}}}\,\bigg{[}\bigg{(}1+\frac{b^{2}\,v^{4}_{{}_{\text{DM}}}}{G^{2}M^{2}_{\odot}}\bigg{)}^{1/2}-1\bigg{]}\,.$
(4)

Setting $R_{\text{min}}\leq R_{\odot}$, we find that such particles will pass
through the Sun for all impact parameters less than

$b\leq
b_{\text{max}}=R_{\odot}\,\bigg{(}1+\frac{v_{\text{esc}}^{2}}{v^{2}_{{}_{\text{DM}}}}\bigg{)}^{1/2}\,.$
(5)

From this, it follows that the effective cross-sectional area for capture in
the Sun will be given by

$\displaystyle A_{\text{cap}}=\pi\,b_{\text{max}}^{2}=\pi
R^{2}_{\odot}\bigg{(}1+\frac{v_{\text{esc}}^{2}}{v^{2}_{{}_{\text{DM}}}}\bigg{)}\,.$
(6)

This leads to a capture rate of

$\displaystyle\Gamma_{\text{cap}}$
$\displaystyle\simeq(f_{{}_{\text{DM}}}\,\rho_{{}_{\text{DM}}}/m_{\chi})\,v_{{}_{\text{DM}}}\,A_{\text{cap}}\,,$
(7)

where $\rho_{{}_{\text{DM}}}\simeq 0.4\ \text{GeV}/\text{cm}^{3}$ is the local
dark matter density. To account for the spread in dark matter velocities, we
average the quantity $v_{{}_{\text{DM}}}A_{\text{cap}}$ in Eq. 7 over the
velocity distribution of the Standard Halo Model, consisting of a boosted-
Maxwellian (with $v_{0}=220\ \text{km}/\text{s}$ and $v_{\odot}=240\
\text{km}/\text{s}$) truncated at the Galactic escape velocity,
$v_{\text{esc}}^{\rm Gal}=550\ \text{km}/\text{s}$. We find that this can be
approximated by simply using Eqs. 6 and 7 with a representative value of
$v_{{}_{\text{DM}}}\simeq 9\times 10^{-4}$.

Figure 1: Left panel: The density profile of millicharged particles
gravitationally bound to the Sun, normalized to the volume-averaged density,
$\langle n_{\chi}\rangle$, for several values of the particle’s mass. For
$m_{\chi}\lesssim 1\ \text{GeV}$, such particles are approximately uniformly
distributed throughout the solar interior. For larger masses, the density
profile is much more highly concentrated near the solar core. Right panel: The
normalized annihilation rate, $\langle n_{\chi}^{2}\rangle/\langle
n_{\chi}\rangle^{2}$, as found in Eq. 8. For $m_{\chi}\ll 1\ \text{GeV}$, the
uniform density profile of millicharged particles implies that $\langle
n_{\chi}^{2}\rangle/\langle n_{\chi}\rangle^{2}\simeq 1$. For larger masses,
the concentration of millicharged particles near the solar core enhances the
likelihood of annihilation.

For a charge-symmetric population of millicharged particles, the quantity
$K_{\text{ann}}$ appearing in Eq. 3 is given by

$\displaystyle K_{\text{ann}}$
$\displaystyle=\frac{1}{2N_{\chi}^{2}}\,\int_{V_{\odot}}n_{\chi}^{2}\,\langle\sigma
v\rangle\,dV\simeq\frac{\langle\sigma
v\rangle}{2\,V_{\odot}}\times\frac{\langle n_{\chi}^{2}\rangle}{\langle
n_{\chi}\rangle^{2}}\,,$ (8)

where the integral in the first equality is performed over the volume of the
Sun, $V_{\odot}$, $n_{\chi}\equiv n_{\chi^{+}}+n_{\chi^{-}}$ is the number
density of millicharged particles, and $\langle\sigma v\rangle$ is the
thermally-averaged $\chi^{+}\chi^{-}$ annihilation cross section. In the
second equality of this expression, we have approximated $\langle\sigma
v\rangle$ as independent of temperature, and the brackets for $\langle
n_{\chi}\rangle$ and $\langle n_{\chi}^{2}\rangle$ denote an average over the
solar volume.

The propagation of charged particles in the Solar System can be impacted by
the turbulent solar wind and its embedded magnetic field, the effects of which
are collectively known as solar modulation Gleeson and Axford (1968); Strauss
_et al._ (2012); Maccione (2013); Cholis  _et al._ (2016, 2022). Solar
modulation significantly suppresses the intensity of the cosmic ray spectrum
at the location of Earth for kinetic energies below $\sim
E_{\text{kin}}\lesssim|Q|\Phi$, where $Q$ is the charge of the cosmic ray and
$\Phi\sim(0.2-1)\ \text{GeV}$ Potgieter (2014); Cholis  _et al._ (2016).

Analogously, it is possible that solar modulation could suppress the flux of
millicharged dark matter particles that reaches the inner Solar System. We
expect these effects to be significant if
$m_{\chi}\,(v_{{{}_{\text{DM}}}}^{2}+v^{2}_{\text{esc}})/2\lesssim
q_{\chi}\,\Phi$, or equivalently,

$m_{\chi}\lesssim 100\
\text{GeV}\times\bigg{(}\frac{q_{\chi}}{10^{-3}}\bigg{)}\bigg{(}\frac{\Phi}{0.2\
\text{GeV}}\bigg{)}\,.$ (9)

This condition will be satisfied in much of the parameter space that we are
interested in, potentially reducing the sensitivity of any solar or
terrestrial-based searches for such particles.

