The Funk Experiment

# Limits from the Funk Experiment on the Mixing Strength of Hidden-Photon Dark
Matter in the Visible and Near-Ultraviolet Wavelength Range

Arnaud Andrianavalomahefa  Institute for Nuclear Physics, Karlsruhe Institute
of Technology , Germany  Christoph M. Schäfer  Institute for Nuclear Physics,
Karlsruhe Institute of Technology , Germany  Darko Veberič
<EMAIL_ADDRESS>Institute for Nuclear Physics, Karlsruhe Institute of
Technology , Germany  Ralph Engel  Institute for Nuclear Physics, Karlsruhe
Institute of Technology , Germany  Thomas Schwetz  Institute for Nuclear
Physics, Karlsruhe Institute of Technology , Germany  Hermann-Josef Mathes
Institute for Nuclear Physics, Karlsruhe Institute of Technology , Germany
Kai Daumiller  Institute for Nuclear Physics, Karlsruhe Institute of
Technology , Germany  Markus Roth  Institute for Nuclear Physics, Karlsruhe
Institute of Technology , Germany  David Schmidt  Institute for Nuclear
Physics, Karlsruhe Institute of Technology , Germany  Ralf Ulrich  Institute
for Nuclear Physics, Karlsruhe Institute of Technology , Germany  Babette
Döbrich  Physics Department, CERN, Geneva, Switzerland  Joerg Jaeckel
Institute for Theoretical Physics, Heidelberg University, Germany  Marek
Kowalski  Deutsches Elektronen Synchrotron (DESY), Hamburg, Germany
Department of Physics, Humboldt University, Berlin, Germany  Axel Lindner
Deutsches Elektronen Synchrotron (DESY), Hamburg, Germany  Javier Redondo
Department of Theoretical Physics, University of Zaragoza, Spain

###### Abstract

We present results from the Funk experiment in the search for hidden-photon
dark matter. Near the surface of a mirror, hidden photons may be converted
into ordinary photons. These photons are emitted perpendicularly to the
surface and have an energy equal to the mass of the dark matter hidden photon.
Our experimental setup consists of a large, spherical mirror with an area of
more than 14 m2, which concentrates the emitted photons into its central
point. Using a detector sensitive to visible and near-UV photons, we can
exclude a kinetic-mixing coupling of stronger than $\chi\approx 10^{-12}$ in
the mass range of 2.5 to 7 eV, assuming hidden photons comprise all of the
dark matter. The experimental setup and analysis used to obtain this limit are
discussed in detail.

###### pacs:

dark matter, hidden photons, spherical mirror, PMT, dark current

##  I Introduction

Constituting approximately 85% of its matter and 25% of its total energy
content Akrami et al. (2018), dark matter (DM) is possibly one of the central
puzzles in our quest to understand the Universe. Structure formation indicates
that DM behaves, to a very good approximation, like a cold and pressureless
gas, which suggests that it is made from slow, non-relativistic particulates.
The mass of these particulates is, however, essentially unknown. They could be
elementary particles with masses as low as ${\sim}10^{-22}$ eV, but they could
also be macroscopic objects, such as black holes, with masses as large as
${\sim}10\,M_{\text{sun}}$.

One of the most promising ways to answer this dilemma is a direct detection of
the DM particles in a laboratory environment. In this paper, we present the
results of exactly such an experiment.

In particular, our experiment, “Finding U(1)’s of a Novel Kind” (Funk),
searches for “dark” or “hidden” photons (HPs) in a light range of masses
around ${\sim}$eV.

Following Occam’s razor, this option is especially appealing since it extends
the Standard Model (SM) gauge group by only a single U(1) gauge factor giving
rise to the HP. Taking all SM particles to be uncharged under the new U(1)
makes the extra gauge boson “hidden”. The only interaction then arises from a
coupling to the ordinary photon through kinetic mixing Holdom (1986).
Analogous to neutrino oscillations, this mixing leads to photon-HP
oscillations. For the oscillations to be observable, the HP needs to have a
non-vanishing mass that can arise either via the Stückelberg or the Higgs
mechanism (see Jaeckel (2012) for a recent review). The relevant Lagrangian is
given by

$\displaystyle\mathcal{L}$
$\displaystyle\supset-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+J^{\mu}A_{\mu}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
-\frac{1}{4}X_{\mu\nu}X^{\mu\nu}-\frac{\chi}{2}F_{\mu\nu}X^{\mu\nu}+\frac{m_{\gamma^{\prime}}^{2}}{2}X_{\mu}X^{\mu}.$
(1)

As usual, $F_{\mu\nu}$ is the field strength of ordinary photons ($A_{\mu}$),
and $X_{\mu\nu}$ is the analogue for hidden photons ($X_{\mu}$). The two new
parameters of the model are the kinetic mixing $\chi$ and the HP mass
$m_{\gamma^{\prime}}$. Explicit models often predict small or even tiny values
for the coupling $\chi$ Holdom (1986); Dienes et al. (1997) (see Jaeckel
(2012) for more references), which is very much in line with this particle
being “hidden”.

Over the past few years, this simple model has received considerable interest
and inspired a number of searches (see, e.g. Hew (2012); Essig et al. (2013);
Battaglieri et al. (2017); Alemany et al. (2019); Beacham et al. (2019)).

In particular, two mass ranges have been focused on. MeV to GeV mass-scale
photons could act as messengers of DM Arkani-Hamed et al. (2009) but have also
been proposed to address the discrepancy between the anomalous magnetic moment
of the muon ($g{-}2$) and its value as predicted by the SM Bennett et al.
(2006); Pospelov (2009).

Another possibility explored in many laboratory-scale experiments, is that the
HP is very light, typically at or below the eV scale. Such HPs are intriguing
not only because of their simplicity but also because they have been shown to
be a viable DM candidate Nelson and Scholtz (2011); Arias et al. (2012);
Graham et al. (2016); Agrawal et al. (2020); Dror et al. (2019); Co et al.
(2019); Bastero-Gil et al. (2019); Long and Wang (2019). This is the case we
are interested in.

As already mentioned, if such HPs were to exist, they would be able to
oscillate into ordinary photons. Thus, even without assuming that HPs are DM,
searches can be performed for HPs using light-shining-through-walls
experiments Woollett et al. (2015); Betz (2014-01-10); Ehret et al. (2010) or
observations of the sun Redondo (2015); Schwarz et al. (2015).

To detect HPs in the MHz and GHz range, “classical” cavity searches Sikivie
(1983) may be performed Hoang Nguyen et al. (2019), or a reinterpretation of
cavity searches can be made when the cavity is inserted into a magnetic field
to search for the axion Arias et al. (2012). At even lower frequencies,
“radio” searches for HP DM have been put forward Silva-Feaver et al. (2017);
Arias et al. (2015); Chaudhuri et al. (2015).

Figure 1: Schematic drawing of the Funk experimental setup. The detector
(photomultiplier) is placed on top of a steel pillar (left) so that the
sensitive surface is located exactly in the central point (CP) of the mirror
(right). The whole experimental area (indicated with a rectangle) is enclosed
with opaque materials. On the right, a human figure is also shown for scale.
(Slightly adapted from Veberič et al. (2018a); Engel et al. (2017).)

Another elegant method to detect HPs by using large, spherical mirrors, the
so-called “dish antennas,” has been proposed Horns et al. (2013); Jaeckel and
Redondo (2013). In contrast to cavity searches, where each single measurement
only probes a mass range equal to the width of the employed cavity, this is,
in principle, a broadband method that can be employed to test a wide parameter
space when appropriate detectors are employed. The search with the dish
antenna can be understood as follows: Owing to their (tiny) effective coupling
to electrons, HPs can induce oscillations of the free electrons in a
conductor. As a consequence of this excitation, ordinary photons can be
emitted from the surface of the conductor. This emission may be increased by
choosing a conductor with high reflectivity and forming it into a spherical
mirror to facilitate the collection of photons. The emitted ordinary photons
can then be collected at the central point (CP) of the spherical mirror,
whereas no such focusing would take place for the background photons with
random arrival directions. This (geometrical) enhancement comes from the fact
that the ordinary photons, generated by HP DM, are emitted almost
perpendicularly to the surface with only a tiny directional modulation Jaeckel
and Knirck (2016) due to Earth’s relative velocity (or the relative velocity
of an experiment) with respect to the rest frame of the HP DM.

