warn luatex=false

The MicroBooNE Collaboration ††thanks<EMAIL_ADDRESS>
# First search for dark-trident processes using the MicroBooNE detector

P. Abratenko  Tufts University, Medford, MA, 02155, USA  O. Alterkait  Tufts
University, Medford, MA, 02155, USA  D. Andrade Aldana  Illinois Institute of
Technology (IIT), Chicago, IL 60616, USA  L. Arellano  The University of
Manchester, Manchester M13 9PL, United Kingdom  J. Asaadi  University of
Texas, Arlington, TX, 76019, USA  A. Ashkenazi  Tel Aviv University, Tel Aviv,
Israel, 69978  S. Balasubramanian  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  B. Baller  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  G. Barr  University of Oxford,
Oxford OX1 3RH, United Kingdom  D. Barrow  University of Oxford, Oxford OX1
3RH, United Kingdom  J. Barrow  Massachusetts Institute of Technology (MIT),
Cambridge, MA, 02139, USA  University of Minnesota, Minneapolis, MN, 55455,
USA  Tel Aviv University, Tel Aviv, Israel, 69978  V. Basque  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  O. Benevides Rodrigues
Illinois Institute of Technology (IIT), Chicago, IL 60616, USA  S. Berkman
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  Michigan
State University, East Lansing, MI 48824, USA  A. Bhanderi  The University of
Manchester, Manchester M13 9PL, United Kingdom  A. Bhat  University of
Chicago, Chicago, IL, 60637, USA  M. Bhattacharya  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  M. Bishai  Brookhaven National
Laboratory (BNL), Upton, NY, 11973, USA  A. Blake  Lancaster University,
Lancaster LA1 4YW, United Kingdom  B. Bogart  University of Michigan, Ann
Arbor, MI, 48109, USA  T. Bolton  Kansas State University (KSU), Manhattan,
KS, 66506, USA  J. Y. Book  Harvard University, Cambridge, MA 02138, USA  M.
B. Brunetti  University of Warwick, Coventry CV4 7AL, United Kingdom  L.
Camilleri  Columbia University, New York, NY, 10027, USA  Y. Cao  The
University of Manchester, Manchester M13 9PL, United Kingdom  D. Caratelli
University of California, Santa Barbara, CA, 93106, USA  F. Cavanna  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  G. Cerati
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  A.
Chappell  University of Warwick, Coventry CV4 7AL, United Kingdom  Y. Chen
SLAC National Accelerator Laboratory, Menlo Park, CA, 94025, USA  J. M. Conrad
Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139, USA  M.
Convery  SLAC National Accelerator Laboratory, Menlo Park, CA, 94025, USA  L.
Cooper-Troendle  University of Pittsburgh, Pittsburgh, PA, 15260, USA  J. I.
Crespo-Anadón  Centro de Investigaciones Energéticas, Medioambientales y
Tecnológicas (CIEMAT), Madrid E-28040, Spain  R. Cross  University of Warwick,
Coventry CV4 7AL, United Kingdom  M. Del Tutto  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  S. R. Dennis  University of
Cambridge, Cambridge CB3 0HE, United Kingdom  P. Detje  University of
Cambridge, Cambridge CB3 0HE, United Kingdom  A. Devitt  Lancaster University,
Lancaster LA1 4YW, United Kingdom  R. Diurba  Universität Bern, Bern CH-3012,
Switzerland  Z. Djurcic  Argonne National Laboratory (ANL), Lemont, IL, 60439,
USA  R. Dorrill  Illinois Institute of Technology (IIT), Chicago, IL 60616,
USA  K. Duffy  University of Oxford, Oxford OX1 3RH, United Kingdom  S. Dytman
University of Pittsburgh, Pittsburgh, PA, 15260, USA  B. Eberly  University of
Southern Maine, Portland, ME, 04104, USA  P. Englezos  Rutgers University,
Piscataway, NJ, 08854, USA  A. Ereditato  University of Chicago, Chicago, IL,
60637, USA  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510,
USA  J. J. Evans  The University of Manchester, Manchester M13 9PL, United
Kingdom  R. Fine  Los Alamos National Laboratory (LANL), Los Alamos, NM,
87545, USA  O. G. Finnerud  The University of Manchester, Manchester M13 9PL,
United Kingdom  W. Foreman  Illinois Institute of Technology (IIT), Chicago,
IL 60616, USA  B. T. Fleming  University of Chicago, Chicago, IL, 60637, USA
D. Franco  University of Chicago, Chicago, IL, 60637, USA  A. P. Furmanski
University of Minnesota, Minneapolis, MN, 55455, USA  F. Gao  University of
California, Santa Barbara, CA, 93106, USA  D. Garcia-Gamez  Universidad de
Granada, Granada E-18071, Spain  S. Gardiner  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  G. Ge  Columbia University, New
York, NY, 10027, USA  S. Gollapinni  Los Alamos National Laboratory (LANL),
Los Alamos, NM, 87545, USA  E. Gramellini  The University of Manchester,
Manchester M13 9PL, United Kingdom  P. Green  University of Oxford, Oxford OX1
3RH, United Kingdom  H. Greenlee  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  L. Gu  Lancaster University, Lancaster LA1
4YW, United Kingdom  W. Gu  Brookhaven National Laboratory (BNL), Upton, NY,
11973, USA  R. Guenette  The University of Manchester, Manchester M13 9PL,
United Kingdom  P. Guzowski  The University of Manchester, Manchester M13 9PL,
United Kingdom  L. Hagaman  University of Chicago, Chicago, IL, 60637, USA  O.
Hen  Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139, USA
C. Hilgenberg  University of Minnesota, Minneapolis, MN, 55455, USA  G. A.
Horton-Smith  Kansas State University (KSU), Manhattan, KS, 66506, USA  Z.
Imani  Tufts University, Medford, MA, 02155, USA  B. Irwin  University of
Minnesota, Minneapolis, MN, 55455, USA  M. S. Ismail  University of
Pittsburgh, Pittsburgh, PA, 15260, USA  C. James  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  X. Ji  Nankai University, Nankai
District, Tianjin 300071, China  J. H. Jo  Brookhaven National Laboratory
(BNL), Upton, NY, 11973, USA  R. A. Johnson  University of Cincinnati,
Cincinnati, OH, 45221, USA  Y.-J. Jwa  Columbia University, New York, NY,
10027, USA  D. Kalra  Columbia University, New York, NY, 10027, USA  N. Kamp
Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139, USA  G.
Karagiorgi  Columbia University, New York, NY, 10027, USA  W. Ketchum  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  M. Kirby
Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  T. Kobilarcik  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  I. Kreslo
Universität Bern, Bern CH-3012, Switzerland  M. B. Leibovitch  University of
California, Santa Barbara, CA, 93106, USA  I. Lepetic  Rutgers University,
Piscataway, NJ, 08854, USA  J.-Y. Li  University of Edinburgh, Edinburgh EH9
3FD, United Kingdom  K. Li  Wright Laboratory, Department of Physics, Yale
University, New Haven, CT, 06520, USA  Y. Li  Brookhaven National Laboratory
(BNL), Upton, NY, 11973, USA  K. Lin  Rutgers University, Piscataway, NJ,
08854, USA  B. R. Littlejohn  Illinois Institute of Technology (IIT), Chicago,
IL 60616, USA  H. Liu  Brookhaven National Laboratory (BNL), Upton, NY, 11973,
USA  W. C. Louis  Los Alamos National Laboratory (LANL), Los Alamos, NM,
87545, USA  X. Luo  University of California, Santa Barbara, CA, 93106, USA
C. Mariani  Center for Neutrino Physics, Virginia Tech, Blacksburg, VA, 24061,
USA  D. Marsden  The University of Manchester, Manchester M13 9PL, United
Kingdom  J. Marshall  University of Warwick, Coventry CV4 7AL, United Kingdom
N. Martinez  Kansas State University (KSU), Manhattan, KS, 66506, USA  D. A.
Martinez Caicedo  South Dakota School of Mines and Technology (SDSMT), Rapid