In the above discussion of solar modulation, we have explicitly assumed that
the millicharged particles couple to electromagnetic fields in the same way as
ordinary charged particles. If these interactions are instead mediated by a
kinetically-mixed dark photon, $A^{\prime}$, their range will be limited to
$\sim m_{A^{\prime}}^{-1}$, where $m_{A^{\prime}}$ is the mass of the dark
photon. In this work, we will focus on charges,
$q_{\chi}=\epsilon\,e^{\prime}/e\gtrsim 10^{-4}$, corresponding to kinetic
mixing parameters of $\epsilon\gtrsim 10^{-5}/e^{\prime}$. For light dark
photons ($m_{A^{\prime}}\lesssim\text{MeV}$), such values of $\epsilon$ are
highly constrained unless $m_{A^{\prime}}\lesssim 10^{-14}\
\text{eV}\sim(10^{4}\ \text{km})^{-1}$ Mirizzi  _et al._ (2009a, b); Kunze and
Vázquez-Mozo (2015); Caputo  _et al._ (2020a, b); Garcia  _et al._ (2020),
corresponding to distances larger than an Earth radius. Hence, within the
viable region of parameter space, $\chi^{\pm}$ efficiently couples to
terrestrial magnetic fields. On the other hand, since the coherence length of
the solar magnetic field is of the same order as the solar radius Longair
(2011), the effects of solar modulation will be suppressed if
$m_{A^{\prime}}\gg R_{\odot}^{-1}\sim 10^{-16}\ \text{eV}$. Throughout the
remainder of this work, we will implicitly assume that this is the case,
allowing us to provide a meaningful comparison between the signals discussed
here and other local searches for millicharged dark matter (all of which are
subject to the effects of solar modulation). Alternatively, solar modulation
may be implicitly incorporated into our analysis by rescaling the incoming
millicharged particle flux, and thereby the definition of
$f_{{}_{\text{DM}}}$.

To evaluate the annihilation rate in Eq. 8, we will need the radial density
profile of millicharged particles in the Sun, $n_{\chi}$. From the condition
of hydrostatic equilibrium, the Euler equation implies that

$m_{\chi}\,n_{\chi}\,\boldsymbol{g}_{\odot}=\nabla P_{\chi}~{},$ (10)

where $\boldsymbol{g}_{\odot}$ is the solar gravitational field and $P_{\chi}$
is the pressure of the millicharged particles. Treating these particles as an
ideal gas, the pressure is given by $P_{\chi}\simeq T\,n_{\chi}$, where $T$ is
the temperature of the Sun. In the spherically-symmetric limit, Eq. 10 can be
written as

$\frac{1}{n_{\chi}(r)}\,\frac{dn_{\chi}(r)}{dr}+\frac{1}{T(r)}\,\frac{dT(r)}{dr}+\frac{m_{\chi}\,g_{\odot}(r)}{T(r)}\simeq
0~{}.$ (11)

For a given value of $m_{\chi}$, we solve Eq. 11 numerically, incorporating
the solar density and temperature profiles from the standard solar model
Bahcall and Pinsonneault (2004). These solutions are presented in Fig. 1. In
the left panel, we show the normalized density profile, $n_{\chi}(r)/\langle
n_{\chi}\rangle$, for several choices of $m_{\chi}$. For $m_{\chi}\lesssim 1\
\text{GeV}$, gravitationally bound millicharged particles are distributed
approximately uniformly throughout the solar interior. For heavier particles,
larger pressure and density gradients are required to offset the gravitational
attraction, leading to a profile that is much more highly concentrated near
the solar core. This behavior is also reflected in the right panel of Fig. 1,
in which we show the normalized annihilation rate, $\langle
n_{\chi}^{2}\rangle/\langle n_{\chi}\rangle^{2}$, as appears in the last
factor of Eq. 8. For $m_{\chi}\ll 1\ \text{GeV}$, the uniform density profile
of millicharged particles implies that $\langle n_{\chi}^{2}\rangle/\langle
n_{\chi}\rangle^{2}\simeq 1$. For larger masses, the concentration of
millicharged particles near the solar core enhances the likelihood of
annihilation, such that $\langle n_{\chi}^{2}\rangle/\langle
n_{\chi}\rangle^{2}\gg 1$.

We are now in a position to calculate the capture and annihilation rates of
millicharged particles in the Sun. Solving Eq. 3 with the initial condition
$N_{\chi}=0$ at $t=0$, the value of $N_{\chi}$ today is given by

$\displaystyle N_{\chi}$
$\displaystyle\simeq\frac{\Gamma_{\text{cap}}}{K_{\text{ann}}}\,\tanh{\big{(}\sqrt{\Gamma_{\text{cap}}\,K_{\text{ann}}}~{}t_{\odot}\big{)}}$
$\displaystyle\simeq\begin{cases}\Gamma_{\text{cap}}\,t_{\odot}&(t_{\odot}\ll\tau)\\\\[3.0pt]
\sqrt{\Gamma_{\text{cap}}/K_{\text{ann}}}&(t_{\odot}\gg\tau)~{},\end{cases}$
(12)

where $t_{\odot}\simeq 4.5\ \text{Gyr}$ is the age of the Sun and
$\tau\equiv(\Gamma_{\text{cap}}\,K_{\text{ann}})^{-1/2}$. From Eqs. II and 3,
the annihilation rate grows as
$N_{\chi}^{2}\,K_{\text{ann}}\sim\Gamma^{2}_{\text{cap}}\,K_{\text{ann}}\,t^{2}$
until $t\sim\tau$, after which equilibrium is reached between the rates of
capture and annihilation,
$N_{\chi}^{2}\,K_{\text{ann}}\simeq\Gamma_{\text{cap}}$.

##  III Millicharged Dark Matter Annihilation

Millicharged particles can annihilate directly into pairs of charged Standard
Model fermions through the exchange of an intermediate photon,
$\chi^{+}\chi^{-}\to\gamma^{*}\to f^{+}f^{-}$. In the non-relativistic limit,
the cross section for this processes is approximately given by

$\langle\sigma v\rangle_{\chi^{+}\chi^{-}\rightarrow
f^{+}f^{-}}\simeq\frac{\pi\alpha^{2}q_{\chi}^{2}}{m_{\chi}^{2}}\sum_{f}\,n_{f}\,q_{f}^{2}~{},$
(13)

where the sum is over all fermions with mass $m_{f}\ll m_{\chi}$ and charge
$q_{f}$, and $n_{f}$ is the number of colors of the fermion. Annihilations to
electroweak bosons and neutrinos are also possible, but depend on the details
of the full theory. For instance, if these interactions are generated by a
kinetically-mixed dark photon, $\chi^{\pm}$ will be millicharged under
Standard Model hypercharge, such that it develops a small coupling to both the
photon and the $Z$ boson. Hence, for $m_{\chi}\gtrsim 10^{2}\ \text{GeV}$,
annihilations to pairs of $W$ bosons and neutrinos are unsuppressed, further
contributing to the signals discussed here. Since such processes are model-
dependent, we have not included their contributions in our final estimate.

