# Signals of Millicharged Dark Matter in Light-Shining-Through-Wall
Experiments

Asher Berlin  Theoretical Physics Division, Fermi National Accelerator
Laboratory, Batavia, IL 60510, USA  Superconducting Quantum Materials and
Systems Center (SQMS), Fermi National Accelerator Laboratory, Batavia, IL
60510, USA  Raffaele Tito D’Agnolo  Université Paris-Saclay, CNRS, CEA,
Institut de Physique Théorique, 91191, Gif-sur-Yvette, France  Sebastian A. R.
Ellis  Département de Physique Théorique, Université de Genève, 24 quai Ernest
Ansermet, 1211 Genève 4, Switzerland  Jury I. Radkovski  Department of Physics
and Astronomy, McMaster University, 1280 Main Street West, Hamilton, ON L8S
4M1, Canada  Université Paris-Saclay, CNRS, CEA, Institut de Physique
Théorique, 91191, Gif-sur-Yvette, France  Perimeter Institute for Theoretical
Physics, Waterloo, Ontario, N2L 2Y5, Canada

###### Abstract

We discuss a novel detection technique for millicharged dark matter that makes
use of existing light-shining-through-wall (LSW) experiments searching for
massive dark photons. Since millicharged particles interact with both the
visible and dark sectors, a small background of such particles enables the
search for visible signals even in the limit of a massless dark photon.
Furthermore, for sufficiently large couplings, a small dark matter
subcomponent consisting of millicharged particles can have a terrestrial
abundance that is significantly enhanced with respect to its galactic
abundance. We leverage this fact to show that in certain parts of parameter
space, future runs of the LSW Dark SRF experiment can be used to set the
strongest limits on millicharged relics.

##  I Introduction

The mystery that is the nature of dark matter (DM) remains unsolved despite
many decades of theoretical and experimental advances. An intriguing
possibility is that DM could be at least partially composed of particles with
extremely small fractional charges under electromagnetism, commonly referred
to as “millicharged particles” (mCPs). Although no mCPs have been observed,
there is strong motivation for their existence, as they naturally arise in
various string theory compactifications and minimal dark sector extensions to
the Standard Model (SM) involving a gauged $U(1)^{\prime}$ symmetry Holdom
(1986); Dienes  _et al._ (1997); Abel and Schofield (2004); Batell and
Gherghetta (2006); Aldazabal  _et al._ (2000); Abel and Santiago (2004); Abel
_et al._ (2008); Acharya  _et al._ (2016, 2018); Gherghetta  _et al._ (2019)
and could also play a role in explaining recent experimental anomalies
Zavattini  _et al._ (2006); Adriani  _et al._ (2009); Chang  _et al._ (2008);
Barkana (2018); Bowman  _et al._ (2018); Berlin  _et al._ (2018); Barkana  _et
al._ (2018); Liu  _et al._ (2019). The boson of this $U(1)^{\prime}$, referred
to as an $A^{\prime}$ or dark photon, may possess a kinetic mixing with the SM
photon, parameterized by a small dimensionless parameter $\epsilon$. In simple
field-theoretic models, this arises from loop diagrams involving fields
charged under both the SM and dark sectors, leading to mixings of
$\epsilon\sim 10^{-6}-10^{-3}$ Holdom (1986), but much smaller mixings are
possible and have been explicitly constructed Arkani-Hamed and Weiner (2008);
Abel  _et al._ (2008); Acharya  _et al._ (2018); Gherghetta  _et al._ (2019).
As a result, particles directly charged in the dark sector under the
$A^{\prime}$ appear as effectively millicharged under normal electromagnetism.

Many experiments have been conducted in an attempt to detect mCPs with
effective charge greater than $\sim 10^{-4}$ Agrawal  _et al._ (2021). These
have all turned up null results, leading to a number of proposals to search
for mCPs with even smaller charges and heavier masses Prinz  _et al._ (1998);
Davidson  _et al._ (2000); Battaglieri  _et al._ (2017); Berlin  _et al._
(2019); Magill  _et al._ (2019); Kelly and Tsai (2019); Chang  _et al._
(2018); Harnik  _et al._ (2019); Acciarri  _et al._ (2020); Ball  _et al._
(2020); Budker  _et al._ (2022). Despite being very feebly-coupled to normal
matter, the long range of the interaction enhances the mCP-SM scattering
cross-section in the non-relativistic limit. As a result, throughout much of
the parameter space, a millicharged DM (mCDM) subcomponent rapidly thermalizes
with the Earth’s environment, cooling down to terrestrial temperatures of
$300\ \text{K}\sim 25\ \text{meV}$ Pospelov and Ramani (2021); Berlin  _et
al._ (2023). Terrestrial direct detection experiments with
$\text{eV}-\text{keV}$ energy thresholds are thus largely insensitive to such
DM subcomponents, motivating the consideration of other approaches Emken  _et
al._ (2019); Pospelov and Ramani (2021); Berlin  _et al._ (2023); Budker  _et
al._ (2022); Berlin  _et al._ (2022a); McKeen  _et al._ (2022); Billard  _et
al._ (2022); Das  _et al._ (2022). This shedding of kinetic energy also
implies a drastic modification to the mCDM phase space; analogous to a
“traffic jam,” conservation of flux implies that the terrestrial density of
thermalized mCDM is drastically enhanced compared to its average galactic
density Wallemacq (2013, 2014); Wallemacq and Cudell (2015); Neufeld  _et al._
(2018); Laletin and Cudell (2019); Pospelov  _et al._ (2020); Pospelov and
Ramani (2021); Berlin  _et al._ (2023); Leane and Smirnov (2022).

The large density and small kinetic energy of such particles makes them ideal
candidates to search for in _direct deflection_ experiments Berlin  _et al._
(2020); Berlin and Schutz (2022). As first pointed out in Ref. Berlin  _et
al._ (2020), it is possible to induce collective effects in the background
fluid of particle-like DM, which can be leveraged to improve detection
prospects in the low momentum-transfer regime. In this work, we investigate
mCDM signals arising from such collective effects in light-shining-through-
wall (LSW) experiments. As shown in Fig. 1, such a setup consists of two
identical radio-frequency (RF) cavities separated by electromagnetic (EM)
shielding. One cavity is loaded in a resonant mode at frequency $\omega\sim 1\
\text{GHz}$. Inside this emitter cavity, the resulting electric field
generates a perturbation in the mCDM density that sources _dark_ electric
fields capable of penetrating through the shielding. The dark electric fields
couple to mCPs inside a second detector cavity, causing them to oscillate and
source _visible_ electric fields that are then resonantly detected. This
effect relies on the presence of a kinetically-mixed $A^{\prime}$, but unlike
direct LSW searches for dark photons, does not decouple in the massless
$A^{\prime}$ limit. Although massless dark photons do not directly couple to
the SM in vacuum, mCPs interact with both sectors, giving rise to rich
phenomenology even in the case that the $U(1)^{\prime}$ symmetry remains
unbroken Berlin  _et al._ (2022b); Vogel and Redondo (2014); Adshead  _et al._
(2022). Furthermore, we find that for coaxially-aligned cavities in the far-
field limit, it is ideal to employ transverse electric modes, which is
suboptimal for typical setups searching for signals arising from massive dark
photons in absence of a mCDM background Betz  _et al._ (2013); Graham  _et
al._ (2014); Romanenko  _et al._ (2023); Berlin  _et al._ (2022c). Regardless,
it is still possible to reinterpret existing limits from experiments performed
at CERN and FNAL Betz  _et al._ (2013); Romanenko  _et al._ (2023) to place
new bounds on mCDM. In this work, we also compute the projected reach of a
future experiment dedicated to enhancing the sensitivity to mCDM. Our main
results are summarized in Figs. 3 and 4, which show that future runs of the
Dark SRF experiment Romanenko  _et al._ (2023) can probe orders of magnitude
of new parameter space.

The remainder of this study is organized as follows. In Section II, we give a
brief overview of mCDM models coupled to massless dark photons. In Section
III, we discuss mCDM signals in LSW experiments. Section IV presents existing
limits and the estimated sensitivity of current and future LSW experiments. We
conclude in Section V.

Figure 1: Cartoon of an LSW experiment with an mCP-induced signal. An emitter
cavity is loaded with EM energy in a resonant mode, which causes mCPs in the
cavity to oscillate. The mCP oscillations generate a small dark current
$\boldsymbol{j}^{\prime}_{\rm em}$ as given in Eq. (18). The invisible
radiation sourced by this current excites the same mode in a shielded receiver
cavity (Eq. (21)), leading to a resonantly-enhanced signal power (Eq. (22)).