City, SD, 57701, USA  S. Martynenko  Brookhaven National Laboratory (BNL),
Upton, NY, 11973, USA  A. Mastbaum  Rutgers University, Piscataway, NJ, 08854,
USA  I. Mawby  University of Warwick, Coventry CV4 7AL, United Kingdom  N.
McConkey  University College London, London WC1E 6BT, United Kingdom  V.
Meddage  Kansas State University (KSU), Manhattan, KS, 66506, USA  J. Micallef
Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139, USA  Tufts
University, Medford, MA, 02155, USA  K. Miller  University of Chicago,
Chicago, IL, 60637, USA  A. Mogan  Colorado State University, Fort Collins,
CO, 80523, USA  T. Mohayai  Fermi National Accelerator Laboratory (FNAL),
Batavia, IL 60510, USA  Indiana University, Bloomington, IN 47405, USA  M.
Mooney  Colorado State University, Fort Collins, CO, 80523, USA  A. F. Moor
University of Cambridge, Cambridge CB3 0HE, United Kingdom  C. D. Moore  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  L. Mora Lepin
The University of Manchester, Manchester M13 9PL, United Kingdom  M. M.
Moudgalya  The University of Manchester, Manchester M13 9PL, United Kingdom
S. Mulleriababu  Universität Bern, Bern CH-3012, Switzerland  D. Naples
University of Pittsburgh, Pittsburgh, PA, 15260, USA  A. Navrer-Agasson  The
University of Manchester, Manchester M13 9PL, United Kingdom  N. Nayak
Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  M. Nebot-Guinot
University of Edinburgh, Edinburgh EH9 3FD, United Kingdom  J. Nowak
Lancaster University, Lancaster LA1 4YW, United Kingdom  N. Oza  Columbia
University, New York, NY, 10027, USA  O. Palamara  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  N. Pallat  University of Minnesota,
Minneapolis, MN, 55455, USA  V. Paolone  University of Pittsburgh, Pittsburgh,
PA, 15260, USA  A. Papadopoulou  Argonne National Laboratory (ANL), Lemont,
IL, 60439, USA  V. Papavassiliou  New Mexico State University (NMSU), Las
Cruces, NM, 88003, USA  H. B. Parkinson  University of Edinburgh, Edinburgh
EH9 3FD, United Kingdom  S. F. Pate  New Mexico State University (NMSU), Las
Cruces, NM, 88003, USA  N. Patel  Lancaster University, Lancaster LA1 4YW,
United Kingdom  Z. Pavlovic  Fermi National Accelerator Laboratory (FNAL),
Batavia, IL 60510, USA  E. Piasetzky  Tel Aviv University, Tel Aviv, Israel,
69978  I. Pophale  Lancaster University, Lancaster LA1 4YW, United Kingdom  X.
Qian  Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  J. L. Raaf
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  V.
Radeka  Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  A.
Rafique  Argonne National Laboratory (ANL), Lemont, IL, 60439, USA  M.
Reggiani-Guzzo  University of Edinburgh, Edinburgh EH9 3FD, United Kingdom
The University of Manchester, Manchester M13 9PL, United Kingdom  L. Ren  New
Mexico State University (NMSU), Las Cruces, NM, 88003, USA  L. Rochester  SLAC
National Accelerator Laboratory, Menlo Park, CA, 94025, USA  J. Rodriguez
Rondon  South Dakota School of Mines and Technology (SDSMT), Rapid City, SD,
57701, USA  M. Rosenberg  Tufts University, Medford, MA, 02155, USA  M. Ross-
Lonergan  Los Alamos National Laboratory (LANL), Los Alamos, NM, 87545, USA
C. Rudolf von Rohr  Universität Bern, Bern CH-3012, Switzerland  I. Safa
Columbia University, New York, NY, 10027, USA  G. Scanavini  Wright
Laboratory, Department of Physics, Yale University, New Haven, CT, 06520, USA
D. W. Schmitz  University of Chicago, Chicago, IL, 60637, USA  A. Schukraft
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  W.
Seligman  Columbia University, New York, NY, 10027, USA  M. H. Shaevitz
Columbia University, New York, NY, 10027, USA  R. Sharankova  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  J. Shi  University of
Cambridge, Cambridge CB3 0HE, United Kingdom  E. L. Snider  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  M. Soderberg  Syracuse
University, Syracuse, NY, 13244, USA  S. Söldner-Rembold  The University of
Manchester, Manchester M13 9PL, United Kingdom  J. Spitz  University of
Michigan, Ann Arbor, MI, 48109, USA  M. Stancari  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  J. St. John  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  T. Strauss  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  A. M. Szelc
University of Edinburgh, Edinburgh EH9 3FD, United Kingdom  W. Tang
University of Tennessee, Knoxville, TN, 37996, USA  N. Taniuchi  University of
Cambridge, Cambridge CB3 0HE, United Kingdom  K. Terao  SLAC National
Accelerator Laboratory, Menlo Park, CA, 94025, USA  C. Thorpe  The University
of Manchester, Manchester M13 9PL, United Kingdom  D. Torbunov  Brookhaven
National Laboratory (BNL), Upton, NY, 11973, USA  D. Totani  University of
California, Santa Barbara, CA, 93106, USA  M. Toups  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  Y.-T. Tsai  SLAC
National Accelerator Laboratory, Menlo Park, CA, 94025, USA  J. Tyler  Kansas
State University (KSU), Manhattan, KS, 66506, USA  M. A. Uchida  University of
Cambridge, Cambridge CB3 0HE, United Kingdom  T. Usher  SLAC National
Accelerator Laboratory, Menlo Park, CA, 94025, USA  B. Viren  Brookhaven
National Laboratory (BNL), Upton, NY, 11973, USA  M. Weber  Universität Bern,
Bern CH-3012, Switzerland  H. Wei  Louisiana State University, Baton Rouge,
LA, 70803, USA  A. J. White  University of Chicago, Chicago, IL, 60637, USA
S. Wolbers  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510,
USA  T. Wongjirad  Tufts University, Medford, MA, 02155, USA  M. Wospakrik
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  K.
Wresilo  University of Cambridge, Cambridge CB3 0HE, United Kingdom  W. Wu
University of Pittsburgh, Pittsburgh, PA, 15260, USA  E. Yandel  University of
California, Santa Barbara, CA, 93106, USA  T. Yang  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  L. E. Yates  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  H. W. Yu  Brookhaven
National Laboratory (BNL), Upton, NY, 11973, USA  G. P. Zeller  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  J. Zennamo  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  C. Zhang
Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA

###### Abstract

We present a first search for dark-trident scattering in a neutrino beam using
a data set corresponding to $7.2\times 10^{20}$ protons on target taken with
the MicroBooNE detector at Fermilab. Proton interactions in the neutrino
target at the Main Injector produce $\pi^{0}$ and $\eta$ mesons, which could
decay into dark-matter (DM) particles mediated via a dark photon $A^{\prime}$.
A convolutional neural network is trained to identify interactions of the DM
particles in the liquid-argon time projection chamber (LArTPC) exploiting its
image-like reconstruction capability. In the absence of a DM signal, we
provide limits at the $90\%$ confidence level on the squared kinematic mixing
parameter $\varepsilon^{2}$ as a function of the dark-photon mass in the range
$10\leq M_{A^{\prime}}\leq 400$ MeV. The limits cover previously unconstrained
parameter space for the production of fermion or scalar DM particles $\chi$
for two benchmark models with mass ratios $M_{\chi}/M_{A^{\prime}}=0.6$ and
$2$ and for dark fine-structure constants $0.1\leq\alpha_{D}\leq 1$.