In models that include a dark photon, $A^{\prime}$, the dark matter can also
annihilate to a pair of such particles. The cross section for this process is
given by

$\langle\sigma v\rangle_{\chi^{+}\chi^{-}\rightarrow
A^{\prime}A^{\prime}}\simeq\frac{\pi\alpha^{\prime\,2}}{m_{\chi}^{2}}~{},$
(14)

where $m_{A^{\prime}}$ is the mass of the dark photon and
$\alpha^{\prime}=e^{\prime\,2}/4\pi$. If $\alpha^{\prime}$ is sufficiently
large, annihilations to dark photons could dominate over those to Standard
Model final states, resulting in a suppression of the neutrino flux from the
Sun as calculated in this study. In our analysis, we neglect this process,
which is equivalent to taking $\alpha^{\prime}\ll\alpha$.

##  IV Bound States

Negative-millicharged particles, $\chi^{-}$, can efficiently bind to nuclei,
$N$, in the Sun, with a characteristic binding energy that is given by
$E_{\chi N}\simeq(q_{\chi}Z\alpha)^{2}\,\mu_{\chi N}/2$, where $Z$ is the
atomic number of the nucleus and $\mu_{\chi N}$ is the reduced mass of the
system. Hence, for $m_{\chi}\gtrsim\mu_{\chi N}\sim 10\ \text{GeV}$ and
$q_{\chi}\gtrsim(m_{e}/\mu_{\chi N})^{1/2}\sim 10^{-2}$, millicharged
particles will be much more tightly-bound to nuclei than electrons are.
Furthermore, for $Z\sim 10$ and similar values of the coupling $q_{\chi}$,
$E_{\chi N}$ will exceed the temperature in the solar core, $T\sim 1\
\text{keV}$, in which case the formation of $(\chi^{-}N)$ bound states will be
energetically favored.

Since $(\chi^{-}N)$ has a net positive charge of $Q_{\chi N}=Z-q_{\chi}\simeq
Z$, its formation suppresses the rate at which millicharged particles
annihilate. In particular, such annihilations require that a $(\chi^{-}N)$ and
$\chi^{+}$ overcome the repulsive Coulomb barrier, which is comparable to the
binding energy, $Q_{\chi N}\,q_{\chi}\,\alpha/r_{\chi N}\simeq E_{\chi N}$,
where $r_{\chi N}\simeq(q_{\chi}Z\alpha\,\,\mu_{\chi N})^{-1}$ is the size of
the bound state. As a result, if $E_{\chi N}\gtrsim T$, then $(\chi^{-}N)$ and
$\chi^{+}$ will not possess sufficient kinetic energy to overcome their
electrical repulsion.111For sufficiently large $\alpha^{\prime}$, the
attractive dark Coulomb potential can overcome this repulsion. However, since
this also suppresses the fraction of annihilations to Standard Model states
(as discussed in Sec. III), we will ignore this possibility.

As the rates for the formation and disruption of $(\chi^{-}N)$ bound states in
the Sun are both very rapid Pospelov and Ramani (2021), chemical equilibrium
will be reached and maintained between the bound, $(\chi^{-}N)$, and unbound,
$\chi^{-}$, populations. From the condition of chemical equilibrium,
$\mu_{\chi^{-}}+\mu_{N}=\mu_{(\chi^{-}N)}$, we can use the Maxwell-Boltzmann
distributions of these species to relate the number density of $(\chi^{-}N)$
bound states to the number densities of free nuclei and millicharged
particles:

$\frac{n_{\chi^{-}}\,n_{N}}{n_{(\chi^{-}N)}}=\Big{(}\frac{\mu_{\chi
N}\,T}{2\pi}\Big{)}^{3/2}\,e^{-E_{\chi N}/T}~{}.$ (15)

Since annihilations between $\chi^{+}$ and $(\chi^{-}N)$ are exponentially
suppressed by their repulsive Coulomb barrier, we incorporate the effect of
bound state formation by rescaling the $\chi^{+}\chi^{-}$ annihilation cross
section by $\langle\sigma v\rangle\to\mathcal{R}_{\text{BS}}\,\langle\sigma
v\rangle$, where

$\mathcal{R}_{\text{BS}}\equiv\frac{n_{\chi^{-}}+e^{-E_{\chi
N}/T}\,n_{(\chi^{-}N)}}{n_{\chi^{-}}+n_{(\chi^{-}N)}}~{}.$ (16)

Using Eq. 15, this becomes

$\displaystyle\mathcal{R}_{\text{BS}}$
$\displaystyle=\frac{F_{N}+1}{F_{N}+\,e^{E_{\chi N}/T}}~{},$ (17)

where

$\displaystyle F_{N}$ $\displaystyle\equiv\Big{(}\frac{\mu_{\chi
N}\,T}{2\pi}\Big{)}^{3/2}\,\frac{1}{n_{N}}~{}.$ (18)

From this calculation, we see that bound state formation exponentially
suppresses the annihilation rate when $E_{\chi N}\gtrsim T\,\ln{F_{N}}$. We
incorporate this into our analysis by considering interactions with thorium
(Th) nuclei, the highest-$Z$ element measured in the Sun, which constitutes
$\sim 2\times 10^{-10}$ of the mass density in the photosphere Asplund  _et
al._ (2009). We take this to be representative of the entire solar volume,
with $n_{N}\simeq 7\times 10^{11}\ \text{cm}^{-3}$ independent of radius. We
use this in Eq. 17 along with $T=1\ \text{keV}$, which yields $\ln F_{N}\sim
50$ for $m_{\chi}\gtrsim 1\ \text{GeV}$.