##  II Model Overview of Millicharged Particles

We consider a DM subcomponent consisting of mCPs (denoted as $\chi$), with an
effective charge of $q_{\text{eff}}\ll 1$ under normal electromagnetism. In
this section, we give an overview for how such interactions can arise. If
electric charge is quantized, as expected if $U(1)_{\rm EM}$ originates from
the breaking of a Grand Unified Theory, the simplest realization of mCPs
arises from the kinetic mixing between $U(1)_{\rm EM}$ and an unbroken dark
$U(1)^{\prime}$. The Lagrangian of the theory is then

$\displaystyle\mathcal{L}\supset-\frac{1}{4}\,F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}\,F^{\prime}_{\mu\nu}\,F^{\prime\mu\nu}+\frac{\epsilon}{2}\,F^{\mu\nu}\,F^{\prime}_{\mu\nu}-e\,A_{\mu}\,j^{\mu}-e^{\prime}\,A^{\prime}_{\mu}\,j^{\prime\mu}~{},$
(1)

where $A^{(\prime)\,\mu}$ are the gauge fields, $F_{\mu\nu}^{(\prime)}$ the
field strengths, $e^{(\prime)}$ the gauge couplings, and $j^{(\prime)\,\mu}$
the current densities, such that a prime denotes a dark sector quantity. The
signals discussed in this work dominate over existing LSW searches for massive
dark photons (see, e.g., Ref. Caputo  _et al._ (2021) and references within)
in the limit that the mass is $m_{A^{\prime}}\ll 1/L_{\rm exp}$, where $L_{\rm
exp}$ is the typical size of the experimental apparatus. Therefore, here and
in the following we take the $A^{\prime}$ to be massless. The transformation
$A^{\prime\mu}\to A^{\prime\mu}+\epsilon A^{\mu}$ brings the kinetic terms
into a canonical form to leading order in $\epsilon\ll 1$, such that in this
new basis Eq. (1) becomes

$\displaystyle\mathcal{L}\supset-\frac{1}{4}\,F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}\,F^{\prime}_{\mu\nu}\,F^{\prime\mu\nu}-A_{\mu}\,\big{(}e\,j^{\mu}+\epsilon\,e^{\prime}\,j^{\prime\mu}\big{)}-e^{\prime}\,A^{\prime}_{\mu}\,j^{\prime\mu}~{}.$
(2)

Note that the dark photon field solely couples to the dark sector current. In
other words, massless dark photons do not directly couple to SM sources. As a
result, typical LSW searches are usually thought to lack sensitivity to a
massless $A^{\prime}$. However, this is not the case if mCPs are present. This
can be seen by noting that the SM photon couples to both the SM and dark
sector current densities, with the latter interaction suppressed by
$\epsilon$. Thus, dark sector particles $\chi$ directly charged under the
$A^{\prime}$ couple to the SM photon with an effective charge

$\displaystyle q_{\text{eff}}\equiv\epsilon\,e^{\prime}/e\,.$ (3)

Since dark currents couple to both sectors, the invisible mode _indirectly_
couples to SM currents if it first interacts with an EM-charged mCP
background.

This decoupling in the massless limit can also be understood by phrasing the
choice of basis in terms of the $(A,A^{\prime})$ plane and the $\text{SO}(2)$
symmetry of the kinetic terms. Absent of mCPs ($j^{\prime\mu}=0$), Eq. (1)
implies that the coupling to $j^{\mu}$ identifies a preferred direction in the
$(A,A^{\prime})$ plane, but since the rest of the Lagrangian is $\text{SO}(2)$
symmetric, there is no physical meaning to this specific direction; we can
rotate it at will without changing the rest of the Lagrangian. An observer
will call “the SM photon” any linear combination of fields that couples to
$j^{\mu}$. This is why the massless invisible mode is unobservable in vacuum.
Alternatively, in this language it is simple to see that the massless
invisible mode is instead observable in the presence of mCPs
($j^{\prime\mu}\neq 0$). In this case, there are two preferred directions in
the $(A,A^{\prime})$ plane, given by the linear combinations of fields that
couple to $j^{\mu}$ or $j^{\prime\mu}$. A rotation in the $(A,A^{\prime})$
plane does not change the scalar product between these two directions, so one
can calculate basis-independent quantities such as the degree to which the
linear combination of fields that couples to $j^{\mu}$ also couples to
$j^{\prime\mu}$. Hence, for $j^{\prime\mu}=0$ we have access only to a single
vector and will never know if it is embedded in a plane. But if
$j^{\prime\mu}\neq 0$, we can perform measurements that reveal the existence
of a second vector in the $(A,A^{\prime})$ plane and thus of the plane itself.
The same reasoning can also be applied to the well-studied example of a
massive dark photon absent of mCPs; in this case, the massive $A^{\prime}$ is
observable even if $j^{\prime\mu}=0$ because $j^{\mu}$ and $m_{A^{\prime}}$
specify two preferred directions in the $(A,A^{\prime})$ plane.

The lightest dark sector particles $\chi$ directly charged under the
$A^{\prime}$ are absolutely stable and are therefore natural DM candidates. In
the simplest of cosmologies, such particles arise as DM subcomponents in the
form of thermal relics. For effective charges larger than
$q_{\text{eff}}\gtrsim 10^{-7}\times(m_{\chi}/\text{GeV})^{1/2}$, electron
annihilations $ee\to\chi\chi$ thermalize mCPs with the SM sector in the early
universe, where $m_{\chi}$ is the mCP mass. Such mCPs generically constitute a
small fraction $f_{{}_{\text{DM}}}$ of the total DM energy density, since
freeze-out through $\chi\chi\to A^{\prime}A^{\prime}$ easily depletes the
thermal mCP abundance to $f_{{}_{\text{DM}}}\sim
10^{-8}\times(m_{\chi}/\text{GeV})^{2}\,(1/e^{\prime})^{4}$. Such cosmologies
in which the dark sector efficiently equilibrates with the SM at a temperature
below $\sim 1\ \text{GeV}$ predict a sizable relativistic density of dark
photons, which is tightly constrained by early universe probes of additional
dark radiation Vogel and Redondo (2014); Adshead  _et al._ (2022). Such bounds
are weakened for heavier mCP masses, since the two sectors decouple at
temperatures $T\sim m_{\chi}$; if this occurs at early enough times, the
multiplicity of SM mass thresholds heats the SM relative to the dark sector,
decreasing the relative density in dark radiation. These bounds are also
evaded in cosmological scenarios predicting exponentially small values of
$f_{{}_{\text{DM}}}$, where the reheat temperature of the universe
$T_{\text{RH}}$ is significantly smaller than the mass $m_{\chi}$. In this
case, the dark sector never fully thermalizes with the SM in the early
universe, yielding
$f_{{}_{\text{DM}}}\sim(\alpha_{\text{em}}\,q_{\text{eff}})^{2}\,e^{-2m_{\chi}/T_{\text{RH}}}\,m_{\chi}m_{\text{pl}}/(T_{\text{RH}}\,T_{\text{eq}})\ll
1$ Berlin  _et al._ (2022a), where $m_{\text{pl}}$ is the Planck mass and
$T_{\text{eq}}\sim 1\ \text{eV}$ is the temperature at matter-radiation
equality.