A wealth of astronomical data at different scales provide evidence for the
existence of dark matter (DM): the motion of galaxies and the stars within
them, gravitational lensing, the cosmic microwave background, and the large-
scale structure of the universe [1]. The nature of dark matter, however,
remains elusive. Non-baryonic particles predicted by dark-sector models are
candidates for dark matter [2]. The search for their production at
accelerators is a focus of the high-energy hadron collider program at the LHC
[3] and of fixed-target experiments exposed to high-intensity beams [4].

The dark-trident process has been proposed as a new way to search for low-mass
dark-matter particles in neutrino beams [5]. In this letter, we report a first
search for such dark tridents with the MicroBooNE liquid-argon time projection
chamber (LArTPC) [6]. In the future, similar searches can be performed with
the DUNE near detector [7] and the detectors of the Fermilab short-baseline
program [8].

A pair of DM particles, $\chi\bar{\chi}$, is produced in the dark-trident
process through the decay of neutral $\pi^{0}$ or $\eta$ mesons, which was
created by the interactions of the protons and by secondary interactions in
the neutrino target (Fig. 1a). The decays
$\pi^{0},\eta\rightarrow\gamma\chi\bar{\chi}$ are mediated by a virtual, off-
shell dark photon $A^{\prime*}$. The masses of the dark photon,
$M_{A^{\prime}}$, and of the dark fermion (or scalar), $M_{\chi}$, are
parameters of the model.

Figure 1: (a) A pair of DM particles, $\chi\bar{\chi}$, is produced in a
$\pi^{0}$ or $\eta^{0}$ decay; (b) in the dark-trident process, $\chi$ (or
$\bar{\chi}$) scatters off an argon nucleus to produce a dark photon
$A^{\prime}$ decaying into an $e^{+}e^{-}$ pair with a branching ratio of $1$.
The rate depends on the kinematic mixing parameter $\varepsilon$ and the dark
fine-structure constant $\alpha_{D}$.

The DM particle $\chi$ (or $\bar{\chi}$) then travels uninterrupted to the
MicroBooNE detector where it could scatter off argon nuclei through the
trident process $\chi+\textrm{Ar}\rightarrow\chi+\textrm{Ar}+A^{\prime}$ (see
Fig. 1b). The dark photon $A^{\prime}$ promptly decays inside the detector
into an $e^{+}e^{-}$ pair. The $\chi$ production rate depends on the kinematic
mixing parameter $\varepsilon$ and a dark fine-structure constant
$\alpha_{D}$, which is defined in terms of the dark-photon gauge coupling
$g_{D}$ as $\alpha_{D}=g_{D}^{2}/(4\pi)$. We consider the mass ratios
$M_{\chi}/M_{A^{\prime}}=0.6$ and $2$ in this search as proposed in Ref. [5].
Since $M_{\chi}/M_{A^{\prime}}>0.5$, the dark photons need to be off-shell to
decay into $\chi\bar{\chi}$ pairs, and, when on-shell, they will exclusively
decay to $e^{+}e^{-}$. The signal rate therefore scales with
$\varepsilon^{4}\alpha_{D}^{3}$. Other recent experimental searches cover the
mass range where $A^{\prime}$ decays invisibly [9, 10, 11].

We use data recorded with the MicroBooNE detector [6] between 2015 and 2018.
The detector’s LArTPC has an instrumented volume of $85$ tonnes of liquid
argon inside a cryostat. Ionization charge drifts across an electric field of
$273$ V/cm and is read out by one charge collection and two induction planes
forming the anode. The LArTPC was simultaneously exposed to the on-axis
booster neutrino beam (BNB) [12] and the off-axis beam of neutrinos from the
main injector (NuMI) [13]. Only NuMI data are used in this search, as the
higher average energy of the NuMI beam gives access to higher values of
$M_{A^{\prime}}$.

The NuMI data used here correspond to $7.2\times 10^{20}$ protons on target
(POT), which were taken in two operating modes – forward horn current (FHC)
with $2.2\times 10^{20}$ POT (Run 1) and reverse horn current (RHC) with
$5.0\times 10^{20}$ POT (Run 3). This data set has previously been used to
search for heavy neutral leptons [14, 15] and Higgs portal scalars [15, 16],
and to measure neutrino cross sections [17, 18].

We simulate the dark-trident process with a dedicated generator in three
steps: the neutral meson flux in the beamline, the decay of the neutral
mesons, and the scattering of the DM particles on argon. First, the kinematics
of the $\pi^{0}$ and $\eta$ mesons for both beam configurations, FHC and RHC,
are obtained using the g4NuMI simulation [19], which is based on a full GEANT4
description of the beamline geometry. The full simulation results in a
significantly higher meson rate compared to Ref. [5] since it includes mesons
produced within the $\approx 1$ m long graphite target by secondary
interactions and within other beamline components.

We then simulate the radiative decays
$\pi^{0},\eta^{0}\rightarrow\gamma\chi\bar{\chi}$ with BdNMC [20]. In addition
to the scalar DM production supported by BdNMC, we added the option to
generate fermions. We calculate the rate of the scattering process
$\chi+\textrm{Ar}\rightarrow\chi+\textrm{Ar}+A^{\prime}$ inside the LArTPC as
a function of the energy of the DM particle and the path traveled inside the
detector [21]. We compare our signal simulation to the calculations of Ref.
[5] and find good agreement in the kinematics, e.g., the distribution of the
$e^{+}e^{-}$ opening angle as a function of the energy of each lepton. The
cross section of the process shown in Fig. 1b is calculated using GenExLight
[21]. We find an agreement better than $1\%$ when comparing these cross
sections to calculations obtained with MadGraph [22].

We use a “beam-on” data sample to search for the dark-trident signal where the
event triggers coincide with the arrival time of neutrinos from the NuMI beam.
The background is modeled considering three contributions. Beam-on background
events that are triggered by a cosmic ray and not a neutrino interaction are
modeled by a “beam-off” sample collected under identical trigger conditions
but when no neutrino beam is present. The “beam-off” sample is scaled so that
its normalization corresponds to the number of beam spills of the beam-on
sample. Neutrino-induced background from the NuMI beam is modeled using a
GENIE Monte Carlo simulation [23] embedded in the LArSoft software framework
[24]. The “in-cryostat $\nu$” sample contains interactions of neutrinos with
the argon inside the cryostat, and the “out-of-cryostat $\nu$” sample
describes interactions with the material surrounding the detector.