The time-averaged power $P$ of ordinary photons emitted by the HPs emanating
from the mirror towards the CP is

$P=\chi^{2}\,\langle\cos^{2}\theta\rangle\,\rho_{\text{DM}}\,A_{\text{mirr}},$
(2)

where $\theta$ is the angle of polarization of the HPs with respect to the
mirror plane, and the brackets $\langle\cdot\rangle$ indicate the average over
the local population. Moreover, $\rho_{\text{DM}}$ is the local DM density,
and $A_{\text{mirr}}$ is the area of the mirror. Accordingly, a large
$A_{\text{mirr}}$ can greatly enhance the collected power and increase the
sensitivity of the setup.

The Funk experiment exploits exactly the aforementioned dish-antenna method.
In the following, we report our results from an ${\sim}1$ month-long data-
acquisition campaign, which took place in 2019. Preliminary results and
calibrations were presented in a number of proceedings Veberič et al. (2016);
Döbrich et al. (2015); Veberič et al. (2018a); Engel et al. (2017), some of
which we repeat here for convenience and self-containment purposes.

Besides our experiment, three other experiments Suzuki et al. (2015); Knirck
et al. (2018); Brun et al. (2019) have published results based on the dish-
antenna method. However, the surface of our mirror is about one order of
magnitude larger, which on the one hand strongly enhances the experimental
sensitivity while on the other hand increases the complexity of this
experiment. Even though the principles behind the dish-antenna method are
quite simple, Funk required a careful design and a number of iterations to
arrive at the final sensitivity reported in this paper.

Owing to a comparatively large mirror size and relatively low noise of the
chosen detector, we are able to set limits on HP coupling on the order of
$\chi<10^{-13}$, which is comparable to the strongest astrophysical
constraints existing for HP DM in this mass range.

The paper is structured as follows: Section II describes the experimental
setup including mirror alignment, external-light shielding of the experimental
area, mechanics, and the data-acquisition and muon-monitoring systems. Section
III describes the data analysis, including the observation of a “memory
effect” in the photon sensor, the understanding of which crucially affected
the analysis procedure. In Section IV, based on the non-observation of a
(significant) signal above background, we set upper limits on the mixing
parameter $\chi$ for the specific HP mass range corresponding to our detector.

##  II Experimental setup

###  II.1 Mirror

Figure 2: _Left:_ Placement of the assembled mirror (blue) and its support
structure (black) in the experimental area (green). _Right:_ Illustration of
the dimensions of the spherical mirror, where $r=3.40$ m, $a=3.72$ m, $b=3.70$
m, and $\phi=66^{\circ}$, resulting in a total mirror area of
$A_{\text{mirr}}=14.56$ m2. Adapted from Veberič et al. (2016).

As a suitable surface for the conversion of HPs, we employ a prototype mirror
(see Fig. 1) of the Pierre Auger Observatory Abraham et al. (2004); Aab et al.
(2015). As shown in Fig. 2, the spherical mirror is built from $6\times 6$
approximately rectangular pieces, each consisting of an Al base with a thin
surface of an Al-Mg-Si-alloy. The dimensions of the mirror are indicated in
Fig. 2. For our purposes, the most relevant feature is the large reflecting
area $A_{\text{mirr}}=14.56$ m2. See Veberič et al. (2016) for more details.

We located the experiment in a windowless, air-conditioned experimental hall
with the approximate dimensions $18\,\text{m}\times 14\,\text{m}\times 9$ m
and with 2 m thick concrete walls. The main experimental zone takes up a
volume of $4.35{\times}4.98{\times}4.30$ m3 (see Fig. 3) and has been for
light-shielding purposes enveloped with a double-layer of black polyethylene
sheets with a thickness of $120\,\upmu$m stretched over thick cotton curtains.
Furthermore, we treated the floor within this confined area with a non-
reflective and non-glossy black paint.

The assembly of the spherical mirror first followed the standard procedure for
construction of the Auger fluorescence telescopes Abraham et al. (2010). Each
mirror segment has an adjustable pointing direction and can be radially moved
forward/backward by a few centimeters. To capture as many of the produced
photons as possible, the spherical mirror must focus them into the CP with a
small spot size111We stress that the produced photons are concentrated in the
CP and not in the usual focal point of the mirror. This is due to the
perpendicular emission from the spherical surface Horns et al. (2013).. The
process to quantify the quality of this focus was already described in Veberič
et al. (2016); Veberič et al. (2018a); however, we briefly summarize it again
here.

Any light ray emanating from a source located at the CP of an ideal spherical
mirror hits the mirror surface at a right angle. The ray is therefore by the
spherical mirror reflected exactly back into the CP. On the other hand, a
slight lateral offset of the source results in reflected rays focusing at a
point mirrored across the CP. With a suitably small source, the mirror
segments can therefore be aligned by focusing their reflected light. For this
purpose, we used a platform consisting of a frosted-glass pane, which served
as a suitable projection screen, and a yellow-green LED as the light source.

Figure 3: Photograph of the experimental area, where the black-painted floor
can be seen with the PMT-camera pillar on the left side and the spherical
mirror on the right side. The black-cotton curtain, shown here behind the
mirror in a partially installed state, eventually enclosed the whole area and
was covered with another layer of black-polyethylene foil. Taken from Veberič
et al. (2018a); Engel et al. (2017). Figure 4: Illustration of the point-
spread function of the mirror before and after the realignment of the mirror
segments: _Left:_ Contours of radial distance (i.e. perpendicular to the
optical axis of the mirror) from the image center of mass (CM) containing
certain fractions of the total light (from 20% to 90% in steps of 10%). The
blue lines mark the longitudinal distance (i.e. along the optical axis) of the
three images of the spot, shown as insets to the plot. _Right:_ Similar to the
plot on the left, but after the dedicated realignment campaign. The radial
size of the spot containing 90% of the light has shrunk from 6 to 2 mm (see
the two middle insets). Taken from Veberič et al. (2016), see also Veberič et
al. (2018a).

We employed a standard CCD camera to observe this reflection pattern on the
frosted-glass screen. By moving the screen along the optical axis of the
mirror, we then obtained cross-sectional images of the converging light beams
from the individual mirror segments. The position corresponding to the
smallest spot size and best sharpness of these images is then taken as the
nominal CP of the mirror, where we locate the detector.

In Fig. 4-(left), we present the result of such a scan around the CP of the
initial setup. The insets show the images obtained at different longitudinal
positions of the screen, where the central inset corresponds to the position
with the smallest spot size. This is confirmed by the contour lines, which
indicate the fraction of the light contained within a circle of a given radius
from the image center-of-mass (CM). The smallest spot for the 90% quantile is
observed at a radius $r\approx 6$ mm. Potential for improvement can be seen
when observing the lower inset of Fig. 4-(left). Here, individual mirror
segments are in good focus but unfortunately with all the foci in different
positions. Therefore, a fine realignment of the pointing and the radial
distance of each individual segment was required to achieve the smallest
possible spot size (more details can be found in Veberič et al. (2016)). This
procedure enabled us also to compensate for the manufacturing variance in the
radius of curvature of individual mirror segments. The improvement of the spot
size (and thus the accuracy of the CP position) is illustrated in Fig. 4,
where the size of the spot has been reduced by a factor of 3 (to only 2 mm for
the 90% quantile). Note that the spot size is now much smaller than the active
area of the detector described in the next section.