As a reference point for the experimental signals considered in this work, we
can consider a model of mCDM with $m_{\chi}\gtrsim\text{few}\times\text{MeV}$,
where $\chi$ is a small fraction of DM, $f_{{}_{\text{DM}}}\ll 1$. For such
masses, we can invoke a small reheat temperature $T_{\text{RH}}$ to evade
cosmological bounds on dark radiation. Small mCDM subcomponents are observable
in the laboratory today because interactions with terrestrial matter can
significantly enhance their local density $n_{\chi}^{\oplus}$ compared to
their mean number density throughout the galaxy
$n_{\chi}^{(\text{gal})}\sim\mathcal{O}{(1)}\ \text{cm}^{-3}\times
f_{{}_{\text{DM}}}\,(\text{GeV}/m_{\chi})$. Specifically, for
$q_{\text{eff}}\gg 10^{-7}\times(m_{\chi}/\text{GeV})^{1/2}$, galactic mCPs
bombarding the Earth rapidly equilibrate to terrestrial temperatures after
scattering with nuclei in the atmosphere and crust. Various dynamics tied to
this process can give rise to large local overdensities. For instance,
“strongly-coupled” mCPs much heavier than $\sim 1\ \text{GeV}$ accumulate on
Earth over geological timescales, since their characteristic thermal velocity
is well below the terrestrial escape velocity. More generally, the large
collision rate with normal matter increases the residence time spent by mCPs.
Therefore, as such particles diffuse past the surface, eventually escaping the
Earth entirely or settling into hydrostatic equilibrium elsewhere,
conservation of flux implies a local enhancement to their density. Such
effects have been studied in Refs. Wallemacq (2013, 2014); Wallemacq and
Cudell (2015); Neufeld  _et al._ (2018); Laletin and Cudell (2019); Pospelov
_et al._ (2020); Pospelov and Ramani (2021); Leane and Smirnov (2022); Berlin
_et al._ (2023), which have showed that the terrestrial mCDM density can be as
large as $n_{\chi}^{\oplus}\sim\mathcal{O}{(10^{15})}\ \text{cm}^{-3}\times
f_{{}_{\text{DM}}}$.

A dedicated analysis of the formation of terrestrial mCP densities (required
to map $f_{{}_{\text{DM}}}$ onto $n_{\chi}^{\oplus}$) is beyond the scope of
this work. We calculate limits and projections on the _laboratory_ density
$n_{\chi}$ of mCPs instead of $f_{{}_{\text{DM}}}$, remaining agnostic
regarding the formation of such a population and possible modifications from
mCP self-interactions. Beyond simplifying the study, there are various reasons
to remain agnostic. First, we will focus exclusively on mCP densities many
orders of magnitude smaller than the density of normal matter, in which case
the timescale associated with self-interactions is often longer than the time
it takes for such terrestrial densities to develop Pospelov and Ramani (2021).
Furthermore, processes in the early universe or at late times may lead to a
sizable mCP asymmetry, such that some of these interactions are forbidden. For
instance, for $q_{\text{eff}}\gtrsim m_{e}/\min{(m_{\chi},m_{N})}$ negatively-
charged mCPs efficiently form bound states with atomic nuclei $N$, such that
the freely propagating mCP population on Earth consists dominantly of
positively-charged particles Pospelov  _et al._ (2020); Berlin  _et al._
(2022a). The signal we discuss here remains observable even in the case of an
asymmetric population of mCPs.

##  III Light Shining Through a Wall

LSW experiments searching for direct signals of dark photons include ALPS
Ehret  _et al._ (2010); Ortiz  _et al._ (2022); Hallal  _et al._ (2022), CROWS
Betz  _et al._ (2013), and Dark SRF Romanenko  _et al._ (2023). In this work,
we show that these same searches are also sensitive to mCPs, and, furthermore,
that this sensitivity can be enhanced with small modifications to the
experimental geometry. We solely focus on LSW setups employing RF cavities,
since this technology has been firmly established by the CROWS experiment, and
will be developed to a much greater degree by the Dark SRF collaboration in
the coming years.

###  III.1 Review of Massive Dark Photon Signals

CROWS and Dark SRF search for massive dark photons by first driving fields of
amplitude $E_{\text{em}}\sim 10\ \text{MV}/\text{m}$ and frequency $\omega\sim
1\ \text{GHz}$ in an emitter cavity. The SM photons source a dark photon
field, which travels unimpeded into a shielded receiver cavity, exciting a
small visible EM field of the same frequency. In Ref. Graham  _et al._ (2014),
it was pointed out that cavities driven in the $\text{TM}_{010}$ mode in a
longitudinal arrangement (where the cavities are aligned along the
polarization axis of the electric field) have optimal sensitivity to
ultralight dark photons.

This result will be useful also in our setup. To understand it we can follow
Ref. An  _et al._ (2013) and rewrite the interaction between the dark photon
and the ordinary photon in the original gauge basis such that

$\displaystyle\mathcal{L}=-\frac{1}{4}\,F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}\,F^{\prime}_{\mu\nu}\,F^{\prime\mu\nu}-eA_{\mu}\,\,j^{\mu}-\epsilon\,m_{A^{\prime}}^{2}\,A^{\prime}_{\mu}\,A^{\mu}\,.$
(4)

Dark photons are emitted as in Fig. 2, with an amplitude

$\displaystyle\mathcal{M}_{i\to
f+A^{\prime}}=\epsilon\,e\,m_{A^{\prime}}^{2}\,(j_{\mu})_{if}\,\langle
A^{\mu},A^{\nu}\rangle\,\epsilon_{\nu}^{\prime}\,.$ (5)

$A$$A^{\prime}$$\epsilon\,m^{2}_{A^{\prime}}$$j_{\mu}$ Figure 2: Leading
diagram for the emission of a massive kinetically-mixed dark photon
$A^{\prime}$ from a SM electromagnetic current $j_{\mu}$.

Here, $(j_{\mu})_{if}$ is the current matrix element between initial and final
states, $\langle A^{\mu},A^{\nu}\rangle$ is the photon propagator in the
cavity, and $\epsilon_{\nu}^{\prime}$ the emitted dark photon polarization.
The cavity walls break translational invariace, giving the photon an effective
mass that corresponds to the normal modes (labelled by $n$) of the EM field in
the cavity. In Coulomb gauge ($\nabla\cdot\mathbf{A}=0$), we have

$\displaystyle\langle
A_{i},A_{j}\rangle=\sum_{n=0}^{\infty}\frac{\delta_{ij}}{\omega^{2}-\omega_{n}^{2}}~{}~{},~{}~{}\langle
A_{0},A_{0}\rangle=0\,,$

where $\omega_{n}$ are the wavenumbers of the normal modes of the cavity
(eigenvalues of $\nabla^{2}$). Using the well-known expressions of
$\epsilon^{\prime}_{\nu}$ for a massive vector, we can conclude that in the
limit of interest ($m_{A^{\prime}}\ll 1/L_{\rm cav}$), longitudinal modes
$A_{L}^{\prime}$ are emitted with a higher rate than transverse ones
$A_{T}^{\prime}$,

$\displaystyle\Gamma_{i\to
f+A^{\prime}_{L}}\propto\epsilon^{2}\,\frac{m_{A^{\prime}}^{2}}{\omega^{2}}\,,\quad\Gamma_{i\to
f+A^{\prime}_{T}}\propto\epsilon^{2}\,\frac{m_{A^{\prime}}^{4}}{\omega^{4}}\,,$
(6)

where our expressions are valid for $\omega>\omega_{0}$, such that
$\omega_{0}$ is the lowest-lying mode of the cavity $\omega_{0}\sim 1/L_{\rm
cav}$. This reverses the intuition of emission in vacuum where $\Gamma_{i\to
f+A^{\prime}_{L}}\propto\epsilon^{2}m_{A^{\prime}}^{2}/\omega^{2}$ and
$\Gamma_{i\to f+A^{\prime}_{T}}\propto\epsilon^{2}$. Thus, in the limit that
$m_{A^{\prime}}\ll\omega\sim 1\ \mu\text{eV}$, the longitudinal LSW
arrangement has optimal sensitivity to massive dark photons. On resonance
($\omega\simeq\omega_{0}$), the finite widths of the modes need to be to
included in Eq. (III.1), $\langle
A_{i},A_{j}\rangle\sim\delta_{ij}(\omega^{2}-\omega_{0}^{2}-i\omega\omega_{0}/Q_{0})^{-1}$,
but this does not modify the overall scalings of $\Gamma_{i\to
f+A^{\prime}_{L}}\sim\epsilon^{2}m_{A^{\prime}}^{2}$ and $\Gamma_{i\to
f+A^{\prime}_{T}}\sim\epsilon^{2}m_{A^{\prime}}^{4}$.

###  III.2 Massless Dark Photons in a Millicharged Plasma

Similar signals arise for massless dark photons provided that there is an
ambient density of mCPs, even though there is no mass splitting between the
visible and dark transverse modes. Qualitatively, this originates from the
plasma mass of the dark photon induced by the mCP density in the cavities.
However, this does not constitute a physical mass and care must be taken when
recasting the reach of LSW experiments to mCPs. For simplicity, we will take
the mCDM to be equally dense in the emitter and receiver cavities. In
principle, this need not be the case if the two cavities are kept in separate
environments with different temperatures, which can alter the local mCDM
density in either location. More generally, our adopted values of $n_{\chi}$
should be interpreted as the geometric mean of the mCP density in the two
cavities.