We reconstruct neutrino interactions and cosmic rays within the argon with a
chain of pattern-recognition algorithms, implemented using the Pandora
software development kit [25, 26]. The algorithms use hits that are formed
from the waveforms read out by the charge collection plane and the two
induction planes. Collections of hits are reconstructed as a track, as
expected for a minimum ionizing particle, or a shower, consistent with being
an electron or photon conversion.

Table 1: Numbers of events that remain after preselection normalized to POT for the data and the background model. Sample | Run 1 (FHC) | Run 3 (RHC)
---|---|---
POT | $2.2\times 10^{20}$ | $5.0\times 10^{20}$
Beam-off | $2410$ | $4826$
In-cryostat $\nu$ | $1262$ | $2759$
Out-of-cryostat $\nu$ | $\phantom{0}354$ | $\phantom{0}402$
Sum of predictions | $4026$ | $7987$
Beam-on (data) | $4021$ | $7684$

We use the results of the Pandora reconstruction to select events that are
consistent with the signal hypothesis. Dark-trident events are frequently
reconstructed as a single shower due to the small opening angle of the
$e^{+}e^{-}$ pairs and, in a few cases, as two showers arising from a common
vertex. Background processes that can mimic the signal topology are neutral
current (NC) $\nu$ interactions with $\pi^{0},\eta\to\gamma\gamma$ decays in
the final state or with emission of single photons that are reconstructed as
an $e^{+}e^{-}$ pair. Each event is therefore required to have at least one
vertex, at least one shower, and no tracks. The efficiency of this
preselection for a DM signal lies in the range of $(32$–$40)\%$ for masses in
the range $(10$–$400)$ MeV. We find good agreement between the number of data
events and the sum of the predictions for the background processes after this
preselection (Table 1).

We use a convolutional neural network (CNN) for discriminating signal and
background based on the previous development of such algorithms in MicroBooNE
for multiple particle identification (MPID) [27]. Convolutional neural
networks (CNNs) are deep learning networks that are ideally suited for images
reconstructed from LArTPC data [28, 29, 30]. The CNN architecture is based on
a model for dense images with adaptations for LArTPCs. Convolution filters of
size $3\times 3$ allow scanning of the information contained in showers. The
output layer has two neurons that correspond to the probability for signal or
background.

We only consider images from the charge collection plane, as it has the best
signal to noise ratio [27]. The size of each image in pixels corresponds to
$3456$ wires multiplied by $6048$ time ticks. To improve processing time, we
first compress the time axis by a factor of $6$ and then crop the images
around the interaction vertex producing a region of interest (ROI) of
$512\times 512$ pixels. After compression, each pixel has a resolution of
$\approx 3\times 3$ mm2.

Figure 2:  Distribution of the $y$ coordinate of the reconstructed vertices
for the Run 3 data after preselection compared to the background model. The
positive direction of the $y$ axis points vertically upwards. The gray band
represents the systematic uncertainty in the background model.

We validate the agreement of the vertex reconstruction by comparing data and
the background model after the preselection (Fig. 2). The increase of beam-off
events towards the top of the detector due to cosmic rays is reproduced by the
background model. While we use the reconstructed vertices for the data and
background samples, the true vertex location is used for the training. This
prevents the CNN from training on an ROI that does not contain the interaction
of interest, which can occur when a vertex is reconstructed at a large
distance from the true interaction vertex.

For the training of the CNN we prepare a dedicated training data set. We use a
single signal sample with the parameters $\alpha_{D}=0.1$, $M_{A^{\prime}}=50$
MeV, and $M_{A^{\prime}}/M_{\chi}=0.6$. As background samples we use cosmic
rays simulated with CORSIKA [31] and $\nu$ interactions leading to $\pi^{0}$
mesons simulated with GENIE [23]. In addition, we overlay the hits of cosmic
rays simulated with CORSIKA to the $\nu$ interaction background and the signal
samples. A test set, comprising $10\%$ of the events included in the training
set, is used to evaluate the progress of the CNN training.

Figure 3:  Comparison of the CNN signal score distribution for Run 3 data with
the background model after the preselection. The gray band corresponds to the
total systematic uncertainty in the background. The signal distribution for
$\alpha_{D}=0.1$, $M_{A^{\prime}}=50$ MeV, and $M_{A^{\prime}}/M_{\chi}=0.6$
is superimposed, scaled by an arbitrary factor. Figure 4:  A dark-trident
candidate with a CNN score of $6.4$, within the ROI of $512\times 512$ pixels
($\approx 1.5\times 1.5$ m2). A cosmic ray crosses in the lower right-hand
corner.

The CNN model is trained during $\approx 10{,}000$ iterations ($\approx 5$
epochs) with a batch size of $32$ images and a learning rate of $0.001$ [28].
Dropout layers, regularization terms, and batch normalization are implemented
during the CNN training to prevent overfitting. The training progress is
monitored with a Binary Cross Entropy (BCE) loss function and using the
accuracy, which is defined as the fraction of correctly classified images over
the total number of images processed by the CNN. As an additional figure of
merit, we use the receiver operating characteristic curve (ROC) to decide the
number of training steps where the CNN model is frozen. Figure 3 shows the
discrimination between signal and background for CNN signal scores $>0$.

Figure 5:  The $90\%$ CL observed limits on $\varepsilon^{2}$ as a function of
$M_{A^{\prime}}$ for $\alpha_{D}=0.1$ and $\alpha_{D}=1$, and (a)
$M_{\chi}/M_{A^{\prime}}=0.6$ and (b) $M_{\chi}/M_{A^{\prime}}=2$, together
with the $1$ and $2$ standard deviation bands around the median expected
limits. We use a linear interpolation between the mass points. A total of $13$
mass values have been simulated for $M_{\chi}/M_{A^{\prime}}=0.6$, equally
spaced between $10$–$100$ MeV and between $100$–$400$ MeV and a total of $19$
mass values for $M_{\chi}/M_{A^{\prime}}=2$, an additional $6$ mass values are
added at higher $M_{A}$. A table of the limits at each point is provided as
supplementary material.

(a) (b)

Figure 6:  The $90\%$ CL limits on $\varepsilon^{2}$ as a function of
$M_{A^{\prime}}$ for (a) scalar DM with $\alpha_{D}=1.0$,
$M_{\chi}/M_{A^{\prime}}=0.6$; (b) fermion DM with $\alpha_{D}=1.0$,
$M_{\chi}/M_{A^{\prime}}=0.6$; and (c) for fermion DM with $\alpha_{D}=1.0$,
$M_{\chi}/M_{A^{\prime}}=2.0$. The constraints provided by the NA48/2 [32],
BaBar [33], NA64 [34], and LHCb collaborations [35], and by beam dump
experiments [36, 37, 38] are displayed as shaded regions. The
reinterpretations of LSND results [5, 39] and the unpublished FASER [40]
limits are shown as dashed lines. The two isolated contours at
$M_{A^{\prime}}\approx 200$–$300$ MeV are also excluded by LHCb data. The
upper limits on $\varepsilon^{2}$ from Planck data [41, 42] apply for fermion
DM with $M_{\chi}/M_{A^{\prime}}=0.6$.