Figure 2: In shaded red, regions of the millicharged dark matter parameter
space that is excluded by our analysis, based on data from the IceCube
Neutrino Observatory Aartsen  _et al._ (2017); Abbasi  _et al._ (2022), and
for several choices of the fractional abundance, $f_{{}_{\text{DM}}}$. The
dashed red lines denote the largest couplings that would have been excluded by
our analysis if we had considered bound states with oxygen nuclei, rather than
thorium (see Sec. IV). Also shown in gray are previous constraints from
accelerator searches Davidson  _et al._ (2000); Chatrchyan  _et al._ (2013a,
b); Hayrapetyan  _et al._ (2024), direct detection experiments Emken  _et al._
(2019), XQC Erickcek  _et al._ (2007); Mahdawi and Farrar (2018), and RRS Rich
_et al._ (1987); Emken  _et al._ (2019). For RRS and XQC, we have extrapolated
previous results to higher masses and rescaled to smaller fractional
abundances; we have refrained from showing these limits when the gyroradius of
millicharged dark matter in Earth’s magnetic field is smaller than one tenth
of an Earth radius.

##  V Constraints From IceCube

The IceCube Neutrino Observatory consists of several thousand optical modules
distributed throughout a cubic kilometer of Antarctic ice. In the energy range
of interest, this instrument is primarily sensitive to muon tracks which
result from the charged-current interactions of muon neutrinos in or near the
detector’s instrumented volume. Dark matter annihilating in the Sun could
produce a potentially detectable flux of energetic neutrinos, allowing IceCube
to place constraints on the particle nature of dark matter.

The IceCube Collaboration has placed stringent limits on the flux of high-
energy neutrinos that reaches us from the direction of the Sun Abbasi  _et
al._ (2022); Aartsen  _et al._ (2017, 2016, 2013); Abbasi  _et al._ (2012,
2009). In this study, we recast the limits presented in Refs. Aartsen  _et
al._ (2017); Abbasi  _et al._ (2022), which constrain the following quantity:

$\Gamma_{\text{ann}}\equiv\frac{1}{2}\,N_{\chi}^{2}\,K_{\text{ann}}^{\text{SM}}=\frac{1}{4}\,\int_{V_{\odot}}n_{\chi}^{2}\,\langle\sigma
v\rangle_{\text{SM}}\,dV~{},$ (19)

where $K_{\text{ann}}^{\text{SM}}$ is defined analogously to Eq. 8, but in
terms of the cross section for annihilations to a particular Standard Model
final state, $\langle\sigma v\rangle_{\text{SM}}$. In this study, we focus on
the case of $\tau^{+}\tau^{-}$ final states, as these lead to strong bounds
and do not depend on the complete model details of the theory (see the
discussion in Sec. III).

Before proceeding, it is worthwhile to use the above results to estimate the
signal rate in Eq. 19. This is simplest in the large coupling regime, in which
case the capture and annihilation rate have reached steady state. In this
case, Eqs. 6, 7, and II can be used to evaluate this quantity:

$\Gamma_{\text{ann}}\simeq 5\times 10^{29}\ \text{Hz}\times
f_{{}_{\text{DM}}}\,\bigg{(}\frac{\langle\sigma
v\rangle_{\text{SM}}}{\langle\sigma
v\rangle}\bigg{)}\,\bigg{(}\frac{\text{GeV}}{m_{\chi}}\bigg{)}\,.$ (20)

Note that $\Gamma_{\text{ann}}$ is independent of $q_{\chi}$, provided that
the annihilations are dominated by Standard Model final states. As an example,
for the annihilations of $m_{\chi}=100\ \text{GeV}$ millicharged particles to
$\tau^{+}\tau^{-}$, the search in Ref. Abbasi  _et al._ (2022) places an upper
limit of $\Gamma_{\text{ann}}\lesssim 4\times 10^{21}\ \text{Hz}$. Using the
fact that $\langle\sigma v\rangle_{\tau^{+}\tau^{-}}\sim
10^{-1}\times\langle\sigma v\rangle$, Eq. 20 then implies that IceCube is
sensitive to fractional abundances as small as
$f_{{}_{\text{DM}}}\sim\text{few}\times 10^{-5}$.

This is confirmed in Fig. 2, which presents our limits in the
$q_{\chi}-m_{\chi}$ plane for various representative choices of the fractional
abundance, $f_{{}_{\text{DM}}}$. Also shown are previous constraints from
accelerator searches Davidson  _et al._ (2000); Chatrchyan  _et al._ (2013a,
b); Hayrapetyan  _et al._ (2024), direct detection experiments Emken  _et al._
(2019), the rocket-based XQC experiment Erickcek  _et al._ (2007); Mahdawi and
Farrar (2018), and the balloon-based RRS experiment Rich  _et al._ (1987);
Emken  _et al._ (2019). For these latter two searches, we have extrapolated to
higher masses and rescaled to smaller fractional abundances. We note that
terrestrial dark matter searches cannot be simply extended to regions of
parameter space in which millicharged particles strongly couple to Earth’s
magnetic field. In particular, we refrain from showing these limits when the
gyroradius of $\chi^{\pm}$ is smaller than $10^{-1}\,R_{\oplus}$, where
$R_{\oplus}\simeq 6400\ \text{km}$ is the radius of Earth, since the
millicharged particles will be significantly deflected before reaching
terrestrial-based detectors in this case Emken  _et al._ (2019). We find that
IceCube is sensitive to millicharged dark matter for masses in the range of
$5\ \text{GeV}-10\ \text{TeV}$ for fractional abundances as small as
$f_{{}_{\text{DM}}}\sim 3\times 10^{-5}$. This allows us to constrain
unexplored parameter space for $m_{\chi}\lesssim 10^{2}\ \text{GeV}$ and
$q_{\chi}\gtrsim 10^{-3}$, for which the couplings are too large to be probed
by terrestrial-based experiments.