Before providing a detailed derivation of such signals, we first give a
heuristic discussion of the effect, working in the basis of Eq. (2) in which
$\chi$ couples to both visible and invisible fields. In the emitter cavity,
mCPs are accelerated by the driven electric field $E_{\text{em}}$, setting up
a dark current oscillating at the same frequency with amplitude
$j_{\text{em}}^{\prime}\sim(e^{\prime}n_{\chi}/m_{\chi})\,(eq_{\chi}E_{\text{em}}/\omega)$.
This dark current sources a dark EM field of $E_{\text{em}}^{\prime}\sim
j_{\text{em}}^{\prime}/\omega$, which penetrates into the receiver cavity,
accelerating the mCPs contained there and inducing a visible current of
$j_{\text{rec}}\sim(eq_{\chi}n_{\chi}/m_{\chi})\,(e^{\prime}E_{\text{em}}^{\prime}/\omega)$.
This visible current thus resonantly excites the receiver mode to a level of

$\displaystyle E_{\text{sig}}\sim Q\,j_{\text{rec}}/\omega\sim
Q\,\epsilon^{2}\,(\omega_{p}^{\prime}/\omega)^{4}\,E_{\text{em}}~{},$ (7)

where

$\displaystyle\omega_{p}^{\prime}\equiv\sqrt{e^{\prime\,2}\,n_{\chi}/m_{\chi}}$
(8)

is the in-medium dark plasma frequency as sourced by the mCP background.
Comparing Eq. (7) to the signal from massive dark photons in Eq. (6), we see
that the mCP-induced signal scales similarly to setups measuring the
_transverse_ mode of a massive $A^{\prime}$, with the dark plasma frequency
$\omega_{p}^{\prime}$ playing the role of an effective dark photon mass. We
now substantiate this estimate with a detailed calculation.

An effective coarse-grained description of a dense mCP background is that of a
fluid. The equations of motion of this mCP fluid can be derived starting from
the non-relativistic Boltzmann equation,

$\displaystyle\frac{df_{\pm}}{dt}\simeq\frac{1}{m_{\chi}}\,C[f_{\pm}]~{},$ (9)

where $f_{\pm}$ is the phase space density for positively- or negatively-
charged mCPs and $C[f_{\pm}]$ is the collision operator. The LHS of Eq. (9)
models the evolution of phase space elements arising from density/temperature
gradients and background fields, whereas the RHS incorporates additional
effects from, e.g., collisions between mCPs and nuclei or other mCPs. In the
presence of background visible or invisible electric fields $\boldsymbol{E}$
and $\boldsymbol{E}^{\prime}$, the time-derivative in Eq. (9) can be expanded
as

$\displaystyle\frac{df_{\pm}}{dt}\simeq\frac{\partial f_{\pm}}{\partial
t}+{\bf v}_{\pm}\cdot\frac{\partial f_{\pm}}{\partial{\bf
x}}\pm\frac{1}{m_{\chi}}\,\big{(}eq_{\text{eff}}\,\boldsymbol{E}+e^{\prime}\boldsymbol{E}^{\prime}\big{)}\cdot\frac{\partial
f_{\pm}}{\partial{\bf v}_{\pm}}~{},$ (10)

where ${\bf v}_{\pm}$ is the $\chi^{\pm}$ particle velocity. In Eq. (10), we
have ignored magnetic fields since their effect is suppressed in the non-
relativistic limit. Using this in Eq. (9) and taking the first moment over the
mCP phase space yields

$\displaystyle\partial_{t}{\bf V}_{\pm}+({\bf V}_{\pm}\cdot\nabla)\,{\bf
V}_{\pm}+\frac{\nabla
P_{\pm}}{n_{\pm}\,m_{\chi}}\simeq\pm\frac{1}{m_{\chi}}\,\big{(}eq_{\text{eff}}\,\boldsymbol{E}+e^{\prime}\boldsymbol{E}^{\prime}\big{)}+\frac{m_{\chi}^{2}}{n_{\pm}}\int\frac{d^{3}{\bf
v}_{\pm}}{(2\pi)^{3}}~{}{\bf v}_{\pm}\,C[f_{\pm}]~{},$ (11)

where $n_{\pm}$, ${\bf V}_{\pm}$, and $P_{\pm}$ are the number density, bulk
velocity, and pressure of the positively- or negatively-charged mCP fluid. In
general, the collision term $C[f_{\pm}]$ includes contributions from
scattering processes involving same-charge and opposite-charge mCPs. However,
when integrated over the velocity as in the last term of Eq. (11), only
scattering between oppositely-charged particles contributes, since
interactions between identical particles do not change the overall bulk
velocity ${\bf V}_{\pm}$ of either the positively- or negatively-charged
fluid.

In evaluating the LHS of Eq. (11), we have ignored terms arising from higher-
order viscosity moments of the mCP phase space. This is justified in the limit
that the mCP phase space is well-approximated by a boosted Maxwellian, whose
only moments are $n_{\chi}$ (the monopole) and ${\bf V}_{\pm}$ (the dipole).
Indeed, since in the parameter space of interest the ambient matter
efficiently thermalizes the mCPs before they enter the cavity, higher-order
moments (e.g., quadrupole, octupole, etc.) are parametrically suppressed by
the large mCP-atomic scattering rate Dodelson (2003). Under this same
approximation and also taking the bulk velocity to be small compared to the
thermal velocity ($V_{\pm}\sim
eq_{\text{eff}}\,E/(m_{\chi}\omega)\ll\sqrt{T_{\chi}/m_{\chi}}$), the
collision term on the RHS of Eq. (11) has been evaluated in Refs. Dvorkin  _et
al._ (2014); Boddy  _et al._ (2018); Becker  _et al._ (2021); Dvorkin  _et
al._ (2021) within the context of DM-baryon scattering in the early
Universe.111Note that for a receiver cavity cooled down to $T_{\text{cav}}=10\
\text{mK}$, the approximation $V_{\pm}\ll\sqrt{T_{\chi}/m_{\chi}}$ does not
hold for sufficiently large couplings. In this case, the exact form of the
collision term is modified compared to Eqs. (12) and (13). However, this does
not affect our results since collisions have a negligible impact on the signal
within the entirety of our parameter space, as discussed below Eq. (17). These
results can be easily adapted to incorporate $\chi^{+}-\chi^{-}$ and
$\chi^{\pm}-\text{atomic}$ scattering to give

$\displaystyle\partial_{t}{\bf V}_{\pm}+({\bf V}_{\pm}\cdot\nabla)\,{\bf
V}_{\pm}+\frac{\nabla
P_{\pm}}{n_{\pm}\,m_{\chi}}\simeq\pm\frac{1}{m_{\chi}}\,\big{(}eq_{\text{eff}}\,\boldsymbol{E}+e^{\prime}\boldsymbol{E}^{\prime}\big{)}-\Gamma_{\chi}\,({\bf
V}_{\pm}-{\bf V}_{\mp})-\Gamma_{\text{SM}}\,({\bf V}_{\pm}-{\bf
V}_{\text{SM}})~{}.$ (12)

Above, $\Gamma_{\chi}$ and $\Gamma_{\text{SM}}$ are the relevant momentum-
exchange rates from mCP self-scattering and scattering with SM matter,
respectively,

$\displaystyle\Gamma_{i}\simeq\frac{\mu_{\chi
i}}{3m_{\chi}\,\bar{v}_{0}^{2}}\,n_{i}\,\langle\sigma_{T}\,v_{\text{rel}}^{3}\rangle~{},$
(13)

where $i=\chi$ or $i=\text{SM}$, $v_{\text{rel}}$ is the relative velocity
between $\chi$ and the target scatterer, $\mu_{\chi i}$ the $\chi-i$ reduced
mass, and $\sigma_{T}$ the momentum-transfer cross-section. The brackets
correspond to a thermal average over $v_{\text{rel}}$ with the distribution

$\displaystyle
f_{\text{rel}}(v_{\text{rel}})=\frac{1}{(2\pi)^{3/2}\,\bar{v}_{0}^{3}}\,e^{-v_{\text{rel}}^{2}/2\bar{v}_{0}^{2}}~{}~{},~{}~{}\bar{v}_{0}\equiv\sqrt{T_{\chi}/m_{\chi}+T_{i}/m_{i}}~{},$
(14)

with species $i$ at temperature $T_{i}$.