###  II.2 Detector and calibration

The measurements reported in this paper were made by a low-noise
photomultiplier tube 9107QB produced by ET Enterprises ET Enterprises (a).
This photomultiplier tube (PMT) has a sensitivity in the optical and near-
ultraviolet range (wavelengths between 150 and 630 nm) and reaches peak
quantum efficiency at $q_{\text{eff}}(330\,\text{nm})=24\%$ (see Fig. 5).
Additionally, when operating at high gain, it exhibits low levels of noise and
good single-photoelectron (SPE) resolution, thus making it an excellent
detector for photon-counting applications.

The photocathode has an effective area with a diameter of ${\sim}25$ mm, which
is large enough to contain the spot and capture all of the potential HP signal
emitted from the surface of the mirror, even in the presence of the seasonal
movement of the spot Döbrich et al. (2014). The PMT was placed inside of a
cooled housing FACT50 ET Enterprises (b) (see also Fig. 6), which is able to
reduce the internal temperature to 50 K below the ambient temperature.
Nevertheless, for the measurements reported here, the cooling was disabled,
mostly to avoid build-up of ice on the PMT pins and to remove the additional
insulating double-pane window the housing had installed in front of the PMT.
Cosmic muons passing through this volume of glass could produce a significant
quantity of Cherenkov photons, which would increase the levels of background.

The operating voltage of the PMT was set to 1050 V which was determined from
the observation of a break in the dark-count rate of a voltage scan Döbrich et
al. (2015). We then performed an operational test and a calibration of the PMT
with a faint blue LED flasher emitting light pulses of adjustable strength. In
Fig. 7, we observe how the distributions of collected anode charge $Q$ (black
lines) at particular pulse strengths develop a peak at large charges (i.e. at
$Q>10^{9}$ e) as the intensity of the LED is gradually increased and
increasingly more photons are emitted in each pulse. For the estimation of the
SPE charge, the optimal setting of the LED was chosen such that _(i)_ 95% of
the captured traces ($4\,\upmu$s long) contained only one single pulse, _(ii)_
the PMT saw pulses only 20% of the time, and _(iii)_ the pulse was observed in
a narrow time interval ${\sim}290$ ns after the flasher trigger. From the
charge distribution at this chosen LED intensity and high-voltage setting, we
determined the gain of the PMT to be $G\approx 10^{8}$.

The _internal background_ of the PMT was measured by attaching a light-tight
metal lid onto the FACT50 housing, essentially isolating the PMT from all
external light sources. The trigger rates captured in this sealed-PMT
condition for more than a month are shown in Fig. 8, where a relatively stable
and low trigger rate can be observed with a mean of 1.55 Hz and a standard
deviation of 0.20 Hz. The isolated peaks, seen at certain times, are most
likely the result of cosmic-ray muons directly hitting the glass envelope of
the PMT and thus releasing many Cherenkov photons.

Figure 5: Quantum efficiency of our specific ET 9107QB photomultiplier, as
obtained by a dedicated measurement of the manufacturer’s laboratory. Figure
6: Photograph of the camera setup. The PMT is housed in the FACT50 enclosure
(gray box). The photocathode of the PMT can be seen through the iris shutter
with an inner diameter of 65 mm. Most of the front-facing surfaces were
additionally covered with black tape to reduce potential reflections. The tape
was removed for this photograph. Figure 7: A pseudo-3D plot of charge $Q$
collected at the anode of the PMT as a function of the intensity of the LED
flasher (pulse strength in arbitrary units). The red line indicates the
setting used for the SPE study (see text for details) and the blue line is the
higher setting used to acquire pulses shown in Fig. 14. Figure 8: Measurement
of the internal background rate of the PMT. Data were taken in 60 s intervals,
with the PMT inside the sealed FACT50 housing.

###  II.3 Mechanical parts

Figure 9: Schematic drawing of the four modes in a measurement cycle. For the
two modes on the left, the PMT is placed _in_ the central point (CP) of the
mirror and the shutter in front of the PMT is _open_ (top) and _closed_
(bottom). The same shutter configurations are repeated with the PMT displaced
by 71.5 mm _out_ of the CP (right). Simplified from Veberič et al. (2018a);
Engel et al. (2017).

The PMT camera was placed in the CP and directed along the optical axis of the
mirror, see Figs. 1 and 3. Our setup included motorized components that
enabled us to obscure and move the PMT. We installed a plate wrapped in black
felt as a shutter in front of the PMT in a way that, when closed, it obscured
the view of the PMT towards the mirror (see the drawing in Fig. 9). This
shutter was operated by a Thorlabs MFF101 motorized flipper. In addition to
that, we placed the whole camera (PMT, FACT50 housing, and shutter) on an OWIS
Precision Linear Stage LTM60-150-HSM with a total travel distance of 145 mm.
The linear stage was installed horizontally and orthogonal to the optical axis
of the mirror, enabling us to drive the PMT onto positions inside (_in_) and
outside (_out_) of the CP, where the latter was at a distance of 71.5 mm from
the CP. Since the experimental area is locked down during data acquisition, we
had to control these mechanical components remotely. For this task we chose a
Raspberry Pi (mini)computer, which could be accessed and programmed remotely
through a network connection. As the PMT was exhibiting a non-negligible
memory effect (see Section III.5) we decided to replace the Thorlabs shutter,
which could obscure the PMT with only ${\lesssim}80\%$ efficiency, with a
perfectly light-tight 6-blade-teflon iris-diaphragm Uniblitz NS65B with a bi-
stable optical shutter with an aperture of 65 mm.

Since the PMT is observing slightly different scenery behind the camera
(through the reflection in the mirror) when it is moved in and out of the CP,
we installed a felt baffle around the camera opening so that this view stays
approximately the same in both positions, see Fig. 6.

The standard measurement procedure we used was cycling between the positions
_out_ of the CP (where no potential signal can be seen) and _in_ the CP (where
we expect to see the potential DM signal). Additionally, in both positions we
measured with an _opened_ and _closed_ shutter. This results in the following
four modes per cycle: (a) _out/open_ , (b) _out/closed_ , (c) _in/open_ , and
(d) _in/closed_ , as is schematically illustrated in Fig. 9. As will be
discussed later in Section III.5, this cycling between the _open_ and _closed_
shutter also reduces the exposure of the PMT to the remaining ambient light
and thus keeps the background rate of the PMT smaller.

###  II.4 Data acquisition

The output current of the PMT was captured by a PicoScope 6404 digitizer with
an 8-bit analog-to-digital converter (ADC). For the sampling period of the
ADC, we choose $\Delta t=0.8$ ns (i.e. sampling rate of 1.25 GS/s), and we
captured traces with $\pm 1000$ samples around the trigger, thus storing
traces with a total length of 1.6 $\upmu$s. The voltage range of the ADC was
set to $\pm 1$ V at $50\,\Omega$ input impedance. An internal trigger was set
to a threshold of 8 ADC counts (${\sim}63$ mV) below a baseline (since PMT
pulses are negative). The ADC itself was then connected through a USB
connection to a data-acquisition (DAQ) computer, which controlled the whole
data-acquisition procedure. The captured traces from the ADC were stored with
additional data including the state of the measurement cycle (PMT position,
shutter status), the PMT and ambient temperatures, air pressure etc. obtained
from the slow control system. The DAQ also controlled the movement of the
mechanical components of the experiment.

The stored traces were analyzed off-line by running a baseline estimation
algorithm first followed by a pulse-finding algorithm. Most of the traces
contained only one pulse that triggered the ADC; nevertheless, some traces
contained additional pulses. In the following, rates were obtained by counting
all pulses in an event.