For sufficiently large couplings, the formation of $(\chi^{-}N)$ bound states
suppresses the annihilation signals discussed here. This dictates the top of
the regions probed by IceCube shown in Fig. 2. As discussed in Sec. IV, these
limits are conservative, since we have incorporated thorium, the highest-$Z$
element measured in the photosphere, and have taken its photospheric density
to be representative of its overall solar abundance. In contrast, the
highest-$Z$ element for which we have an accurate understanding of its radial
profile is oxygen, since it is actively produced in the Sun. The dashed lines
shown in Fig. 2 represent the largest couplings that would be excluded by
IceCube if only bound states with oxygen nuclei had been included in our
analysis.

Although not shown in Fig. 2, other bounds on millicharged dark matter
subcomponents include those derived from ion traps Budker  _et al._ (2022)
(which are subject to the effect of Earth’s magnetic field, as discussed
above), spectral distortions of the CMB Berlin  _et al._ (2023b) (which apply
solely to models involving dark photons for parameter space that is
complementary to that considered here), and white dwarf/neutron stars Gould
_et al._ (1990); Fedderke  _et al._ (2020) (which are most relevant for masses
larger than those considered in this work).

##  VI Summary and Conclusions

In this study, we have calculated the capture and annihilation rates of
millicharged particles in the Sun, assuming that these particles make up a
subdominant fraction of the total dark matter abundance. We have taken into
account the effects of solar modulation, and the formation of $(\chi^{-}N)$
bound states in the solar environment. From this annihilation rate and the
resulting high-energy neutrino flux, we use previously published results from
the IceCube Collaboration to constrain this class of models. We find that the
IceCube data rules out significant regions of previously unexplored parameter
space, featuring $m_{\chi}\sim(5-100)\ \text{GeV}$, $q_{\chi}\sim
10^{-3}-10^{-2}$, and fractional abundances as small as
$f_{{}_{\text{DM}}}\sim 3\times 10^{-5}$.

## Acknowledgements

Fermilab is operated by the Fermi Research Alliance, LLC under Contract DE-
AC02-07CH11359 with the U.S. Department of Energy. This material is based upon
work supported by the U.S. Department of Energy, Office of Science, National
Quantum Information Science Research Centers, Superconducting Quantum
Materials and Systems Center (SQMS) under contract number DE-AC02-07CH11359.
This work was completed in part at the Perimeter Institute. Research at
Perimeter Institute is supported in part by the Government of Canada through
the Department of Innovation, Science and Economic Development Canada and by
the Province of Ontario through the Ministry of Colleges and Universities.