We reduce Eq. (12) further still by making several additional simplifying
approximations. First, the low density of normal matter inside the RF cavities
implies that $\Gamma_{\chi}\gg\Gamma_{\text{SM}}$. Second, we drop the terms
involving $({\bf V}_{\pm}\cdot\nabla){\bf V}_{\pm}$ and $\nabla P_{\pm}$ since
both are higher-order in the non-relativistic (small temperature) limit. Also
note that such gradient terms are additionally suppressed if the emitter
cavity is driven in its lowest-lying transverse-magnetic mode, in which case
the visible electric field $E$ is spatially uniform and is thus not expected
to lead to significant spatial gradients in the mCP phase space. With these
simplifications, Eq. (12) reduces to

$\displaystyle\partial_{t}{\bf
V}_{\pm}\simeq\pm\frac{1}{m_{\chi}}\,\big{(}eq_{\text{eff}}\,\boldsymbol{E}+e^{\prime}\boldsymbol{E}^{\prime}\big{)}-\Gamma_{\chi}\,({\bf
V}_{\pm}-{\bf V}_{\mp})~{}.$ (15)

The electric fields in Eq. (15) are sourced by the oscillating driven modes of
the emitter cavity. Taking these to be oscillating with frequency $\omega$,
Eq. (15) can be solved for the dark mCP current,

$\displaystyle\boldsymbol{j}^{\prime}=e^{\prime}\,(n_{+}{\bf V}_{+}-n_{-}{\bf
V}_{-})\simeq-\frac{i}{\omega}\,\frac{(e^{\prime}/eq_{\text{eff}})\,\omega_{p}^{2}\,\boldsymbol{E}+\omega_{p}^{\prime\,2}\,\boldsymbol{E}^{\prime}}{1-2i\,\Gamma_{\chi}/\omega}~{},$
(16)

where we took $n_{+}=n_{-}=n_{\chi}/2$ and defined the mCP contribution to the
SM plasma frequency

$\displaystyle\omega_{p}\equiv\sqrt{(eq_{\text{eff}})^{2}\,n_{\chi}/m_{\chi}}~{}.$
(17)

Note that Eq. (16) implies that $j^{\prime}$ is suppressed in the strong-
coupling limit, $\Gamma_{\chi}\gg\omega$. Numerically, we find that this is
not the case and that $\Gamma_{\chi}/\omega\ll 1$ in all of the parameter
space of interest. Thus, we set $\Gamma_{\chi}\to 0$ in Eq. (16). In this
case, the dark electric field sourced by $j^{\prime}$ is $E^{\prime}\sim
j^{\prime}/\omega$. For sufficiently large dark coupling, this dark electric
field can backreact on the mCPs, inhibiting growth of the current
$j^{\prime}$. To see when this occurs, take the mCP current to be initially
excited by the visible field of the emitter cavity, i.e.,
$j^{\prime}\sim(e^{\prime}/eq_{\text{eff}})\,\omega_{p}^{2}\,E/\omega$. Thus,
the dark field sourced by this current is
$E^{\prime}\sim(e^{\prime}/eq_{\text{eff}})\,(\omega_{p}/\omega)^{2}E$. From
Eq. (16), we see that this contribution dominates over that of the visible
field if $\omega_{p}^{\prime}\gtrsim\omega$. Note that this agrees with the
intuition that if the timescale for the mCP fluid to screen the emitter
cavity, $\sim 1/\omega_{p}^{\prime}$, is longer than the oscillation timescale
$\sim 1/\omega$ of the driven cavity, then backreactions can be ignored. We
have checked that $\omega_{p}^{\prime}\lesssim\omega$ for all of the parameter
space of interest, in which case we can ignore such processes.

Eq. (16) can also be used to determine the corresponding dark charge density
$\rho^{\prime}$, which is related via continuity
$\rho^{\prime}=(i/\omega)\,\nabla\cdot\boldsymbol{j}^{\prime}$. Hence, at the
level of these approximations, the dark current and charge induced in the
emitter cavity are

$\displaystyle\boldsymbol{j}_{\text{em}}^{\prime}\simeq-\frac{i}{\omega}\,\Big{(}\frac{eq_{\text{eff}}\,e^{\prime}\,n_{\chi}}{m_{\chi}}\,\boldsymbol{E}_{\text{em}}+\omega_{p}^{\prime\,2}\,\boldsymbol{E}_{\text{em}}^{\prime}\Big{)}~{}~{},~{}~{}\rho_{\text{em}}^{\prime}\simeq\frac{eq_{\text{eff}}\,e^{\prime}\,n_{\chi}}{m_{\chi}}\,\frac{\nabla\cdot\boldsymbol{E}_{\text{em}}}{\omega^{2}-\omega_{p}^{\prime\,2}}~{},$
(18)

where we used that
$\nabla\cdot\boldsymbol{E}_{\text{em}}^{\prime}=\rho_{\text{em}}^{\prime}$.
Note that since
$\nabla\cdot\boldsymbol{E}_{\text{em}}=\rho+\mathcal{O}{(\epsilon)}$ vanishes
in the interior vacuum of the cavity to leading order in $\epsilon\ll 1$, non-
zero dark charge density $\rho_{\text{em}}^{\prime}$ can only be induced in
the cavity walls. The dark charge and current densities in Eq. (18) source an
oscillating invisible EM field emanating from the emitter cavity. In
particular, using Eq. (18) in the standard form of the wave equation,
$(\nabla^{2}+\omega^{2})\boldsymbol{E}_{\text{em}}^{\prime}=\partial_{t}\boldsymbol{j}_{\text{em}}^{\prime}+\nabla\rho_{\text{em}}^{\prime}$,
this field is

$\displaystyle\boldsymbol{E}_{\text{em}}^{\prime}({\bf
x},t)=-\frac{eq_{\text{eff}}\,e^{\prime}\,n_{\chi}}{4\pi\,m_{\chi}}\,e^{i\omega
t}\int_{\text{em}}d^{3}{\bf x}^{\prime}~{}\frac{e^{-ik|{\bf x}-{\bf
x}^{\prime}|}}{|{\bf x}-{\bf
x}^{\prime}|}\,\Big{[}\boldsymbol{E}_{\text{em}}({\bf
x}^{\prime})+\frac{1}{\omega^{2}-\omega_{p}^{\prime\,2}}\,\nabla\nabla\cdot\boldsymbol{E}_{\text{em}}({\bf
x}^{\prime})\Big{]}~{},$ (19)

where the integral is performed over the volume of the emitter cavity and we
defined the wavenumber

$\displaystyle
k=\begin{cases}\sqrt{\omega^{2}-\omega_{p}^{\prime\,2}}&(\omega_{p}^{\prime}<\omega)\\\
-i\,\sqrt{\omega_{p}^{\prime\,2}-\omega^{2}}&(\omega_{p}^{\prime}>\omega)~{}.\end{cases}$
(20)

Note that for $\omega_{p}^{\prime}\gg\omega$, $E_{\text{em}}^{\prime}$ is
exponentially suppressed, as expected from the heuristic discussion above.

This invisible field propagates unattenuated into the receiver cavity where it
drives a corresponding _visible_ mCP current density, which is determined
analogously to Eqs. (16) and (18),

$\displaystyle\boldsymbol{j}_{\text{rec}}=eq_{\text{eff}}\,(n_{+}{\bf
V}_{+}-n_{-}{\bf
V}_{-})\simeq-i\,\frac{eq_{\text{eff}}\,e^{\prime}\,n_{\chi}}{\omega\,m_{\chi}}\,\boldsymbol{E}^{\prime}_{\text{em}}~{}.$
(21)

This visible mCP current $j_{\text{rec}}$ acts as a source of SM EM fields,
and thus can resonantly excite the receiver cavity tuned to the same
frequency, depositing a total signal power of Hill (2009)

$\displaystyle
P_{\text{sig}}\simeq\frac{Q}{\omega}\,\frac{\big{|}\int_{\text{rec}}d^{3}{\bf
x}~{}\boldsymbol{E}_{\text{rec}}^{*}\cdot\boldsymbol{j}_{\text{rec}}\big{|}^{2}}{\int_{\text{rec}}d^{3}{\bf
x}~{}|\boldsymbol{E}_{\text{rec}}|^{2}}~{},$ (22)

where $\boldsymbol{E}_{\text{rec}}({\bf x})$ is the spatial profile of the
excited receiver cavity mode and the integrals are performed over the volume
of the receiver cavity. Note that we have not included the effect of an
associated mCP visible charge density $\rho_{\text{rec}}$ in Eq. (22);
although it can produce small irrotational electric fields, it cannot excite
resonant cavity modes, which are purely solenoidal Condon (1941); Smythe
(1988); Collin (1990).