(a) (b) (c)

We use a single CNN model that has been optimized with a benchmark signal
point trained against the NC $\pi^{0}$ and cosmic-ray background samples. The
areas under the curve for the ROCs of the different signal points relative to
the full background sample (see Table 1) agree within ($1$–$2)\%$ with the
benchmark CNN model.

A data event with a high CNN signal score is shown in Fig. 4, where the shower
points in the direction of the NuMI beam. By modifying the training events, we
determine that the CNN learns about the kinematics (angles, energies) of the
scattering process through the number of pixels and the orientation of pixel
clusters.

We evaluate systematic uncertainties that could modify the CNN score
distributions for signal and background. For the in-cryostat $\nu$ background,
we consider the impact of the neutrino flux simulation $(10$–$20)\%$ [19], the
neutrino-argon cross-section modeling $(12$–$20)\%$ [43], hadron interactions
with argon ($\approx 1\%$) [44], and detector modeling ($\approx 30\%$) [45].
The beam-off sample is taken from data and therefore has no associated
systematic uncertainties other than statistical fluctuations. The impact of
the normalization uncertainty on the out-of-cryostat sample and of the POT
counting is negligible [15].

The sum of the detector-related systematic uncertainties on the signal is in
the range $(10$–$20)\%$. A form factor accounts for the spatial distribution
of the argon nucleus in the $\chi$-Ar scattering [5]. Recalculating the cross
sections with different form factors [46, 47] yields uncertainties in the
range $(2$–$20)\%$ in the mass range $(10$–$200)$ MeV.

The signal rate also depends on the NuMI $\pi^{0}$ and $\eta$ flux simulated
by g4NuMI. We confirm that the ratio of $\pi^{0}$ production relative to
$\pi^{\pm}$ production in g4NuMI is consistent with expectations of isospin
symmetry. We therefore use the beam flux uncertainty of $22\%$ determined for
the charged meson flux [17], which includes hadron production and beam line
modeling uncertainties.

The CNN score distributions are found to be consistent with the background
expectation and used to derive limits on the squared mixing parameter
$\varepsilon^{2}$ as a function of $M_{A^{\prime}}$. The limit setting is done
with the pyhf algorithm [48], which is an implementation of a statistical
model to estimate confidence intervals [49]. Systematic uncertainties are
treated through profile likelihood ratios. The results are validated with the
modified frequentist CLs calculation of the COLLIE program [50].

The observed limits of Fig. 5 are shown at the $90\%$ confidence level
(CL${}_{s}=0.1$) for several benchmark points. Since we use a single CNN model
for all signal points, the background CNN score distributions are highly
correlated between the different mass hypotheses $M_{A^{\prime}}$. All
observed limits are therefore consistently within the $1$ and $2$ standard
deviation ranges around the median expected limit.

In Fig. 6, we compare the results for a scalar dark matter particle $\chi$
with existing constraints on dark-trident processes from rare pion decays
measured by the NA48/2 collaboration [32], beam dump experiments [36, 37, 38],
and searches for promptly decaying dark photons into $e^{+}e^{-}$ pairs by the
BaBar [33], FASER [40], and NA64 [51] collaborations. The limits obtained by
the LHCb collaboration [35] apply for higher masses $M_{A^{\prime}}>200$ MeV.
The most sensitive constraints are obtained for $\alpha_{D}=1$ and
$M_{\chi}/M_{A^{\prime}}=0.6$.

For the fermion model, we also compare to reinterpretations of LSND results
[5, 39]. Cosmological constraints on $\chi\bar{\chi}$ annihilation in the
early universe are obtained using Planck measurements on the cosmic microwave
background [41, 42]. The $\chi\bar{\chi}$ annihilation cross section is only
relevant for a fermion $\chi$ and $M_{\chi}/M_{A^{\prime}}=0.6$. The
cosmological data constrain $\varepsilon^{2}$ from below, since the thermal
relic dark-matter density becomes too small for larger $\varepsilon^{2}$ [5].

In summary, we apply convolutional neural networks to obtain first constraints
on the production of dark matter in a liquid argon detector exposed to a
neutrino beam. We consider fermion and boson dark-matter particles $\chi$
produced in a dark-trident process with $M_{\chi}/M_{A^{\prime}}=0.6$ and
$M_{\chi}/M_{A^{\prime}}=2$, and dark fine-structure constants in the range
$0.1\leq\alpha_{D}\leq 1$. The constraints in the plane of the squared
kinematic mixing parameter $\varepsilon^{2}$ and the dark-photon mass
$M_{A^{\prime}}$ exclude previously unexplored regions of parameter space in
the range $10\leq M_{A^{\prime}}\leq 400$ MeV.

This document was prepared by the MicroBooNE collaboration using the resources
of the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of
Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi
Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359.
MicroBooNE is supported by the following: the U.S. Department of Energy,
Office of Science, Offices of High Energy Physics and Nuclear Physics; the
U.S. National Science Foundation; the Swiss National Science Foundation; the
Science and Technology Facilities Council (STFC), part of the United Kingdom
Research and Innovation; the Royal Society (United Kingdom); the UK Research
and Innovation (UKRI) Future Leaders Fellowship; and the NSF AI Institute for
Artificial Intelligence and Fundamental Interactions. Additional support for
the laser calibration system and cosmic ray tagger were provided by the Albert
Einstein Center for Fundamental Physics, Bern, Switzerland. We also
acknowledge the contributions of technical and scientific staff to the design,
construction, and operation of the MicroBooNE detector as well as the
contributions of past collaborators to the development of MicroBooNE analyses
without whom this work would not have been possible. We also thank the authors
of Ref. [5] for useful discussions about the dark-trident model. For the
purpose of open access, the authors have applied a Creative Commons
Attribution (CC BY) public copyright license to any Author Accepted Manuscript
version arising from this submission.

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Supplemental material: first search for dark-trident processes using the
MicroBooNE detector

  * Figure 7:  The $90\%$ CL limits on $\varepsilon^{2}$ as a function of $M_{A^{\prime}}$. The first row corresponds to scalar DM models and the second row to fermion DM models. The constraints provided by the NA48/2 [32], BaBar [33], NA64 [51], and LHCb collaborations [35], and by beam dump experiments [36, 37, 38] are displayed as shaded regions. The reinterpretations of LSND results [5, 39] and the unpublished FASER [40] limits are shown as dashed lines. The two isolated contours at $M_{A^{\prime}}\approx 200$–$300$ MeV are also excluded by LHCb data. The upper limits on $\varepsilon^{2}$ from Planck data [41, 42] apply for fermion DM with $M_{\chi}/M_{A^{\prime}}=0.6$.