#### Timing accuracy.

The data acquisition of our experiment was split into events, each with a
duration of 60 seconds. With typical trigger rates of several Hz, this
duration was short enough to have all the triggered traces stored in the
internal memory of the ADC so that the DAQ would not be interrupted by trace
transfers over the relatively slow USB connection. In this way, no irregular
dead time was introduced, and all the traces were read-out only at the end of
the event duration. Since the time delay between the call of the acquisition
start function and the actual begin of the data acquisition by the ADC was not
known, we implemented two timers within our DAQ software to obtain the
smallest possible uncertainty on the exact duration of the measurement
intervals (the _on_ time). The first timer $t_{1}$ starts right before the
call to the start function and ends right after the call to the stop function.
The second timer $t_{2}$ is started immediately after the call to the start
function and stops right before the call to the stop function. The true
measuring time $t_{\text{meas}}$ thus lies somewhere between the two timer
readings $t_{2}\leqslant t_{\text{meas}}\leqslant t_{1}$, as illustrated in
Fig. 10-(top).

The distributions of the two times as well as their difference are shown in
Fig. 10-(bottom). The duration of an event is estimated to be close to the
mean of both timers, $t_{\text{event}}\approx(t_{1}+t_{2})/2$, and its
uncertainty on the order of their difference, $\Delta t=t_{1}-t_{2}$.

For the particular run with $1.44{\times}10^{7}$ events, used for the analysis
below, the mean event duration was $\langle t_{\text{event}}\rangle=60.00138$
s, the mean time difference (and thus the uncertainty of the mean event
duration) was $\langle\Delta t\rangle=0.00237$ s, resulting in a relative time
uncertainty of $\langle\Delta t\rangle/\langle
t_{\text{event}}\rangle=4{\times}10^{-5}$.

Figure 10:  _Top:_ Schematic drawing of the implementation of the two timers
within the DAQ procedure. _Bottom:_ Distributions of the two timers $t_{1}$
and $t_{2}$ (left) and their difference $\Delta t$ (right). Since $t_{1}$
includes the calls to the start and stop functions, its distribution is more
spread out than that of $t_{2}$ which only measures the 60 s of the sleep
function. Note that the difference $\Delta t$ is shown in milliseconds.

#### Comparison of digitizers.

For the purposes of systematic cross-checks of our DAQ, a different digitizer
was employed in addition to the PicoScope described above. The second
digitizer (CAEN DT5751) had a higher ADC resolution of 10 bits and a fixed
input range of 1 V at $50\,\Omega$ impedance. In a special run, we installed
the digitizers in parallel so that they both triggered on the same output of
the PMT (via an impedance-matched signal splitter), and the trigger thresholds
for both devices were set to equal levels. Furthermore, in this run a rubber-
sealed metal lid was used to completely obscure the PMT. With this setup, we
determined that the mean difference in the relative rate is ${\sim}(3.05\pm
0.04)\%$.

###  II.5 Muon-monitoring system

Figure 11: _Top:_ Schematic diagram of the muon-monitoring system. The system
counts vertical coincidences of ionizing particles (red), when PMT1 and PMT2
are triggered, and lateral coincidence (blue), when PMT2 and PMT3 are
triggered. _Bottom:_ Measured mean rates for the vertical (red) and lateral
(blue) coincidences.

Since cosmic muons can easily penetrate thick layers of rock, they also reach
our experimental hall through its ${\sim}2$ m of concrete. They therefore
contribute to our background by emitting Cherenkov photons either in the glass
enclosure of the PMT or in the air surrounding the experiment. In order to
systematically monitor variations in this potential source of background, we
built a dedicated muon-monitoring system which was placed in a corner of the
experimental hall. The system consists of three scintillator bars
($62.5\,\mathrm{cm}\times 25\,\mathrm{cm}\times 2.2\,\mathrm{cm}$), each of
which is connected to a PMT via a light guide. The outputs of the PMTs are
connected to discriminators, sending triggers to coincidence units which are
in turn connected to counters, read out periodically by a slow-control
computer (Raspberry Pi). The scintillators and coincidence units were arranged
so that we could monitor the temporal and spatial coincidence of muons, as
schematically depicted in Fig. 11-(top). The captured coincidence count rates
of our muon-monitoring system for a time period of approximately one year are
shown in Fig. 11-(bottom). The typical vertical-coincidence rate for the muon
monitor was ${\sim}130\,\mathrm{Hz/m^{2}}$. Note that the muon monitor was not
deployed as any kind of a veto system, nevertheless, in the next Section this
coincidence data was used for correlation studies with measured rates to
search for potential sources of background.

##  III Analysis

We reported some preliminary results from previous measurement runs Engel et
al. (2017); Veberič et al. (2018b). For the analysis discussed here, we use
run number v35, which was taken from March 07 to April 04, 2019 (with a
duration of 27.5 days). To minimize eventual systematic drifts due to changing
environmental conditions, after 60 s of data taking we switch between each of
the modalities in the measurement cycle shown in Fig. 9. This corresponds to
an effective life-time of $4\times 145.3\,\mathrm{hours}$ for the entire
duration of run v35.

###  III.1 Sources of background

Fig. 12 shows the averaged trigger rates in this dataset for the PMT operating
in the four possible modes. It is interesting to note that the rate observed
with the _closed_ shutter is much higher than the dark-count rate (shown in
Fig. 8) and follows the variations of the rates with the _open_ shutter rather
closely. The origin of this phenomenon is discussed in Section III.5. In the
following, we make a distinction between internal and external sources of the
background, based on whether their origins are the internal parts of the PMT
or external factors.

As is clear from Fig. 9, a positive difference between the _open_ rates,
$\Delta r_{\emph{open}}=r_{\emph{in/open}}-r_{\emph{out/open}}$, is a proxy
for the HP signal. We found that this difference is uncorrelated with the
internal background as measured in the _closed_ modes, $\Delta
r_{\emph{closed}}=r_{\emph{in/closed}}-r_{\emph{out/closed}}$, implying that
the potential detection of a HP signal is mostly limited by external
nuisances. Moreover, as can be seen in Fig. 12, the typical rates are 3 to 4
times higher than the internal-background rate measured with the sealed PMT
(see Fig. 8).

Figure 12: Trigger rates recorded in run v35. For this plot, the rates are
averaged over 10 cycles (i.e. over $44$ min). The dashed line indicates the
level of the internal dark-count rate (with the sealed PMT, see Fig. 8).

In the following, we consider three main contributions to the dark-count rate,
which originate from thermal and cosmic-ray-induced backgrounds, and summarize
the conservative estimates of the size of these effects. Rather than
attempting to give an absolute size of each of the effects, we compare the
relative variation of the rate, $v=\sigma(r)/\langle r\rangle$, with the
relative variations of the background sources in question.

In the two _out_ modes, where no HP signal is expected, we observe relative
variations on the order of $v_{\emph{out/open}}\approx 20\%$ and
$v_{\emph{out/closed}}\approx 23\%$. In the following, we will consider three
possible sources of the background. Nevertheless, none of them (nor their
combination) can account for the variability observed in our data.

#### Thermal emission from the PMT photocathode.

The rate of thermal emission of our PMT is estimated by rescaling existing
measurements of the thermal response of bialkali photocathodes Coates (1972),
in order to account for the difference in the photocathode area and collection
efficiency. Assuming that the dark-count rate observed in the sealed PMT is
mostly of thermal origin, we obtain a linear dependence of the rate on the
temperature with an average gradient of 0.07 Hz/K, albeit only within the
range of temperatures encountered during the measurement ($10^{\circ}$ to
$20^{\circ}$C). We monitored the standard deviation of the temperature to 0.2
K, over the entire run, which ultimately yields $v_{\text{temp}}\lesssim 1\%$
for the dark-count rate due to thermal emission.

#### Thermal emission from the environment.

For simplicity we assume that the PMT in the _open_ mode, which has a field of
view of $2\pi$ sr towards the mirror, observes an isotropic blackbody-like
source with a temperature of 300 K. The rate of photons from such a thermal
source, that possess wavelengths within the range where the PMT has non-
negligible quantum efficiency, is on the order of $10^{-7}$ Hz. Clearly, this
is insufficient to explain the observed background. Furthermore, the quantum
efficiency of the PMT decreases rapidly in the infrared wavelengths, where the
thermal Planck spectrum at room temperatures actually starts to rise.