###  III.3 Calculation of the Millicharged DM Signal for a Pair of Cylindrical
Cavities

Eqs. (19), (21), and (22) can be evaluated for any choice of cavity geometry
and emitter mode. Here, we evaluate these expressions for two different
representative mode choices in a setup involving coaxially-aligned cylindrical
cavities of radius $R$ and length $L$ separated by a distance $d\gg R,L$. In
particular, we calculate the signal power in the case that both cavities are
operated in either the $\text{TM}_{010}$ or $\text{TE}_{011}$ mode. In
cylindrical coordinates, the electric fields profiles in either configuration
are given by

$\displaystyle\boldsymbol{E}_{\text{em,rec}}({\bf
x})=E_{\text{em,rec}}\times\begin{cases}J_{0}(\alpha_{0}\rho/R)\,\Theta(z)\,\Theta(L-z)\,\hat{\boldsymbol{z}}&(\text{TM}_{010})\\\
J_{1}(\alpha_{1}\rho/R)\,\sin{(\pi
z/L)}\,\hat{\boldsymbol{\phi\,}}&(\text{TE}_{011})~{},\end{cases}$ (23)

where $\rho\in[0,R]$, $z\in[0,L]$, $E_{\text{em,rec}}$ controls the
normalization of the respective field, and $\alpha_{0}\simeq 2.40$,
$\alpha_{1}\simeq 3.83$ are the first zeroes of $J_{0}$, $J_{1}$,
respectively. In the first line of Eq. (23), we have included the Heaviside
step functions $\Theta$ enforcing that the EM fields are confined to
$z\in[0,L]$ in order to account for the discontinuity in the electric field at
the cavity endcaps located at $z=0$ and $z=L$. The corresponding resonant
frequency of either mode is given by $\omega=\alpha_{0}/R$ and
$\omega=\sqrt{(\alpha_{1}/R)^{2}+(\pi/L)^{2}}$ for $\text{TM}_{010}$ and
$\text{TE}_{011}$, respectively.

Let us begin by evaluating the invisible field generated by mCPs in an emitter
cavity driven in the $\text{TM}_{010}$ mode. Taking the emitter cavity to span
$z^{\prime}\in[0,L]$, the relevant factor in the integrand of Eq. (19) is

$\displaystyle\boldsymbol{E}_{\text{em}}({\bf
x}^{\prime})+\frac{1}{\omega^{2}-\omega_{p}^{\prime\,2}}\,\nabla\nabla\cdot\boldsymbol{E}_{\text{em}}({\bf
x}^{\prime})=E_{\text{em}}\,J_{0}(\alpha_{0}\rho^{\prime}/R)\,\hat{\boldsymbol{z}}\,\Big{(}1+\frac{1}{\omega^{2}-\omega_{p}^{\prime\,2}}\,\partial_{z^{\prime}}\big{[}\delta(z^{\prime})-\delta(z^{\prime}-L)\big{]}\Big{)}~{},$
(24)

where the factor involving delta functions in the expression above arises from
the step functions in Eq. (23) and effectively incorporates the presence of
the mCP density $\rho^{\prime}$ generated in the emitter cavity near the
endcaps. Using this in Eq. (19), the invisible field sourced by mCPs in the
emitter cavity is approximately

$\displaystyle\boldsymbol{E}_{\text{em}}^{\prime}({\bf
x},t)\simeq\frac{eq_{\text{eff}}\,e^{\prime}\,n_{\chi}}{\alpha_{0}\,m_{\chi}}~{}\frac{e^{-ikz}\,(1-e^{ikL})}{(kz)^{2}}~{}\bar{E}_{\text{em}}\,R^{2}\,e^{i\omega
t}\,\hat{\boldsymbol{z}}~{},$ (25)

in the far-field limit $z\gg R,L$, where
$\bar{E}_{\text{em}}=E_{\text{em}}J_{1}(\alpha_{0})$ is the RMS volume-
averaged emitter field and we assumed that $k>0$ (corresponding to
$\omega_{p}^{\prime}<\omega$). Using the above expression in Eqs. (21) and
(22) yields the signal power deposited into the receiver cavity placed a
distance $d\gg R,L$ away,

$\displaystyle P_{\text{sig}}(\text{TM}_{010})\simeq
64\pi\,\Big{(}\frac{eq_{\text{eff}}\,e^{\prime}\,n_{\chi}}{\alpha_{0}\,m_{\chi}}\Big{)}^{4}\,\frac{Q\,\bar{E}_{\text{em}}^{2}\,R^{6}}{d^{4}\,L\,\omega^{3}\,k^{6}}\,\sin^{4}{(kL/2)}~{}.$
(26)

Instead, for cavities operated in the $\text{TE}_{011}$ mode,
$\nabla\cdot\boldsymbol{E}_{\text{em}}=0$, such that the second term in the
integrand of Eq. (19) can be dropped. In the far-field limit, we find that the
invisible field sourced by mCPs in the emitter cavity is approximately

$\displaystyle\boldsymbol{E}_{\text{em}}^{\prime}({\bf
x},t)\simeq\frac{\pi^{2}\,H_{1}(\alpha_{1})}{2\sqrt{2}\,\alpha_{1}}\,\frac{eq_{\text{eff}}\,e^{\prime}\,n_{\chi}}{m_{\chi}}~{}\frac{e^{-ikz}\,(1+e^{ikL})}{z}~{}\bar{E}_{\text{em}}\,\frac{R^{2}\,L}{\pi^{2}-k^{2}\,L^{2}}\,e^{i\omega
t}\,\hat{\boldsymbol{\phi\,}}~{},$ (27)

where $\bar{E}_{\text{em}}=E_{\text{em}}\,|J_{0}(\alpha_{1})|\,/\sqrt{2}$ is
the $\text{TE}_{011}$ RMS volume-averaged emitter field. Comparing Eqs. (25)
and (27), we see that the $\text{TE}_{011}$ configuration leads to enhanced
fields in the far-field limit, since $E_{\text{em}}^{\prime}\propto 1/z$,
instead of $E_{\text{em}}^{\prime}\propto 1/z^{2}$ as in the $\text{TM}_{010}$
configuration. For the $\text{TE}_{011}$ driven mode, the corresponding signal
is then evaluated to be

$\displaystyle P_{\text{sig}}(\text{TE}_{011})\simeq
4\pi^{9}\,\Big{(}\frac{eq_{\text{eff}}\,e^{\prime}\,n_{\chi}}{\alpha_{1}\,m_{\chi}}\Big{)}^{4}\,\frac{Q\,\bar{E}_{\text{em}}^{2}\,R^{6}\,L^{3}}{d^{2}\,\omega^{3}\,(\pi^{2}-k^{2}\,L^{2})^{4}}\,H_{1}^{4}(\alpha_{1})\,\cos^{4}{(kL/2)}~{}.$
(28)

The enhancement of the $\text{TE}_{011}$ configuration in the far-field limit
is to be expected. Just as in normal electromagnetism, if the dark photon is
massless then $\boldsymbol{E}_{\text{em}}^{\prime}$ is dominantly emitted in
directions transverse to the direction of the oscillating dark current
$\boldsymbol{j}_{\text{em}}^{\prime}$, which in turn is aligned with the
polarization of the driven emitter’s electric field
$\boldsymbol{E}_{\text{em}}$. Thus, the “longitudinal” $\text{TM}_{010}$
configuration has suppressed sensitivity in the far-field limit. However, as
we show below, since $d\sim 1/\omega$ in typical RF LSW experiments, this
amounts to only a small penalty in sensitivity, and as a result both setups
have comparable sensitivity to the mCDM coupling $q_{\text{eff}}$.

In our estimates above, we have only investigated coaxially-aligned cavities,
finding that the signal in a TE configuration is enhanced compared to that of
a TM setup in the far-field limit. We expect similar conclusions to hold for
different choices of the cavity alignment. For instance, in the alternative
case that the cavities are instead kept parallel and separated in the radial
direction, then TM modes involve electric fields which are purely transverse
(with respect to the axis connecting the two cavities), while TE modes possess
both transverse and longitudinal components. Hence, applying the lessons
learned from the above examples, we would expect both mode choices to possess
optimal scaling in the far-field limit. Indeed, an explicit calculation
confirms this intuition, although we do not present the detailed analysis
here.