Scalar Dark Matter, $\alpha_{D}=0.1$, $M_{\chi}/M_{A^{\prime}}=2.0$

$M_{A^{\prime}}$ (MeV) | Observed | Expected |  Expected $1\sigma$ |  Expected $2\sigma$
---|---|---|---|---
10 |  7.19$\cdot 10^{-8}$ |  7.40$\cdot 10^{-8}$ |  (6.04–9.29)$\cdot 10^{-8}$ |  (0.51–1.17)$\cdot 10^{-7}$
20 |  4.93$\cdot 10^{-7}$ |  5.01$\cdot 10^{-7}$ |  (4.10–6.30)$\cdot 10^{-7}$ |  (3.47–7.92)$\cdot 10^{-7}$
30 |  1.50$\cdot 10^{-6}$ |  1.51$\cdot 10^{-6}$ |  (1.24–1.90)$\cdot 10^{-6}$ |  (1.05–2.39)$\cdot 10^{-6}$
40 |  3.07$\cdot 10^{-6}$ |  3.09$\cdot 10^{-6}$ |  (2.53–3.89)$\cdot 10^{-6}$ |  (2.14–4.89)$\cdot 10^{-6}$
50 |  6.30$\cdot 10^{-6}$ |  6.18$\cdot 10^{-6}$ |  (5.04–7.78)$\cdot 10^{-6}$ |  (4.26–9.81)$\cdot 10^{-6}$
60 |  1.32$\cdot 10^{-5}$ |  1.33$\cdot 10^{-5}$ |  (1.09–1.68)$\cdot 10^{-5}$ |  (0.92–2.11)$\cdot 10^{-5}$
65 |  2.17$\cdot 10^{-5}$ |  2.19$\cdot 10^{-5}$ |  (1.80–2.76)$\cdot 10^{-5}$ |  (1.52–3.47)$\cdot 10^{-5}$
70 |  2.86$\cdot 10^{-5}$ |  2.92$\cdot 10^{-5}$ |  (2.39–3.67)$\cdot 10^{-5}$ |  (2.02–4.62)$\cdot 10^{-5}$
75 |  4.35$\cdot 10^{-5}$ |  4.38$\cdot 10^{-5}$ |  (3.58–5.51)$\cdot 10^{-5}$ |  (3.03–6.93)$\cdot 10^{-5}$
80 |  6.21$\cdot 10^{-5}$ |  6.28$\cdot 10^{-5}$ |  (5.13–7.90)$\cdot 10^{-5}$ |  (4.34–9.95)$\cdot 10^{-5}$
85 |  9.87$\cdot 10^{-5}$ |  1.01$\cdot 10^{-4}$ |  (0.82–1.26)$\cdot 10^{-4}$ |  (0.70–1.59)$\cdot 10^{-4}$
90 |  1.46$\cdot 10^{-4}$ |  1.47$\cdot 10^{-4}$ |  (1.20–1.84)$\cdot 10^{-4}$ |  (1.01–2.32)$\cdot 10^{-4}$
95 |  2.41$\cdot 10^{-4}$ |  2.45$\cdot 10^{-4}$ |  (2.00–3.08)$\cdot 10^{-4}$ |  (1.69–3.88)$\cdot 10^{-4}$
100 |  3.55$\cdot 10^{-4}$ |  3.59$\cdot 10^{-4}$ |  (2.93–4.51)$\cdot 10^{-4}$ |  (2.49–5.67)$\cdot 10^{-4}$
105 |  5.74$\cdot 10^{-4}$ |  5.80$\cdot 10^{-4}$ |  (4.74–7.29)$\cdot 10^{-4}$ |  (4.02–9.18)$\cdot 10^{-4}$
110 |  1.19$\cdot 10^{-3}$ |  1.22$\cdot 10^{-3}$ |  (1.00–1.53)$\cdot 10^{-3}$ |  (0.84–1.93)$\cdot 10^{-3}$
115 |  2.18$\cdot 10^{-3}$ |  2.19$\cdot 10^{-3}$ |  (1.79–2.76)$\cdot 10^{-3}$ |  (1.52–3.48)$\cdot 10^{-3}$
120 |  5.63$\cdot 10^{-3}$ |  5.69$\cdot 10^{-3}$ |  (4.65–7.15)$\cdot 10^{-3}$ |  (3.94–9.00)$\cdot 10^{-3}$
125 |  1.74$\cdot 10^{-2}$ |  1.80$\cdot 10^{-2}$ |  (1.47–2.27)$\cdot 10^{-2}$ |  (1.25–2.86)$\cdot 10^{-2}$

Scalar Dark Matter, $\alpha_{D}=1.0$, $M_{\chi}/M_{A^{\prime}}=2.0$

$M_{A^{\prime}}$ | Observed | Expected |  Expected $1\sigma$ |  Expected $2\sigma$
---|---|---|---|---
(MeV) |  |  |  |
10 |  2.27$\cdot 10^{-9}$ |  2.34$\cdot 10^{-9}$ |  (1.91–2.94)$\cdot 10^{-9}$ |  (1.62–3.69)$\cdot 10^{-9}$
20 |  1.56$\cdot 10^{-8}$ |  1.59$\cdot 10^{-8}$ |  (1.30–1.99)$\cdot 10^{-8}$ |  (1.10–2.51)$\cdot 10^{-8}$
30 |  4.74$\cdot 10^{-8}$ |  4.79$\cdot 10^{-8}$ |  (3.91–6.02)$\cdot 10^{-8}$ |  (3.31–7.57)$\cdot 10^{-8}$
40 |  9.71$\cdot 10^{-8}$ |  9.77$\cdot 10^{-8}$ |  (0.80–1.23)$\cdot 10^{-7}$ |  (0.68–1.55)$\cdot 10^{-7}$
50 |  1.99$\cdot 10^{-7}$ |  1.95$\cdot 10^{-7}$ |  (1.59–2.46)$\cdot 10^{-7}$ |  (1.35–3.10)$\cdot 10^{-7}$
60 |  4.18$\cdot 10^{-7}$ |  4.21$\cdot 10^{-7}$ |  (3.44–5.30)$\cdot 10^{-7}$ |  (2.91–6.67)$\cdot 10^{-7}$
65 |  6.85$\cdot 10^{-7}$ |  6.94$\cdot 10^{-7}$ |  (5.68–8.73)$\cdot 10^{-7}$ |  (0.48–1.10)$\cdot 10^{-6}$
70 |  9.04$\cdot 10^{-7}$ |  9.24$\cdot 10^{-7}$ |  (0.75–1.16)$\cdot 10^{-6}$ |  (0.64–1.46)$\cdot 10^{-6}$
75 |  1.37$\cdot 10^{-6}$ |  1.38$\cdot 10^{-6}$ |  (1.13–1.74)$\cdot 10^{-6}$ |  (0.96–2.19)$\cdot 10^{-6}$
80 |  1.96$\cdot 10^{-6}$ |  1.99$\cdot 10^{-6}$ |  (1.62–2.50)$\cdot 10^{-6}$ |  (1.37–3.15)$\cdot 10^{-6}$
85 |  3.12$\cdot 10^{-6}$ |  3.18$\cdot 10^{-6}$ |  (2.60–4.00)$\cdot 10^{-6}$ |  (2.20–5.03)$\cdot 10^{-6}$
90 |  4.63$\cdot 10^{-6}$ |  4.63$\cdot 10^{-6}$ |  (3.78–5.83)$\cdot 10^{-6}$ |  (3.21–7.33)$\cdot 10^{-6}$
95 |  7.63$\cdot 10^{-6}$ |  7.74$\cdot 10^{-6}$ |  (6.32–9.73)$\cdot 10^{-6}$ |  (0.54–1.23)$\cdot 10^{-5}$
100 |  1.12$\cdot 10^{-5}$ |  1.14$\cdot 10^{-5}$ |  (0.93–1.43)$\cdot 10^{-5}$ |  (0.79–1.79)$\cdot 10^{-5}$
105 |  1.82$\cdot 10^{-5}$ |  1.84$\cdot 10^{-5}$ |  (1.50–2.31)$\cdot 10^{-5}$ |  (1.27–2.90)$\cdot 10^{-5}$
110 |  3.75$\cdot 10^{-5}$ |  3.85$\cdot 10^{-5}$ |  (3.15–4.84)$\cdot 10^{-5}$ |  (2.67–6.09)$\cdot 10^{-5}$
115 |  6.88$\cdot 10^{-5}$ |  6.94$\cdot 10^{-5}$ |  (5.66–8.73)$\cdot 10^{-5}$ |  (0.48–1.10)$\cdot 10^{-4}$
120 |  1.78$\cdot 10^{-4}$ |  1.80$\cdot 10^{-4}$ |  (1.47–2.26)$\cdot 10^{-4}$ |  (1.25–2.85)$\cdot 10^{-4}$
125 |  5.50$\cdot 10^{-4}$ |  5.71$\cdot 10^{-4}$ |  (4.66–7.17)$\cdot 10^{-4}$ |  (3.95–9.03)$\cdot 10^{-4}$