#### Cherenkov photons from cosmic-ray muons.

The threshold energy of muons for production of Cherenkov photons in air under
normal conditions ($T=15^{\circ}$C, $P=1013.25$ hPa, dry) is ${\sim}4$ GeV.
The energy loss of muons in the concrete shielding of the building is
estimated to be ${\sim}1$ GeV. The muon monitor will thus be triggered by
muons with energies above ${\sim}1$ GeV, while the relevant contribution to
the Cherenkov background comes from the muons with energies above ${\sim}5$
GeV (before the concrete shielding). The vertical muon intensities for the two
energies are
$\mathrm{d}N/\mathrm{d}A\,\mathrm{d}\Omega\,\mathrm{d}t|_{\theta=0^{\circ}}=73$
and 23 / m2 sr s, respectively, the former being compatible with the average
vertical-coincidence rate observed in our muon monitors. We estimate the rate
of associated Cherenkov photons with wavelengths between 150 and 640 nm as
around 90 photons / m of muon travel in air and 120 photons / mm in the fused-
silica window of the PMT. This corresponds roughly to a flux density of
${\sim}3000$ photons / s m3 being emitted within the experimental area (see
Badino (2000) for comparisons). Direct muon hits can be identified in the
event selection, as will be discussed later. Since the rate of Cherenkov
photons is directly proportional to the muon rate, we estimate the relative
variation of the Cherenkov background $v_{\text{ch}}$ from the relative
variation of the muon rate, $v_{\mu}\approx 2.6\%$ as estimated from the muon
monitoring. Nevertheless, this variation is an order of magnitude smaller than
that observed in the data. Therefore, the background cannot be predominantly
driven by the Cherenkov photons produced by cosmic muons.

###  III.2 Volume effect

Figure 13: Distribution of the trigger rates obtained during the measurements
of the dark counts with the sealed PMT (black), the dark counts in the reduced
volume box (blue), and counts during the regular run (dark red). For the last
two, the data were taken from the measurement with an _open_ shutter. The
inset shows their mean rates $\mu$ and variances $\sigma^{2}$. Note that for
independent counts (homogeneous Poissonian process) $\sigma^{2}$ should be
close to $\mu$.

The PMT is set up within a light-tight experimental area with an effective
volume of ${\sim}100$ m3. In the previous section, we argued that our
background fluctuations are mostly driven by external sources, although we
could not identify a strong correlation with a particular source of
background. We thus investigated the dependence of the dark-count rate on the
volume surrounding the PMT, motivated by the fact that the amount of detected
Cherenkov light in underground caves scales with their dimensions Badino
(2000).

We performed a series of dark-count measurements with a progressively smaller
light-tight box built around the PMT while operating the shutter in the same
way as for a regular run. Each of these runs lasted for 2 days. Initially, the
box had a volume ${\sim}500{\times}$ smaller than the experimental area. This
volume was then increasingly reduced to ${\sim}2500{\times}$ smaller than the
experimental area. As an immediate effect, we observed a significant reduction
of the trigger rates and its clear proportionality to the volume observed by
the PMT.

The distribution of the trigger rates recorded with the installed box is shown
in Fig. 13 and compared to the results from run v35 and the internal-
background rate measured with the sealed PMT.

This leads us to the conclusion that our background is, to some extent,
dependent on the effective volume of the region surrounding or observed by the
PMT. Nevertheless, since we already rejected Cherenkov or thermal photons as
our main background, we can at this point only speculate about other possible
sources Helbing et al. (2003); Aartsen et al. (2017).

###  III.3 Event selection

Figure 14: _Top:_ Distribution of the information entropy and duration of
recorded pulses. The black line indicates the relation between these two
parameters for pulses with a Gaussian shape in time. The red-dashed line
indicates the SPE-like selection applied to the pulses (ROI). _Middle:_ Same
as above but for the two settings of the LED-flasher strength. The information
entropy for the weak strength, which produces mostly SPE-like pulses, is shown
on the left, while the same for the higher setting producing pulses with
multiple photons is shown on the right, illustrating the particular choice for
ROI above. _Bottom:_ Distribution of the pulse charge $Q$ before and after
event selection.

The data acquisition is performed in a self-triggering mode, as described in
Section II.4, whereas pulse finding in recorded traces and subsequent event-
building is performed with an off-line algorithm. The HP signal is expected to
appear as an increased rate of SPE-like pulses (in the _in/open_ mode).
Therefore, a mild event selection is applied to achieve the best possible
single-photon counting accuracy.

Pulses are selected according to the following two properties: duration of the
pulse and its information (Shannon) entropy. These properties were chosen to
_(i)_ avoid direct selection based on the pulse amplitude or charge (since
they both have standard deviations comparable to their means, and thus, have
poor discrimination power) and _(ii)_ suppress long-duration pulses which are
most likely associated with photon bursts from non-HP sources.

The chosen selection criteria define a region of interest (ROI), the
parameters of which were determined and refined through a series of
calibrations with the single-photoelectron dataset obtained with an adjustable
blue-LED source (red line in Fig. 7). The ROI and charge distributions are
shown in Fig. 14. Note that the ROI contains ${\sim}90\%$ of all the pulses,
which we refer to as _events_ from now on.

###  III.4 Deviations from homogeneous Poissonian process

The recorded events exhibit two distinct features which deviate from standard
Poisson statistics: _(i)_ an overdispersion (i.e. the variance is many times
larger than the mean), and _(ii)_ the existence of correlations on short-time
scales.

The overdispersion is illustrated in Fig. 15 and can be successfully described
by allowing for nonhomogeneity in the counting process, e.g. with the Gamma
heterogeneity of the Poissonian rate Molenberghs et al. (2010). The resulting
marginal distribution is a negative binomial (NBD) whose mean always exceeds
the variance Boziev (1993) and fits the observed distribution of rates rather
well.

We investigated the distribution of the pulse arrival-times in an independent
run during which the PMT was kept at a fixed position and the shutter remained
_open_ (mode _out/open_). The data were captured in intervals of 60 s and the
complete measurement lasted for 10 days. The recorded sample contains
$4.95{\times}10^{6}$ pulses, which were tagged and processed off-line. The
pulses were then selected according to the SPE-like criteria discussed in
Section III.3.

In a homogeneous Poissonian process with a mean rate $\lambda$, the
distribution of the waiting time $t$ between two consecutive events is
exponential, $\mathrm{d}N/\mathrm{d}t=\lambda\exp(-\lambda t)$. A useful
transformation of the waiting times, $\Theta(t)=1-\exp(-\lambda t)$, maps
short and large times towards 0 and 1, respectively, to give us a uniform
distribution $\mathrm{d}N/\mathrm{d}\Theta=1$ for $\Theta$ in the half-open
interval $[0,1)$. We can use the deviation from this uniform distribution as a
suitable measure of heterogeneity of the arrival process.

Figure 15: _Top:_ Distribution of event counts in the internal-background run
with the sealed PMT. The distribution is quite close to the expected
homogeneous Poissonian (dashed gray) and the negative-binomial (NBD) fit is
slightly better (green). _Bottom:_ The same for the regular run. Here, the
homogeneous Poisson distribution is a much worse description than the NBD.

Figure 16: The dashed gray line in all the plots above represents a
corresponding homogeneous Poissonian reference, deviations from which
illustrate the heterogeneity of the process. _Top:_ Distribution of the
$\Theta$-transformed waiting times (see text for more details). _Middle:_
Mean-squared deviation (MSD) of the $\Theta$-transformed distribution (top) to
a uniform distribution (homogeneous Poissonian) as function of
$t_{\text{wait}}$. _Bottom:_ Remaining fraction of data as a function of
$t_{\text{wait}}$ cut. See Fig. 14 for the definition of “events”.