## References

  * Haas  _et al._ (2015) Andrew Haas, Christopher S. Hill, Eder Izaguirre, and Itay Yavin, “Looking for milli-charged particles with a new experiment at the LHC,” Phys. Lett. B 746, 117–120 (2015), arXiv:1410.6816 [hep-ph] .
  * Izaguirre and Yavin (2015) Eder Izaguirre and Itay Yavin, “New window to millicharged particles at the LHC,” Phys. Rev. D 92, 035014 (2015), arXiv:1506.04760 [hep-ph] .
  * Magill  _et al._ (2019) Gabriel Magill, Ryan Plestid, Maxim Pospelov, and Yu-Dai Tsai, “Millicharged particles in neutrino experiments,” Phys. Rev. Lett. 122, 071801 (2019), arXiv:1806.03310 [hep-ph] .
  * Kelly and Tsai (2019) Kevin J. Kelly and Yu-Dai Tsai, “Proton fixed-target scintillation experiment to search for millicharged dark matter,” Phys. Rev. D 100, 015043 (2019), arXiv:1812.03998 [hep-ph] .
  * Acciarri  _et al._ (2020) R. Acciarri _et al._ (ArgoNeuT), “Improved Limits on Millicharged Particles Using the ArgoNeuT Experiment at Fermilab,” Phys. Rev. Lett. 124, 131801 (2020), arXiv:1911.07996 [hep-ex] .
  * Harnik  _et al._ (2019) Roni Harnik, Zhen Liu, and Ornella Palamara, “Millicharged Particles in Liquid Argon Neutrino Experiments,” JHEP 07, 170 (2019), arXiv:1902.03246 [hep-ph] .
  * Perez  _et al._ (2024) Santiago Perez _et al._ (Oscura), “Searching for millicharged particles with 1 kg of Skipper-CCDs using the NuMI beam at Fermilab,” JHEP 02, 072 (2024), arXiv:2304.08625 [hep-ex] .
  * Plestid  _et al._ (2020) Ryan Plestid, Volodymyr Takhistov, Yu-Dai Tsai, Torsten Bringmann, Alexander Kusenko, and Maxim Pospelov, “New Constraints on Millicharged Particles from Cosmic-ray Production,” Phys. Rev. D 102, 115032 (2020), arXiv:2002.11732 [hep-ph] .
  * Bloch  _et al._ (2021) Itay M. Bloch, Andrea Caputo, Rouven Essig, Diego Redigolo, Mukul Sholapurkar, and Tomer Volansky, “Exploring new physics with O(keV) electron recoils in direct detection experiments,” JHEP 01, 178 (2021), arXiv:2006.14521 [hep-ph] .
  * Foroughi-Abari  _et al._ (2021) Saeid Foroughi-Abari, Felix Kling, and Yu-Dai Tsai, “Looking forward to millicharged dark sectors at the LHC,” Phys. Rev. D 104, 035014 (2021), arXiv:2010.07941 [hep-ph] .
  * Harnik  _et al._ (2021) Roni Harnik, Ryan Plestid, Maxim Pospelov, and Harikrishnan Ramani, “Millicharged cosmic rays and low recoil detectors,” Phys. Rev. D 103, 075029 (2021), arXiv:2010.11190 [hep-ph] .
  * Marocco and Sarkar (2021) Giacomo Marocco and Subir Sarkar, “Blast from the past: Constraints on the dark sector from the BEBC WA66 beam dump experiment,” SciPost Phys. 10, 043 (2021), arXiv:2011.08153 [hep-ph] .
  * Berlin  _et al._ (2023a) Asher Berlin, Raffaele Tito D’Agnolo, Sebastian A. R. Ellis, and Jury I. Radkovski, “Signals of millicharged dark matter in light-shining-through-wall experiments,” JHEP 08, 017 (2023a), arXiv:2305.05684 [hep-ph] .
  * Chen  _et al._ (2023) Chuan-Ren Chen, Bui Hong Nhung, and Chrisna Setyo Nugroho, “Millicharged dark matter detection with Mach-Zehnder interferometer,” Phys. Lett. B 839, 137764 (2023), arXiv:2212.13017 [hep-ph] .
  * Budker  _et al._ (2022) Dmitry Budker, Peter W. Graham, Harikrishnan Ramani, Ferdinand Schmidt-Kaler, Christian Smorra, and Stefan Ulmer, “Millicharged Dark Matter Detection with Ion Traps,” PRX Quantum 3, 010330 (2022), arXiv:2108.05283 [hep-ph] .
  * Argüelles Delgado  _et al._ (2021) Carlos Alberto Argüelles Delgado, Kevin James Kelly, and Víctor Muñoz Albornoz, “Millicharged particles from the heavens: single- and multiple-scattering signatures,” JHEP 11, 099 (2021), arXiv:2104.13924 [hep-ph] .
  * Afek  _et al._ (2021) Gadi Afek, Fernando Monteiro, Jiaxiang Wang, Benjamin Siegel, Sumita Ghosh, and David C. Moore, “Limits on the abundance of millicharged particles bound to matter,” Phys. Rev. D 104, 012004 (2021), arXiv:2012.08169 [hep-ex] .
  * Holdom (1986) Bob Holdom, “Two U(1)’s and Epsilon Charge Shifts,” Phys. Lett. B 166, 196–198 (1986).
  * Cohen  _et al._ (2010) Timothy Cohen, Daniel J. Phalen, Aaron Pierce, and Kathryn M. Zurek, “Asymmetric Dark Matter from a GeV Hidden Sector,” Phys. Rev. D 82, 056001 (2010), arXiv:1005.1655 [hep-ph] .
  * Baumgart  _et al._ (2009) Matthew Baumgart, Clifford Cheung, Joshua T. Ruderman, Lian-Tao Wang, and Itay Yavin, “Non-Abelian Dark Sectors and Their Collider Signatures,” JHEP 04, 014 (2009), arXiv:0901.0283 [hep-ph] .
  * Dubovsky  _et al._ (2004) S. L. Dubovsky, D. S. Gorbunov, and G. I. Rubtsov, “Narrowing the window for millicharged particles by CMB anisotropy,” JETP Lett. 79, 1–5 (2004), arXiv:hep-ph/0311189 .
  * de Putter  _et al._ (2019) Roland de Putter, Olivier Doré, Jérôme Gleyzes, Daniel Green, and Joel Meyers, “Dark Matter Interactions, Helium, and the Cosmic Microwave Background,” Phys. Rev. Lett. 122, 041301 (2019), arXiv:1805.11616 [astro-ph.CO] .
  * Kovetz  _et al._ (2018) Ely D. Kovetz, Vivian Poulin, Vera Gluscevic, Kimberly K. Boddy, Rennan Barkana, and Marc Kamionkowski, “Tighter limits on dark matter explanations of the anomalous EDGES 21 cm signal,” Phys. Rev. D 98, 103529 (2018), arXiv:1807.11482 [astro-ph.CO] .
  * Buen-Abad  _et al._ (2022) Manuel A. Buen-Abad, Rouven Essig, David McKeen, and Yi-Ming Zhong, “Cosmological constraints on dark matter interactions with ordinary matter,” Phys. Rept. 961, 1–35 (2022), arXiv:2107.12377 [astro-ph.CO] .