Table 2:  The $90\%$ CL observed limits on $\varepsilon^{2}$ for a scalar DM
particle $\chi$ as a function of $M_{A^{\prime}}$ for $\alpha_{D}=0.1$ and
$\alpha_{D}=1$, and $M_{\chi}/M_{A^{\prime}}=2$, together with the $1$ and $2$
standard deviation bands around the median expected limits. The results are
obtained with a data set of $7.2\times 10^{20}$ POT.

Fermion Dark Matter, $\alpha_{D}=0.1$, $M_{\chi}/M_{A^{\prime}}=2.0$

$M_{A^{\prime}}$ | Observed | Expected |  Expected $1\sigma$ |  Expected $2\sigma$
---|---|---|---|---
(MeV) |  |  |  |
10 |  2.17$\cdot 10^{-8}$ |  2.22$\cdot 10^{-8}$ |  (1.81–2.82)$\cdot 10^{-8}$ |  (1.53–3.58)$\cdot 10^{-8}$
20 |  1.36$\cdot 10^{-7}$ |  1.38$\cdot 10^{-7}$ |  (1.13–1.74)$\cdot 10^{-7}$ |  (0.96–2.18)$\cdot 10^{-7}$
30 |  4.53$\cdot 10^{-7}$ |  4.58$\cdot 10^{-7}$ |  (3.74–5.75)$\cdot 10^{-7}$ |  (3.16–7.24)$\cdot 10^{-7}$
40 |  9.39$\cdot 10^{-7}$ |  9.45$\cdot 10^{-7}$ |  (0.77–1.19)$\cdot 10^{-6}$ |  (0.65–1.49)$\cdot 10^{-6}$
50 |  1.77$\cdot 10^{-6}$ |  1.74$\cdot 10^{-6}$ |  (1.42–2.19)$\cdot 10^{-6}$ |  (1.20–2.76)$\cdot 10^{-6}$
60 |  3.38$\cdot 10^{-6}$ |  3.40$\cdot 10^{-6}$ |  (2.78–4.28)$\cdot 10^{-6}$ |  (2.35–5.39)$\cdot 10^{-6}$
65 |  4.63$\cdot 10^{-6}$ |  4.69$\cdot 10^{-6}$ |  (3.84–5.90)$\cdot 10^{-6}$ |  (3.25–7.42)$\cdot 10^{-6}$
70 |  6.66$\cdot 10^{-6}$ |  6.81$\cdot 10^{-6}$ |  (5.56–8.56)$\cdot 10^{-6}$ |  (0.47–1.08)$\cdot 10^{-5}$
75 |  9.05$\cdot 10^{-6}$ |  9.12$\cdot 10^{-6}$ |  (0.75–1.15)$\cdot 10^{-5}$ |  (0.63–1.44)$\cdot 10^{-5}$
80 |  1.25$\cdot 10^{-5}$ |  1.27$\cdot 10^{-5}$ |  (1.04–1.59)$\cdot 10^{-5}$ |  (0.88–2.01)$\cdot 10^{-5}$
85 |  1.90$\cdot 10^{-5}$ |  1.94$\cdot 10^{-5}$ |  (1.58–2.44)$\cdot 10^{-5}$ |  (1.34–3.07)$\cdot 10^{-5}$
90 |  2.54$\cdot 10^{-5}$ |  2.54$\cdot 10^{-5}$ |  (2.07–3.19)$\cdot 10^{-5}$ |  (1.76–4.02)$\cdot 10^{-5}$
95 |  4.16$\cdot 10^{-5}$ |  4.21$\cdot 10^{-5}$ |  (3.44–5.30)$\cdot 10^{-5}$ |  (2.91–6.67)$\cdot 10^{-5}$
100 |  5.54$\cdot 10^{-5}$ |  5.61$\cdot 10^{-5}$ |  (4.58–7.04)$\cdot 10^{-5}$ |  (3.88–8.85)$\cdot 10^{-5}$
105 |  9.22$\cdot 10^{-5}$ |  9.32$\cdot 10^{-5}$ |  (0.76–1.17)$\cdot 10^{-4}$ |  (0.64–1.47)$\cdot 10^{-4}$
110 |  1.53$\cdot 10^{-4}$ |  1.57$\cdot 10^{-4}$ |  (1.29–1.98)$\cdot 10^{-4}$ |  (1.09–2.49)$\cdot 10^{-4}$
115 |  2.69$\cdot 10^{-4}$ |  2.71$\cdot 10^{-4}$ |  (2.21–3.41)$\cdot 10^{-4}$ |  (1.88–4.30)$\cdot 10^{-4}$
120 |  5.61$\cdot 10^{-4}$ |  5.67$\cdot 10^{-4}$ |  (4.63–7.13)$\cdot 10^{-4}$ |  (3.92–8.97)$\cdot 10^{-4}$
125 |  1.43$\cdot 10^{-3}$ |  1.49$\cdot 10^{-3}$ |  (1.22–1.87)$\cdot 10^{-3}$ |  (1.03–2.35)$\cdot 10^{-3}$