In Fig. 16-(top), we plot the distribution of the $\Theta$-transformed
intervals between successive events, referred to as waiting times. The
$\Theta$ transformation clearly reveals the deviation from the homogeneous
Poisson process, especially the substantial clustering of the time-correlated
events with small waiting times, either by PMT after-pulses or photon bursts.

To investigate the time scales on which the clustering occurs, we apply a
waiting-time selection where consecutive events closer than
$t<t_{\text{wait}}$ are counted only as one. Selected waiting times from a
homogeneous Poisson process can also be transformed to a uniform distribution
with the transformation
$\Theta^{\prime}(t)=1-\exp[-\lambda(t-t_{\text{wait}})]$.

In Fig. 16-(middle), we plot the mean-squared deviation (MSD) of the uniform
distribution and the distribution of $\Theta^{\prime}$-transformed waiting
times as a function of $t_{\text{wait}}$. The MSD gives a value of zero for a
uniform distribution and is normalized such that the maximum possible
deviation gives a value of 1. We observe two significant departure points from
homogeneous Poisson statistics. The first break occurs on a millisecond scale
($10^{-3}$ s) and is most likely related to cosmic-ray showers. The second
break appears on a microsecond scale ($10^{-6}$ s) and is probably related to
PMT after-pulsing since this can occur within a time window of 100 ns to a few
$\upmu$s. Nevertheless, for our PMT, after-pulses normally amount to only 1%
of the count rate ET Enterprises (a). In Fig. 16-(bottom), we show the
efficiency of the waiting-time selection as a function of $t_{\text{wait}}$.
Roughly 23% of events are clustered on scales below 1 ms; 14%, below
$1\,\upmu$s.

We do not expect that HPs can induce such time-correlated events. Nonetheless,
in order to derive a conservative limit in Section IV, we do not perform any
waiting-time selection or any other rejection based on the arrival time of the
pulses.

###  III.5 Memory effect

When comparing the sealed-PMT rates (Fig. 8) with the _closed_ rates during
the measurement run (Fig. 12), we immediately see that the latter are two to
five times higher. Furthermore, the _closed_ rates in Fig. 12 also tightly
follow the _open_ ones, indicating a certain level of correlation between the
_open_ and _closed_ measurements. Additionally, we also noticed that after
each visit to the experimental area, which unavoidably brings a certain level
of illumination, the PMT needed several hours to reduce and stabilize its
rate. To quantify the dependence of the rate on the historical levels of
illumination, we devised a special run where the shutter was initially kept
closed for 200 min, after which it was open for a duration of 30 min. This
sequence was then repeated several times. Identically to the regular runs, we
recorded the number of triggers in the same intervals of 60 s.

This measurement is plotted in Fig. 17. After the initial fast rise, the
photon rate continues to increase slowly as the PMT photocathode continues to
be exposed to the external sources of background light. After the shutter is
closed, the rate slowly decays, on the timescale of an hour, back to the
typical value of the internal background as seen with the sealed PMT. From
this experiment, we conclude that in this particular low-noise PMT a _memory_
of the past history of the illumination exists. It is striking that this
effect extends over such long periods and even when the photocathode has been
exposed only to a very faint external source of light (few Hz).

Based on these findings, we model the output response $r(t)$ of the PMT to an
external input $s(t)$ as having an instantaneous and historical part,

$r(t)=s(t)+\alpha\int_{-\infty}^{t}\mathrm{d}t^{\prime}\,K(t^{\prime},t)\,s(t^{\prime}),$
(3)

where an exponential decay,

$K(t^{\prime},t)=\frac{1}{\tau}\exp\left(-\frac{t-t^{\prime}}{\tau}\right),$
(4)

is chosen as the memory kernel. The memory efficiency $\alpha$ and the decay
time $\tau$ are characteristics of the PMT and are obtained from fits of the
data in Fig. 17 as

$\displaystyle\begin{split}\alpha&=0.922\pm 0.015,\\\
\tau/\mathrm{min}&=54.124\pm 1.115.\end{split}$ (5)

Figure 17: An example of the trigger rate (black points) measured with a
sequence of 200 min in _closed_ mode and 30 min in _open_ mode. The grey-
shaded region indicates the times at which light came in. The illumination
model (red) is fitted to the data along with the detector response (blue), as
described by Eq. 3. Figure 18: An example of the event-rate reconstruction
(red), showing a one-week sample plotting only the _out/closed_ data. The
measured event rate is shown in black and the contribution from the past
illumination is shown in green.

In this model, the input illumination of the PMT thus influences output only
up to several decay times $\tau$ into the future. Ignoring for the moment the
internal background, a constant input rate $s$ will therefore converge to an
output rate $r=(1+\alpha)s$ in a time span of several $\tau$. In our standard
measurement cycle, where each mode lasts for $\Delta t=60\,$s, this constant
input $s$ is periodically blocked and unblocked by the shutter. With such an
input, the output will converge to the following _open_ and _closed_ expected
rates at the middle of the cycle intervals,

$\displaystyle\begin{split}r_{\text{open}}&=s\left[1+\alpha\left(1-\frac{f}{1+f^{2}}\right)\right],\\\
r_{\text{closed}}&=\alpha\,s\left(\frac{f}{1+f^{2}}\right),\end{split}$ (6)

where we use the shorthand $f=\exp(-\Delta t/2\tau)$. For $\Delta t\ll\tau$
this approximately simplifies to $r_{\text{open}}\approx s(1+\alpha/2)$ and
$r_{\text{closed}}\approx\alpha s/2$ such that the difference is roughly
proportional to the external input, $r_{\text{open}}-r_{\text{closed}}\approx
s$ (c.f. Fig. 12).

With these values known, we can fully correct for the historical memory effect
to recover the instantaneous signal in our actual run. Let us suppose that we
start the data acquisition at $t_{\circ}=0$ and that the modes of the
measurement cycle are changed at discrete intervals $t_{i}=i\Delta t$ for
$i\geqslant 0$. Since $\tau$ is large compared to the sampling interval
$\Delta t$, the average event-count over an $i$th interval $(t_{i-1},t_{i}]$
can be well approximated as in the middle of the interval, i.e.
$\bar{t}_{i}=(t_{i-1}+t_{i})/2$. Eq. 3 can then be written for each time bin
$i$ in terms of discrete sums as

$\begin{split}r_{i}=s_{i}&+\alpha\,e^{-\nicefrac{{\bar{t}_{i}}}{{\tau}}}\times\\\
&\times\left[s_{\circ}+\textstyle{\sum}_{j=1}^{i-1}s_{j}(e^{\nicefrac{{t_{j}}}{{\tau}}}-e^{\nicefrac{{t_{j-1}}}{{\tau}}})+s_{i}(e^{\nicefrac{{\bar{t}_{i}}}{{\tau}}}-e^{\nicefrac{{t_{i}}}{{\tau}}})\right],\end{split}$
(7)

where $r_{i}=r(\bar{t}_{i})$, $s_{i}=s(\bar{t}_{i})$, and $s_{\circ}$
corresponds to the average input signal before the start of the measurement,
$t<0$. Since this state is unknown, we use Eq. 7 in fitting the PMT response
data for the first few hours to obtain the best $s_{\circ}$. Then the true
signal can be iteratively reconstructed as

$s_{i}=\frac{r_{i}-\alpha\,(h_{i}+s_{\circ}e^{-\nicefrac{{\bar{t}_{i}}}{{\tau}}})}{1+\alpha\,(1-f)},$
(8)

where the auxiliary variable $h_{i}$ is obtained as

$\begin{split}h_{\circ}&=0,\\\
h_{i}&=h_{i-1}\,f^{2}+s_{i-1}\,f(1-f^{2}),\qquad i\geqslant 1.\end{split}$ (9)

As is clear from Eq. 8, any bias related to the exact value of $s_{\circ}$ can
be avoided by simply discarding the first few hours of the reconstructed data.
Thereafter, given the parameters $\alpha$ and $\tau$, the reconstruction is
close to exact. An illustration of this procedure is shown in Fig. 18.