As discussed near Eq. (7), the signal power scales as
$P_{\text{sig}}\propto(eq_{\text{eff}}\,e^{\prime}\,n_{\chi}/m_{\chi})^{4}=\epsilon^{4}\,\omega_{p}^{\prime\,8}$.
Interpreting the dark plasma frequency $\omega_{p}^{\prime}$ as an effective
dark photon mass, we see that this result is analogous to direct signals of
transverse massive dark photons in the absence of an mCP background, shown in
Eq. (6). In such searches, the unoptimized “transverse configuration” scales
as $P_{\text{sig}}\propto\epsilon^{4}\,m_{A^{\prime}}^{8}$, whereas the
optimized “longitudinal configuration” scales instead as
$P_{\text{sig}}\propto\epsilon^{4}\,m_{A^{\prime}}^{4}$. However, this analogy
is imperfect, as $\omega_{p}^{\prime}$ does not constitute a true mass for the
dark photon. Indeed, neither of the mCP signals scales as
$P_{\text{sig}}\propto\epsilon^{4}\omega_{p}^{\prime\,4}$. The origin of this
difference is that the plasma frequency and a true mass both modify the
dispersion relation of the transverse mode in a similar manner, i.e.,
$(\omega^{2}-k^{2})\,A^{\prime}_{T}=(m_{A^{\prime}}^{2}+\omega_{p}^{\prime\,2})\,A^{\prime}_{T}$,
whereas the modifications are entirely different for the longitudinal mode
when $k\neq 0$,
$(\omega^{2}-k^{2})\,A^{\prime}_{L}=\big{[}m_{A^{\prime}}^{2}+(1-k^{2}/\omega^{2})\,\omega_{p}^{\prime\,2}\big{]}\,A^{\prime}_{L}$.

##  IV Existing Constraints and Future Sensitivity

The expression for the signal power in Eqs. (26) and (28) can be used to
estimate existing limits and future sensitivity of LSW experiments to a
terrestrial density of mCDM. We begin by recasting the result of the Dark SRF
experiment, which set its first limits in a recent pathfinder run Romanenko
_et al._ (2023). Dark SRF employs two coaxial elliptical cavities operated in
the $\text{TM}_{010}$ mode. In our calculation, we approximate these cavities
as cylinders with radius $R\simeq 10\ \text{cm}$ and length $L\simeq 5\
\text{cm}$, separated by a distance of $d\simeq 50\ \text{cm}$, with the
emitter cavity driven with an RMS-averaged field of $\bar{E}_{\text{em}}\simeq
6\ \text{MV}/\text{m}$, and the receiver cavity possessing a loaded quality
factor of $Q\simeq 3\times 10^{10}$. This pathfinder run placed a limit on the
signal power in the receiver cavity of $P_{\text{sig}}\simeq 2.5\times
10^{-16}\ \text{W}$. From Eq. (26), we estimate that this places an existing
limit of

$\displaystyle q_{\text{eff}}\lesssim
0.1\times\bigg{(}\frac{1}{e^{\prime}}\bigg{)}\,\bigg{(}\frac{m_{\chi}}{1\
\text{GeV}}\bigg{)}\bigg{(}\frac{10^{6}\
\text{cm}^{-3}}{n_{\chi}}\bigg{)}~{}.$ (29)

Future runs of the Dark SRF experiment will strengthen this sensitivity. The
most notable planned improvements are increased frequency matching between the
emitter and receiver cavities (a mismatched frequency in the pathfinder run
limited the attainable signal power by roughly five orders of magnitude),
larger quality factors, and reduced noise. To estimate the reach of a future
experiment, we take the same experimental parameters as assumed above, except
we assume an average field of $\bar{E}_{\text{em}}\simeq 25\
\text{MV}/\text{m}$, a quality factor of $Q=10^{12}$, and take the limiting
noise to be arising from thermal occupation of the EM modes. In this case, if
the emitter field’s phase is actively monitored, an optimized signal analysis
leads to an effective noise power of $P_{\text{noise}}\simeq T/t_{\text{int}}$
Graham  _et al._ (2014), where $T=10\ \text{mK}$ is the temperature of the
receiver cavity, and $t_{\text{int}}=1\ \text{yr}$ is the total experimental
integration time. Setting $P_{\text{sig}}=P_{\text{noise}}$, we estimate a
future sensitivity of

$\displaystyle
q_{\text{eff}}\lesssim\bigg{(}\frac{1}{e^{\prime}}\bigg{)}\,\bigg{(}\frac{m_{\chi}}{1\
\text{GeV}}\bigg{)}\bigg{(}\frac{10^{6}\
\text{cm}^{-3}}{n_{\chi}}\bigg{)}\times\begin{cases}2\times 10^{-6}\
(\text{TM}_{010})\\\ 6\times 10^{-7}\ (\text{TE}_{011})~{}.\end{cases}$ (30)

In the first and second lines in the expression above, we have assumed that
the cavities are operated in the $\text{TM}_{010}$ or $\text{TE}_{011}$ mode,
respectively, which shows that a TE configuration results in an
$\mathcal{O}{(1)}$ enhanced sensitivity to the mCP coupling. We also note that
the CROWS experiment operated in the optimized configuration, employing
coaxially-aligned cavities in the $\text{TE}_{011}$ mode Betz  _et al._
(2013). However, this does not result in a competitive limit compared to that
derived from Dark SRF in Eq. (29), since the mild enhancement due to the TE
mode is not sufficient to overcome the penalty of the smaller quality factor
and field strength.

Figure 3: The future sensitivity (blue lines), as in Eq. (30), of Dark SRF to
mCPs with a laboratory number density of $n_{\chi}=10^{3}\ \text{cm}^{-3}$
(left) or $n_{\chi}=10^{6}\ \text{cm}^{-3}$ (right), fixing $e^{\prime}=1$ and
taking the cavities to be operated in the optimal $\text{TE}_{011}$ (solid) or
the sub-optimal $\text{TM}_{010}$ (dashed) mode-configuration. Limits from
collider searches and SN1987A are shown as shaded grey regions. Model-
dependent bounds from modifications to the cosmological expansion rate are
shown as solid and dotted gray lines. Limits from ion traps Budker  _et al._
(2022) for $n_{\chi}=10^{3}\ \text{cm}^{-3}$ ($10^{6}\ \text{cm}^{-3}$) are
also shown in the left (right) panel. Note that ion traps and Dark SRF are not
sensitive to $q_{\text{eff}}\gtrsim 0.1$, since in this case mCPs are not able
to penetrate typical laboratory devices Budker  _et al._ (2022). Above the
dotted dark red line, mCP relics rapidly thermalize in Earth’s atmosphere and
crust before encountering terrestrial direct detection experiments. Below the
solid light red line, $q_{\text{eff}}$ is not sufficiently large for mCPs to
thermalize within any region of the Earth.

Figure 4: As in Fig. 3, but fixing $m_{\chi}=10\ \text{MeV}$ (left panel) or
$m_{\chi}=10\ \text{GeV}$ (right panel) in the parameter space spanned by
$n_{\chi}$ and $q_{\text{eff}}$. Dark SRF projections are shown as blue lines.
Compared to Fig. 3, we also show existing limits recast from the recent Dark
SRF pathfinder run (light blue) Romanenko  _et al._ (2023) as well as
optically-levitated nanospheres Afek  _et al._ (2021).