  * Stebbins and Krnjaic (2019) Albert Stebbins and Gordan Krnjaic, “New Limits on Charged Dark Matter from Large-Scale Coherent Magnetic Fields,” JCAP 12, 003 (2019), arXiv:1908.05275 [astro-ph.CO] .
  * Griest and Kamionkowski (1990) Kim Griest and Marc Kamionkowski, “Unitarity Limits on the Mass and Radius of Dark Matter Particles,” Phys. Rev. Lett. 64, 615 (1990).
  * Emken  _et al._ (2019) Timon Emken, Rouven Essig, Chris Kouvaris, and Mukul Sholapurkar, “Direct Detection of Strongly Interacting Sub-GeV Dark Matter via Electron Recoils,” JCAP 09, 070 (2019), arXiv:1905.06348 [hep-ph] .
  * Krauss  _et al._ (1985) L. M. Krauss, K. Freese, D. N. Spergel, and W. H. Press, “Cold dark matter candidates and the solar neutrino problem,” Astrophys. J. 299, 1001–1006 (1985).
  * Press and Spergel (1985) William H. Press and David N. Spergel, “Capture by the sun of a galactic population of weakly interacting massive particles,” Astrophys. J. 296, 679–684 (1985).
  * Griest and Seckel (1987) K. Griest and D. Seckel, “Cosmic Asymmetry, Neutrinos and the Sun,” Nucl. Phys. B 283, 681–705 (1987), [Erratum: Nucl.Phys.B 296, 1034–1036 (1988)].
  * Gould (1987a) Andrew Gould, “Resonant Enhancements in WIMP Capture by the Earth,” Astrophys. J. 321, 571 (1987a).
  * Gould (1987b) Andrew Gould, “WIMP Distribution in and Evaporation From the Sun,” Astrophys. J. 321, 560 (1987b).
  * Gould and Raffelt (1990) Andrew Gould and Georg Raffelt, “THERMAL CONDUCTION BY MASSIVE PARTICLES,” Astrophys. J. 352, 654 (1990).
  * Silk  _et al._ (1985) Joseph Silk, Keith A. Olive, and Mark Srednicki, “The Photino, the Sun and High-Energy Neutrinos,” Phys. Rev. Lett. 55, 257–259 (1985).
  * Srednicki  _et al._ (1987) Mark Srednicki, Keith A. Olive, and Joseph Silk, “High-Energy Neutrinos from the Sun and Cold Dark Matter,” Nucl. Phys. B 279, 804–823 (1987).
  * Kamionkowski (1991) Marc Kamionkowski, “Energetic neutrinos from heavy neutralino annihilation in the sun,” Phys. Rev. D 44, 3021–3042 (1991).
  * Barger  _et al._ (2002) Vernon D. Barger, Francis Halzen, Dan Hooper, and Chung Kao, “Indirect search for neutralino dark matter with high-energy neutrinos,” Phys. Rev. D 65, 075022 (2002), arXiv:hep-ph/0105182 .
  * Flacke  _et al._ (2009) Thomas Flacke, Arjun Menon, Dan Hooper, and Katherine Freese, “Kaluza-Klein Dark Matter And Neutrinos From Annihilation In The Sun,” (2009), arXiv:0908.0899 [hep-ph] .
  * Schuster  _et al._ (2010) Philip Schuster, Natalia Toro, and Itay Yavin, “Terrestrial and Solar Limits on Long-Lived Particles in a Dark Sector,” Phys. Rev. D 81, 016002 (2010), arXiv:0910.1602 [hep-ph] .
  * Batell  _et al._ (2010) Brian Batell, Maxim Pospelov, Adam Ritz, and Yanwen Shang, “Solar Gamma Rays Powered by Secluded Dark Matter,” Phys. Rev. D 81, 075004 (2010), arXiv:0910.1567 [hep-ph] .
  * Pospelov and Ramani (2021) Maxim Pospelov and Harikrishnan Ramani, “Earth-bound millicharge relics,” Phys. Rev. D 103, 115031 (2021), arXiv:2012.03957 [hep-ph] .
  * McKeen  _et al._ (2023) David McKeen, David E. Morrissey, Maxim Pospelov, Harikrishnan Ramani, and Anupam Ray, “Dark Matter Annihilation inside Large-Volume Neutrino Detectors,” Phys. Rev. Lett. 131, 011005 (2023), arXiv:2303.03416 [hep-ph] .
  * Mack  _et al._ (2007) Gregory D. Mack, John F. Beacom, and Gianfranco Bertone, “Towards Closing the Window on Strongly Interacting Dark Matter: Far-Reaching Constraints from Earth’s Heat Flow,” Phys. Rev. D 76, 043523 (2007), arXiv:0705.4298 [astro-ph] .
  * Pospelov and Ray (2024) Maxim Pospelov and Anupam Ray, “Neutrinos from Earth-bound dark matter annihilation,” JCAP 01, 029 (2024), arXiv:2309.10032 [hep-ph] .
  * Aartsen  _et al._ (2017) M. G. Aartsen _et al._ (IceCube), “Search for annihilating dark matter in the Sun with 3 years of IceCube data,” Eur. Phys. J. C 77, 146 (2017), [Erratum: Eur.Phys.J.C 79, 214 (2019)], arXiv:1612.05949 [astro-ph.HE] .
  * Abbasi  _et al._ (2022) R. Abbasi _et al._ (IceCube), “Search for GeV-scale dark matter annihilation in the Sun with IceCube DeepCore,” Phys. Rev. D 105, 062004 (2022), arXiv:2111.09970 [astro-ph.HE] .
  * Berlin  _et al._ (2024) Asher Berlin, Hongwan Liu, Maxim Pospelov, and Harikrishnan Ramani, “Terrestrial density of strongly-coupled relics,” Phys. Rev. D 109, 075027 (2024), arXiv:2302.06619 [hep-ph] .
  * Gleeson and Axford (1968) L. J. Gleeson and W. I. Axford, “Solar Modulation of Galactic Cosmic Rays,” Astrophys. J. 154, 1011 (1968).
  * Strauss  _et al._ (2012) R. D. Strauss, M. S. Potgieter, I. Büsching, and A. Kopp, “Modelling heliospheric current sheet drift in stochastic cosmic ray transport models,” Astrophysics and Space Science 339, 223–236 (2012).
  * Maccione (2013) Luca Maccione, “Low energy cosmic ray positron fraction explained by charge-sign dependent solar modulation,” Phys. Rev. Lett. 110, 081101 (2013), arXiv:1211.6905 [astro-ph.HE] .
  * Cholis  _et al._ (2016) Ilias Cholis, Dan Hooper, and Tim Linden, “A Predictive Analytic Model for the Solar Modulation of Cosmic Rays,” Phys. Rev. D 93, 043016 (2016), arXiv:1511.01507 [astro-ph.SR] .