###  III.6 Low-frequency background

Figure 19: _Top:_ Spectral Fourier decomposition of the reconstructed signal
rate. Frequency is plotted from the first harmonic up to the Nyquist
frequency, and the corresponding spectral powers are normalized with respect
to the stationary (constant) component. The shaded region corresponds to the
low-frequency (LF) truncation used in the bottom plot. _Bottom:_ Detrending of
the reconstructed rate as described in Section III.6. Data is taken from the
_out/open_ mode of the measurement. For plotting purposes, the LF harmonics
have been shifted up by the mean event rate $\mu$.

A visualization of the event rate in the frequency domain (see Fig. 19)
reveals a relatively significant modulation of the signal at low frequencies
(i.e. from zero to several cycles per day). Since these harmonics are observed
also in the measurement with the closed shutter, they are treated as
background. Furthermore, for our aimed level of sensitivity (${\sim}10^{-3}$
Hz or ${\sim}1$ detectable HP-to-photon conversion every 15 min), we do not
expect any substantial time modulation of the HP signal.

Modulations on larger time scales are most likely related to modulations in
the cosmic-ray flux, which are in turn modulated by variations in atmospheric
pressure. In our analysis, we detrend the reconstructed dataset by removing
these low-frequency (LF) oscillations up to a certain frequency threshold. The
threshold is chosen such that it does not affect the event distribution of the
higher-frequency (HF) residual. This procedure is illustrated in Fig. 19,
where components of the Fourier spectrum with frequencies less than 4
cycles/day have been removed.

###  III.7 Systematic uncertainties

#### Timing uncertainty.

A systematic shift of the timing estimate $t_{\text{meas}}$ with the
uncertainty quoted in Section II.4 would result in a systematic bias of 0.004%
of the average count rate for a given measurement mode. Given a typical rate
of 4 Hz for the _open_ mode, this implies a systematic uncertainty of
$3{\times}10^{-4}$ Hz for the _in_ – _out_ difference.

Figure 20: Distribution of the internal-background event-rates in a sealed PMT
as a function of the ambient temperature. A linear fit yields a slope of
$(0.030\pm 0.014)\,\mathrm{Hz/^{\circ}C}$, which is compatible with our first
estimation.

#### Temperature drift.

The Funk experiment operates in environmental conditions with excellent
temperature stability, as testified by the continuous monitoring before,
during, and after the measurement runs. Systematic errors can arise from
fluctuations occurring between changes of the measurement configuration. For
our run v35, the typical variation of the temperature difference measured
between the _in_ and _out_ positions is evaluated to be 0.02 K. Based only on
general properties of the bialkali photocathodes, we estimated the temperature
coefficient of the PMT rate to be around 0.07 Hz/K in Section III.1. Here, we
measure this coefficient from the internal-background data of the sealed PMT,
as shown in Fig. 20. Over the whole run, this temperature dependence induces a
systematic uncertainty of $6{\times}10^{-4}\,\mathrm{Hz}$ in the count rate or
3% in the _in_ $-$_out_ difference.

#### Pressure dependence of the Cherenkov rate.

While considering the flux density of Cherenkov photons produced in air in
Section III.1, we estimated that there are roughly 300 kHz of photons produced
within the experimental area. By comparison, the rate difference _open_ –
_closed_ amounts to ${\sim}1\,$Hz, which corresponds to a fraction of
$4{\times}10^{-6}$ of the estimated Cherenkov rate. The latter varies with the
air density and, therefore, with the ambient temperature and pressure. Within
the pressure range of interest, we estimate that this dependence behaves
linearly with a gradient of 0.5 kHz/hPa at $15^{\circ}$C. From the pressure
monitoring, this coefficient yields a typical variation of 30 Hz or 0.01% in
the Cherenkov rate, which translates into a systematic uncertainty of
$1.2{\times}10^{-4}$ Hz for the measured rate difference between the two open
modes in the cycle.

#### Mirror reflection.

When the PMT is positioned at the center of the spherical mirror with the
shutter open, some photons (e.g. from the muon hits) may be refracted towards
the mirror and by design get perfectly reflected back to the PMT. The
magnitude of this effect has been evaluated by measuring time differences
$\Delta t_{\text{ev}}$ between consecutive events and comparing them to the
time a photon requires to cover the distance to the mirror and back
(${\sim}22$ ns). In Fig. 21, we show an observation of two significant peaks
that satisfy the timing requirement for single and double reflection (with a
faint hint of a third). The overall magnitude of this effect is estimated with
a Monte-Carlo simulation, which gives a relative excess of $0.0188\pm 0.0001$
Hz for the event rate in the _in/open_ mode. This excess, however, also
includes contribution from the memory effect addressed in Section III.5. On
account of the event-reconstruction efficiency (57% for the _open_ data), we
find a systematic bias of 0.01 Hz for the _in/open_ mode. We note that this is
of the same order of magnitude as our statistical uncertainty for the
difference _in_ $-$_out_.

Figure 21: Distribution of the time difference $\Delta t_{\text{ev}}$ between two consecutive events recorded within the same trace. The red shaded region corresponds to the event excess due to the reflection off the mirror (see text for more details). Table 1: Summary of the systematic effects introduced by known sources and resulting in different measurement conditions at the positions _in_ and _out_. source | $\sigma_{\text{sys}}/\text{Hz}$
---|---
DAQ measurement time | $3{\times}10^{-4}$
PMT temperature drift | $6{\times}10^{-4}$
Pressure dependence of Cherenkov production | $1{\times}10^{-4}$
Reflections from the mirror | $1{\times}10^{-2}$

##  IV Results

Assuming that the HP particles comprise the whole of the cold DM in the
galactic halo, we can obtain our sensitivity to the kinetic-mixing parameter
directly from Eq. 2 as

$\displaystyle\chi$
$\displaystyle=4.1{\times}10^{-12}\left(\frac{r_{\text{det}}/q_{\text{eff}}}{\mathrm{Hz}}\,\frac{m_{\tilde{\gamma}}}{\mathrm{eV}}\right)^{1/2}\left(\frac{\eta\,A_{\text{mirr}}}{\mathrm{m^{2}}}\right)^{-1/2}$
$\displaystyle\qquad\times\left(\frac{\langle\cos^{2}\theta\rangle}{2/3}\right)^{-1/2}\left(\frac{0.3\,\mathrm{GeV/cm^{3}}}{\rho_{\text{CDM}}}\right)^{1/2},$
(10)

where $r_{\text{det}}$ is the DM-induced photon rate and $q_{\text{eff}}$ the
quantum efficiency of the PMT. The efficiency parameter
$\eta=4(\sqrt{R}-R)^{2}/(1-R)^{2}$ Andrianavalomahefa (2020) accounts for the
reflectivity $R$, where a value of $R=0.7$ is used as the reflectivity does
not exceed this value for all wavelengths in the range of interest. $\theta$
is the angle between the HP field polarization and the mirror plane. The last
two terms are arranged relative to the isotropic polarization distribution of
the DM, where $\langle\cos^{2}\theta\rangle=2/3$, and $\rho_{\text{CDM}}$ is
the local cold DM energy density.

The minimum detectable rate $r_{\text{det}}$ is governed by the dark-count
rate and the total measurement time. In our case, increasing the measurement
time to one year would only improve the limit by a factor of 2.

Before proceeding to the derivation of a limit, we briefly summarize the three
steps of the event selection procedure:

  * _(i)_

individual pulses are tagged and events are selected by the SPE-like filters
based on their duration and informational entropy (see Section III.3),

  * _(ii)_

the total event count for each 60 s measurement mode is corrected for the
memory effect, and the first 50 modes are discarded to avoid any potential
bias (see Section III.5),

  * _(iii)_

the reconstructed rate is detrended in the frequency domain by removing
Fourier components for frequencies of less than 4 cycles/day (see Section
III.6).