The projected sensitivity of Dark SRF is shown as solid (dashed) blue lines in
Figs. 3 and 4, assuming an experimental setup consistent with the second
(first) line of Eq. (30). In Fig. 3, we illustrate the reach in
$q_{\text{eff}}$ as a function of the mCP mass $m_{\chi}$, fixing the dark
gauge coupling to $e^{\prime}=1$ and the ambient mCP density within the
cavities to either $n_{\chi}=10^{3}\ \text{cm}^{-3}$ (left panel) or
$n_{\chi}=10^{6}\ \text{cm}^{-3}$ (right panel). As discussed in Section II,
this density can be parametrically different from the galactic density and
average mCP density near Earth’s surface, since $n_{\chi}$ depends both on the
density and temperature of the surrounding environment Berlin  _et al._
(2023). We discuss this in more detail below. Note that in our projections, we
have adopted a noise temperature of $T=10\ \text{mK}$, which corresponds to
the temperature of the receiver cavity; the emitter cavity, which contains
large driven fields, can only be feasibly cooled to $T\sim 1\ \text{K}$.
Hence, our adopted values of $n_{\chi}$ should be interpreted as the geometric
mean of the mCP density in both cavities. Also shown in solid gray are
existing limits from accelerators Prinz  _et al._ (1998); Acciarri  _et al._
(2020); Ball  _et al._ (2021); Davidson  _et al._ (2000); Argüelles Delgado
_et al._ (2021), observations of SN1987A Chang  _et al._ (2018), and anomalous
heating in ion traps Budker  _et al._ (2022).222We have chosen to show ion
trap limits assuming a comparable fixed density within the trap itself. We
thank Hari Ramani for providing the corresponding limits for us here, which
have been appropriately modified from those presented in Ref. Budker  _et al._
(2022). In Fig. 4, we instead show the mCP parameter space spanned by the
charge $q_{\text{eff}}$ and laboratory density $n_{\chi}$, fixing
$e^{\prime}=1$ and $m_{\chi}=10\ \text{MeV}$ (left panel) or $m_{\chi}=10\
\text{GeV}$ (right panel). From Figs. 3 and 4, we see that LSW signals of mCPs
have a more favorable scaling than other techniques for smaller masses.

Model-dependent cosmological limits are shown as either solid or dotted gray
lines in Fig. 3. Additional dark radiation in the form of mCPs and dark
photons may unacceptably alter the expansion rate in the early Universe. This
is usually parametrized as an increased effective number of neutrino species
$N_{\text{eff}}$. Within parameter space bounded by the contour labelled
“$N_{\text{eff}}(\chi)$,” mCPs efficiently thermalize in the early universe
and directly modify the expansion rate during the time of nucleosynthesis
Creque-Sarbinowski  _et al._ (2019). Within the dotted contour labelled
“$N_{\text{eff}}(A^{\prime})$,” mCPs can indirectly lead to observable
modifications to $N_{\text{eff}}$, since upon thermalizing with the SM bath
they generate an appreciable density in dark photon radiation via $\chi\chi\to
A^{\prime}A^{\prime}$ Vogel and Redondo (2014); Adshead  _et al._ (2022). Such
bounds are weakened for $m_{\chi}\gtrsim 1\ \text{GeV}$, in which case the
relative contribution of the $A^{\prime}$ density is reduced once the SM
decouples from the dark sector at temperatures $T\lesssim m_{\chi}$, before
numerous mass thresholds in the SM heat the visible sector compared to the
dark radiation. However, these limits are alleviated in non-standard
cosmologies. For instance, reheat temperatures of the universe as small as
$T_{\text{reheat}}\sim 5\ \text{MeV}$ are compatible with standard
nucleosynthesis de Salas  _et al._ (2015); Hannestad (2004); Kawasaki  _et
al._ (2000), yet can be invoked to prevent the thermalization of mCPs heavier
than this scale. In this case, non-thermal mechanisms can give rise to a
substantial density of mCPs with $m_{\chi}\gtrsim 10\ \text{MeV}$, alleviating
the bounds labelled as “$N_{\text{eff}}(A^{\prime})$.” While low reheat
temperatures do not prevent the thermalization of sub-MeV mCPs, alleviating
the bounds labelled “$N_{\text{eff}}(\chi)$” is possible for models in which
the millicharge sector’s coupling to SM matter is suppressed at large
densities and early cosmological times (see, e.g., Refs. Masso and Redondo
(2006); DeRocco  _et al._ (2020)).

Above the dotted dark red line labelled “D.D. ceiling” in Fig. 3, mCP relics
rapidly thermalize in Earth’s atmosphere and crust before encountering
terrestrial direct detection experiments searching for DM-SM scattering Emken
_et al._ (2019). Hence, above this contour, scattering-based direct detection
searches are not sensitive to such strongly-coupled DM subcomponents, since
the particles present in such detectors do not have sufficient energy to
scatter and deposit signals above threshold. Below this line, the sensitivity
of these experiments depends on the particular value of the galactic mCP
density $n_{\chi}^{(\text{gal})}$, which can be orders of magnitude smaller
than the fixed laboratory density $n_{\chi}$. For instance, for $1\
\text{GeV}\lesssim m_{\chi}\lesssim 100\ \text{GeV}$, laboratory overdensities
in cryogenic detectors can be as large as
$n_{\chi}/n_{\chi}^{(\text{gal})}\sim 10^{18}$ Berlin  _et al._ (2023). Using
the formalism developed in Ref. Berlin  _et al._ (2023) to translate between
$n_{\chi}$ and $n_{\chi}^{(\text{gal})}$ and the various direct detection
limits presented in Ref. Emken  _et al._ (2019), we find that direct detection
experiments can place limits on minimal models in the parameter space below
the red dotted line in Fig. 3 for $m_{\chi}\lesssim 1\ \text{GeV}$ and
$m_{\chi}\gtrsim 100\ \text{GeV}$, corresponding to where
$n_{\chi}^{(\text{gal})}$ is typically only a few orders of magnitude smaller
than $n_{\chi}$. However, we note that such bounds are alleviated for
$m_{\chi}\lesssim 1\ \text{GeV}$ in non-minimal models where mCPs weakly
couple to an additional long-ranged force, resulting in larger terrestrial
overdensities for sub-GeV masses Acevedo  _et al._ (2023). Finally, below the
solid light red line labelled “inefficient thermalization” in Fig. 3,
$q_{\text{eff}}$ is not sufficiently large for mCPs to thermalize within any
region of the Earth Berlin  _et al._ (2023); thus, in assuming large
terrestrial overdensities, we are restricted to consider couplings above this
contour.

In Fig. 4 we only show the constraints/projections for Dark SRF up to values
of $n_{\chi}$ where $\omega_{p}^{\prime}\sim\omega/10$. From the discussion in
Section III.2, it is evident that when $\omega_{p}^{\prime}$ approaches
$\omega$, some of the simplifications we have made no longer apply. We also
see from Eq. (19) that when $\omega_{p}^{\prime}\gtrsim\omega$, the invisible
dark field produced in the emitter cavity will be exponentially suppressed at
distances $d\gtrsim 1/\omega_{p}^{\prime}$. As a result, the Dark SRF
sensitivity will rapidly deteriorate for higher values of $n_{\chi}$ inside
either cavity.

##  V Conclusion

Additional $U(1)^{\prime}$ gauge groups that are kinetically-mixed with the
visible photon are motivated both by ultraviolet completions of the Standard
Model as well as by their ability to provide a natural dark matter candidate.
Dark states that are directly charged under this additional gauge group would
appear as millicharged particles (mCPs) under electromagnetism. It has
previously been shown that if their effective charge is sufficiently large,
the thermalized terrestrial abundance of such mCPs can be far greater than the
average galactic density, but would have gone undetected by standard direct
detection experiments, motivating alternative experimental techniques.

Although direct signals of dark photons decouple in the massless limit,
indirect signals may still arise in an mCP background. In particular, we have
shown that the light-shining-through-wall (LSW) Dark SRF experiment at FNAL
Romanenko  _et al._ (2023), operated to search for massive dark photons, also
has sensitivity to similar signals of massless dark photons arising from EM-
induced disturbances of mCP dark matter subcomponents. Owing to the nature of
the signal, the optimal geometry for the fields in the emitter and receiver
cavities is one in which the electric fields in both cavities are transverse.
This is unlike typical LSW searches for massive dark photons in which case the
longitudinal arrangement is optimal. However, although the longitudinal
configuration has suppressed sensitivity to mCPs in the far-field limit, the
typical separation of the cavities in LSW experiments is such that this only
results in an $\mathcal{O}{(1)}$ penalty in sensitivity to the mCP charge
$q_{\text{eff}}$. The scaling of the experimental sensitivity is such that
future runs of the Dark SRF experiment can set the best limits on mCPs in
certain regions of parameter space, particularly at low masses and small
terrestrial abundances.

###### Acknowledgements.

We thank Hari Ramani for providing the appropriately re-scaled constraints on
mCPs from ion traps appearing in Fig. 3. 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.
Fermilab is operated by the Fermi Research Alliance, LLC under Contract DE-
AC02-07CH11359 with the U.S. Department of Energy. The work of SARE was
supported by SNF Ambizione grant PZ00P2_193322, New frontiers from sub-eV to
super-TeV. The work of JIR is supported by the Natural Sciences and
Engineering Research Council (NSERC) of Canada. 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.

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