  * Cholis  _et al._ (2022) Ilias Cholis, Dan Hooper, and Tim Linden, “Constraining the charge-sign and rigidity-dependence of solar modulation,” JCAP 10, 051 (2022), arXiv:2007.00669 [astro-ph.HE] .
  * Potgieter (2014) Marius S. Potgieter, “Very local interstellar spectra for galactic electrons, protons and helium,” Braz. J. Phys. 44, 581–588 (2014), arXiv:1310.6133 [astro-ph.SR] .
  * Mirizzi  _et al._ (2009a) Alessandro Mirizzi, Javier Redondo, and Gunter Sigl, “Microwave Background Constraints on Mixing of Photons with Hidden Photons,” JCAP 03, 026 (2009a), arXiv:0901.0014 [hep-ph] .
  * Mirizzi  _et al._ (2009b) Alessandro Mirizzi, Javier Redondo, and Gunter Sigl, “Constraining resonant photon-axion conversions in the Early Universe,” JCAP 08, 001 (2009b), arXiv:0905.4865 [hep-ph] .
  * Kunze and Vázquez-Mozo (2015) Kerstin E. Kunze and Miguel Á. Vázquez-Mozo, “Constraints on hidden photons from current and future observations of CMB spectral distortions,” JCAP 12, 028 (2015), arXiv:1507.02614 [astro-ph.CO] .
  * Caputo  _et al._ (2020a) Andrea Caputo, Hongwan Liu, Siddharth Mishra-Sharma, and Joshua T. Ruderman, “Dark Photon Oscillations in Our Inhomogeneous Universe,” Phys. Rev. Lett. 125, 221303 (2020a), arXiv:2002.05165 [astro-ph.CO] .
  * Caputo  _et al._ (2020b) Andrea Caputo, Hongwan Liu, Siddharth Mishra-Sharma, and Joshua T. Ruderman, “Modeling Dark Photon Oscillations in Our Inhomogeneous Universe,” Phys. Rev. D 102, 103533 (2020b), arXiv:2004.06733 [astro-ph.CO] .
  * Garcia  _et al._ (2020) Andres Aramburo Garcia, Kyrylo Bondarenko, Sylvia Ploeckinger, Josef Pradler, and Anastasia Sokolenko, “Effective photon mass and (dark) photon conversion in the inhomogeneous Universe,” JCAP 10, 011 (2020), arXiv:2003.10465 [astro-ph.CO] .
  * Longair (2011) Malcolm S. Longair,  _High Energy Astrophysics_ , 3rd ed. (Cambridge University Press, 2011).
  * Bahcall and Pinsonneault (2004) John N. Bahcall and M. H. Pinsonneault, “What do we (not) know theoretically about solar neutrino fluxes?” Phys. Rev. Lett. 92, 121301 (2004), arXiv:astro-ph/0402114 .
  * Asplund  _et al._ (2009) Martin Asplund, Nicolas Grevesse, A. Jacques Sauval, and Pat Scott, “The chemical composition of the Sun,” Ann. Rev. Astron. Astrophys. 47, 481–522 (2009), arXiv:0909.0948 [astro-ph.SR] .
  * Davidson  _et al._ (2000) Sacha Davidson, Steen Hannestad, and Georg Raffelt, “Updated bounds on millicharged particles,” JHEP 05, 003 (2000), arXiv:hep-ph/0001179 .
  * Chatrchyan  _et al._ (2013a) S. Chatrchyan _et al._ (CMS), “Search for Fractionally Charged Particles in $pp$ Collisions at $\sqrt{s}=7$ TeV,” Phys. Rev. D 87, 092008 (2013a), [Erratum: Phys.Rev.D 106, 099903 (2022)], arXiv:1210.2311 [hep-ex] .
  * Chatrchyan  _et al._ (2013b) Serguei Chatrchyan _et al._ (CMS), “Searches for Long-Lived Charged Particles in $pp$ Collisions at $\sqrt{s}$=7 and 8 TeV,” JHEP 07, 122 (2013b), [Erratum: JHEP 11, 149 (2022)], arXiv:1305.0491 [hep-ex] .
  * Hayrapetyan  _et al._ (2024) Aram Hayrapetyan _et al._ (CMS), “Search for fractionally charged particles in proton-proton collisions at $\sqrt{s}$ = 13 TeV,” (2024), arXiv:2402.09932 [hep-ex] .
  * Erickcek  _et al._ (2007) Adrienne L. Erickcek, Paul J. Steinhardt, Dan McCammon, and Patrick C. McGuire, “Constraints on the Interactions between Dark Matter and Baryons from the X-ray Quantum Calorimetry Experiment,” Phys. Rev. D 76, 042007 (2007), arXiv:0704.0794 [astro-ph] .
  * Mahdawi and Farrar (2018) M. Shafi Mahdawi and Glennys R. Farrar, “Constraints on Dark Matter with a moderately large and velocity-dependent DM-nucleon cross-section,” JCAP 10, 007 (2018), arXiv:1804.03073 [hep-ph] .
  * Rich  _et al._ (1987) J. Rich, R. Rocchia, and M. Spiro, “A Search for Strongly Interacting Dark Matter,” Phys. Lett. B 194, 173 (1987).
  * Aartsen  _et al._ (2016) M. G. Aartsen _et al._ (IceCube), “Improved limits on dark matter annihilation in the Sun with the 79-string IceCube detector and implications for supersymmetry,” JCAP 04, 022 (2016), arXiv:1601.00653 [hep-ph] .
  * Aartsen  _et al._ (2013) M. G. Aartsen _et al._ (IceCube), “Search for dark matter annihilations in the Sun with the 79-string IceCube detector,” Phys. Rev. Lett. 110, 131302 (2013), arXiv:1212.4097 [astro-ph.HE] .
  * Abbasi  _et al._ (2012) R. Abbasi _et al._ (IceCube), “Multi-year search for dark matter annihilations in the Sun with the AMANDA-II and IceCube detectors,” Phys. Rev. D 85, 042002 (2012), arXiv:1112.1840 [astro-ph.HE] .
  * Abbasi  _et al._ (2009) R. Abbasi _et al._ (IceCube), “Limits on a muon flux from neutralino annihilations in the Sun with the IceCube 22-string detector,” Phys. Rev. Lett. 102, 201302 (2009), arXiv:0902.2460 [astro-ph.CO] .
  * Berlin  _et al._ (2023b) Asher Berlin, Jeff A. Dror, Xucheng Gan, and Joshua T. Ruderman, “Millicharged relics reveal massless dark photons,” JHEP 05, 046 (2023b), arXiv:2211.05139 [hep-ph] .
  * Gould  _et al._ (1990) Andrew Gould, Bruce T. Draine, Roger W. Romani, and Shmuel Nussinov, “Neutron Stars: Graveyard of Charged Dark Matter,” Phys. Lett. B 238, 337–343 (1990).
  * Fedderke  _et al._ (2020) Michael A. Fedderke, Peter W. Graham, and Surjeet Rajendran, “White dwarf bounds on charged massive particles,” Phys. Rev. D 101, 115021 (2020), arXiv:1911.08883 [hep-ph] .