Figure 22: Distribution of the reconstructed rates _in_ and _out_ of the
radius point with the shutter _open_. The light and thicker lines correspond
to the dataset before and after the detrending, respectively.

The rates after this selection are presented in Figs. 22 and 2. The potential
HP “signal” is resolved from the difference of single-photon counts _in_ and
_out_ of the CP with the shutter _open_. For this difference, we find

$\Delta r_{\text{open}}/\mathrm{Hz}=-0.0229\pm 0.0108.$ (11)

Furthermore, it is worth to mention that this negative signal is compatible
with the difference we observe in the background-only measurement with the
shutter _closed_ ,

$\Delta r_{\text{closed}}/\mathrm{Hz}=-0.0267\pm 0.0106.$ (12)

We interpret $\Delta r_{\text{closed}}$ as a correctable systematic effect due
to unknown sources and introduced by driving the PMT _in_ or _out_ of the CP.
Contributions from other known systematic sources are summarized in Table 1.
The dominant effect comes from the mirror reflections $\Delta
r_{\text{refl}}$, which we subtract from $\Delta r_{\text{open}}$. The
uncertainty on these reflections is evaluated to be less than 1%, such that
$\Delta r_{\text{refl}}=(1\pm 0.01){\times}10^{-2}$ Hz. This gives an
additional systematic uncertainty of $10^{-4}$ Hz, which is added in
quadrature with the uncertainties associated with the DAQ timing and
temperature and pressure variations. Our signal is therefore given by

$\displaystyle r_{\text{signal}}/\mathrm{Hz}$ $\displaystyle=\Delta
r_{\text{open}}-\Delta r_{\text{closed}}-\Delta r_{\text{refl}}$
$\displaystyle=-0.0062\pm 0.0151\text{(stat)}\pm 0.0007\text{(sys)},$ (13)

from which we conclude that there is no evidence for the presence of HP DM.
Let us remark that the rather large statistical uncertainty of
$r_{\text{signal}}$ is a direct consequence of the overdispersion of counts
(as discussed in Section III.4), and that assuming homogeneous Poissonian
statistics (as e.g. done by the Tokyo experiment Suzuki et al. (2015)) would
result in a factor 4 smaller uncertainty and thus a factor 2 lower limit for
$\chi$.

Table 2: Rates measured in the four modes of the measurement. $r$/Hz | _in_ | _out_
---|---|---
_open_ | $3.5937\pm 0.0077$ | $3.6166\pm 0.0078$
_closed_ | $2.2034\pm 0.0074$ | $2.2301\pm 0.0077$

To account for the physical bound of the rate being positive-definite, we use
the Feldman-Cousin prescription to derive a positive upper-limit on the
minimum detectable HP signal. We obtain $r_{\text{det}}\lesssim 0.0238$ Hz at
a 95% C.L. Inserting this value into Eq. 11, we obtain an upper bound on the
magnitude of the kinetic-mixing parameter such that $\chi\lesssim
6.65{\times}10^{-13}$ at a 95% C.L. for the range of HP masses
$1.95<m_{\tilde{\gamma}}/\text{eV}<8.55$. The mass-resolved limit is presented
in Figs. 23 and 24, and is given in tabular form in Appendix A.

Figure 23:  Funk exclusion limit at a 95% C.L. (red line). The limit labeled
Tokyo Suzuki et al. (2015) was obtained by another dish antenna experiment
using a smaller mirror. The dashed orange line is a constraint from DAMIC
Aguilar-Arevalo et al. (2019), searching for HP ionization signal in WIMP
detector targets. The dot-dashed green line corresponds to astrophysical
bounds from the solar lifetime Redondo and Raffelt (2013), and the dotted
black line is the limit from Xenon10 Bloch et al. (2017) for model of solar
HPs.

##  V Conclusion

Figure 24: Funk exclusion limit at a 95% C.L. (red-shaded region). The Tokyo
limit was obtained by a similar setup but with a smaller mirror Suzuki et al.
(2015). HP bounds from other direct-detection experiments are also shown for
comparison: DAMIC Aguilar-Arevalo et al. (2019), Xenon10 (both as DM and as
dark radiation emitted by the sun) Bloch et al. (2017), Tokyo2 Knirck et al.
(2018), Tokyo3 Tomita et al. (2020), Shuket Brun et al. (2019), and DM
haloscopes Arias et al. (2012). The regions labeled Solar lifetime, HB
(horizontal-branch stars) and RB (red giants) are indirect constraints derived
from astrophysical considerations Redondo and Raffelt (2013). Finally, the
grey-shaded region corresponds to the allowed parameter-space for HPs to act
as CDM Nelson and Scholtz (2011); Arias et al. (2012).

Understanding the nature and origin of DM in the Universe is possibly one of
the most urgent tasks in particle physics today. Measurements with colliders
and direct searches for WIMPs have imposed strong constraints on the existence
of heavy DM in recent years. The exploration of the frontier of light and
ultralight DM is presently gaining momentum and becoming a crucial direction
of research for the near future Ellis et al. (2019).

With Funk, we have set up an experiment that can search for ultralight hidden
photons as candidates for dark matter, and we have reported an improved limit
in the visible and ultraviolet range of the hidden-photon masses using the
dish-antenna method. In the mass range of 2.5 to 7 eV, we are able to
constrain and exclude HP coupling stronger than $\chi\approx 10^{-12}$ in
kinetic mixing. This limit is competitive with those derived from
astrophysical considerations (e.g. solar lifetime and Xenon10) and, in a
certain HP mass range, exceeds those from direct-detection experiments (e.g.
Tokyo and DAMIC).

One quarter of the background in our measurement is comprised of thermal
emission from the PMT cathode, while external sources in the experimental hall
and within the field of view of the PMT account for the rest. The statistical
uncertainties are not derived or assumed; instead, we obtain them from direct
measurements of the fluctuations in our event rates. The largest sources of
systematic uncertainty arise from reflections on the mirror, a memory effect
in the PMT, and a systematic difference in the rates between two modes of
measurement. The magnitudes of each are well determined. In hindsight, it was
extremely useful to include the _closed_ shutter modes in our measurement
cycle. These enabled the observation and understanding of the memory effect in
our PMT and the development of the correction for this effect, which was
applied in our analysis.

Some guidance is available from theory as to what the hidden photon mass might
be Graham et al. (2016), but the mass-range of viable hidden-photon dark
matter is huge. For this reason, techniques which can scan over a large
parameter space are especially useful even if they cannot explore couplings as
small as is possible with some other methods (e.g. resonant haloscopes). As
Funk is using a broadband search technique, a future exploitation of the setup
in other wavelength ranges is possible, and the Funk setup therefore continues
to carry an enormous physics potential.

## Acknowledgments

We thank the Helmholtz Alliance for Astroparticle Physics (HAP) for partial
funding of the experiment. A. A. and T. S. acknowledge support from the
European Union’s Horizon 2020 research and innovation program under the Marie
Sklodowska-Curie grant agreement No. 674896 (Elusives). A. A. acknowledges the
support from the “Karlsruhe School of Elementary and Astroparticle Physics:
Science and Technology” (KSETA). B. D. acknowledges support through the
European Research Council (ERC) under grant ERC-2018-StG-802836 (AxScale). We
thank the anonymous reviewer for the insightful comments and suggestions.

##  Appendix A Data

Table 3: Funk exclusion-limit data in tabular form. $m_{\tilde{\gamma}}$/eV | $\chi/10^{-13}$
---|---
1.95 | 245
2.17 | 21.2
2.41 | 10.7
2.68 |  008.71
2.97 |  007.39
3.31 |  006.72
3.67 |  006.91
4.08 |  007.47
4.54 |  008.21
5.04 |  009.24
5.60 |  009.57
6.23 |  009.15
6.92 | 10.7
7.69 | 16.1
8.55 | 74.1

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