Fermion Dark Matter, $\alpha_{D}=1.0$, $M_{\chi}/M_{A^{\prime}}=2.0$

$M_{A^{\prime}}$ | Observed | Expected |  Expected $1\sigma$ |  Expected $2\sigma$
---|---|---|---|---
(MeV) |  |  |  |
10 |  6.87$\cdot 10^{-10}$ |  7.03$\cdot 10^{-10}$ |  (5.71–8.91)$\cdot 10^{-10}$ |  (0.48–1.13)$\cdot 10^{-9}$
20 |  4.29$\cdot 10^{-9}$ |  4.37$\cdot 10^{-9}$ |  (3.58–5.49)$\cdot 10^{-9}$ |  (3.03–6.91)$\cdot 10^{-9}$
30 |  1.43$\cdot 10^{-8}$ |  1.45$\cdot 10^{-8}$ |  (1.18–1.82)$\cdot 10^{-8}$ |  (1.00–2.29)$\cdot 10^{-8}$
40 |  2.97$\cdot 10^{-8}$ |  2.99$\cdot 10^{-8}$ |  (2.44–3.76)$\cdot 10^{-8}$ |  (2.07–4.73)$\cdot 10^{-8}$
50 |  5.60$\cdot 10^{-8}$ |  5.49$\cdot 10^{-8}$ |  (4.48–6.92)$\cdot 10^{-8}$ |  (3.79–8.72)$\cdot 10^{-8}$
60 |  1.07$\cdot 10^{-7}$ |  1.08$\cdot 10^{-7}$ |  (0.88–1.35)$\cdot 10^{-7}$ |  (0.74–1.71)$\cdot 10^{-7}$
65 |  1.47$\cdot 10^{-7}$ |  1.48$\cdot 10^{-7}$ |  (1.21–1.87)$\cdot 10^{-7}$ |  (1.03–2.35)$\cdot 10^{-7}$
70 |  2.11$\cdot 10^{-7}$ |  2.15$\cdot 10^{-7}$ |  (1.76–2.71)$\cdot 10^{-7}$ |  (1.49–3.41)$\cdot 10^{-7}$
75 |  2.86$\cdot 10^{-7}$ |  2.88$\cdot 10^{-7}$ |  (2.36–3.63)$\cdot 10^{-7}$ |  (2.00–4.57)$\cdot 10^{-7}$
80 |  3.96$\cdot 10^{-7}$ |  4.01$\cdot 10^{-7}$ |  (3.27–5.04)$\cdot 10^{-7}$ |  (2.77–6.35)$\cdot 10^{-7}$
85 |  6.01$\cdot 10^{-7}$ |  6.13$\cdot 10^{-7}$ |  (5.01–7.71)$\cdot 10^{-7}$ |  (4.24–9.70)$\cdot 10^{-7}$
90 |  8.03$\cdot 10^{-7}$ |  8.03$\cdot 10^{-7}$ |  (0.66–1.01)$\cdot 10^{-6}$ |  (0.56–1.27)$\cdot 10^{-6}$
95 |  1.31$\cdot 10^{-6}$ |  1.33$\cdot 10^{-6}$ |  (1.09–1.67)$\cdot 10^{-6}$ |  (0.92–2.11)$\cdot 10^{-6}$
100 |  1.75$\cdot 10^{-6}$ |  1.77$\cdot 10^{-6}$ |  (1.45–2.23)$\cdot 10^{-6}$ |  (1.23–2.80)$\cdot 10^{-6}$
105 |  2.91$\cdot 10^{-6}$ |  2.95$\cdot 10^{-6}$ |  (2.41–3.70)$\cdot 10^{-6}$ |  (2.04–4.66)$\cdot 10^{-6}$
110 |  4.84$\cdot 10^{-6}$ |  4.97$\cdot 10^{-6}$ |  (4.06–6.25)$\cdot 10^{-6}$ |  (3.45–7.86)$\cdot 10^{-6}$
115 |  8.52$\cdot 10^{-6}$ |  8.58$\cdot 10^{-6}$ |  (0.70–1.08)$\cdot 10^{-5}$ |  (0.59–1.36)$\cdot 10^{-5}$
120 |  1.77$\cdot 10^{-5}$ |  1.79$\cdot 10^{-5}$ |  (1.46–2.25)$\cdot 10^{-5}$ |  (1.24–2.84)$\cdot 10^{-5}$
125 |  4.53$\cdot 10^{-5}$ |  4.70$\cdot 10^{-5}$ |  (3.84–5.91)$\cdot 10^{-5}$ |  (3.26–7.44)$\cdot 10^{-5}$

Table 3:  The $90\%$ CL observed limits on $\varepsilon^{2}$ for a fermion DM
particle $\chi$ as a function of $M_{A^{\prime}}$ for $\alpha_{D}=0.1$ and
$\alpha_{D}=1$, and $M_{\chi}/M_{A^{\prime}}=2$, together with the $1$ and $2$
standard deviation bands around the median expected limits. The results are
obtained with a data set of $7.2\times 10^{20}$ POT.

Scalar Dark Matter, $\alpha_{D}=0.1$, $M_{\chi}/M_{A^{\prime}}=0.6$

$M_{A^{\prime}}$ | Observed | Expected |  Expected $1\sigma$ |  Expected $2\sigma$
---|---|---|---|---
(MeV) |  |  |  |
10 |  2.32$\cdot 10^{-8}$ |  2.36$\cdot 10^{-8}$ |  (1.92–2.98)$\cdot 10^{-8}$ |  (1.63–3.78)$\cdot 10^{-8}$
20 |  5.44$\cdot 10^{-8}$ |  5.56$\cdot 10^{-8}$ |  (4.54–6.99)$\cdot 10^{-8}$ |  (3.85–8.80)$\cdot 10^{-8}$
30 |  1.29$\cdot 10^{-7}$ |  1.31$\cdot 10^{-7}$ |  (1.07–1.64)$\cdot 10^{-7}$ |  (0.91–2.07)$\cdot 10^{-7}$
40 |  2.87$\cdot 10^{-7}$ |  2.91$\cdot 10^{-7}$ |  (2.37–3.66)$\cdot 10^{-7}$ |  (2.01–4.60)$\cdot 10^{-7}$
50 |  5.28$\cdot 10^{-7}$ |  5.39$\cdot 10^{-7}$ |  (4.40–6.77)$\cdot 10^{-7}$ |  (3.73–8.53)$\cdot 10^{-7}$
60 |  9.97$\cdot 10^{-7}$ |  1.01$\cdot 10^{-6}$ |  (0.82–1.27)$\cdot 10^{-6}$ |  (0.70–1.59)$\cdot 10^{-6}$
70 |  1.90$\cdot 10^{-6}$ |  1.90$\cdot 10^{-6}$ |  (1.55–2.39)$\cdot 10^{-6}$ |  (1.31–3.01)$\cdot 10^{-6}$
80 |  2.98$\cdot 10^{-6}$ |  3.02$\cdot 10^{-6}$ |  (2.46–3.80)$\cdot 10^{-6}$ |  (2.08–4.79)$\cdot 10^{-6}$
90 |  5.11$\cdot 10^{-6}$ |  5.11$\cdot 10^{-6}$ |  (4.18–6.43)$\cdot 10^{-6}$ |  (3.54–8.10)$\cdot 10^{-6}$
100 |  7.30$\cdot 10^{-6}$ |  7.33$\cdot 10^{-6}$ |  (5.98–9.22)$\cdot 10^{-6}$ |  (0.51–1.16)$\cdot 10^{-5}$
200 |  1.12$\cdot 10^{-4}$ |  1.15$\cdot 10^{-4}$ |  (0.94–1.44)$\cdot 10^{-4}$ |  (0.79–1.82)$\cdot 10^{-4}$
300 |  2.04$\cdot 10^{-3}$ |  2.08$\cdot 10^{-3}$ |  (1.70–2.62)$\cdot 10^{-3}$ |  (1.43–3.31)$\cdot 10^{-3}$
400 |  9.24$\cdot 10^{-2}$ |  9.19$\cdot 10^{-2}$ |  (0.75–1.16)$\cdot 10^{-1}$ |  (0.63–1.47)$\cdot 10^{-1}$

Scalar Dark Matter, $\alpha_{D}=1.0$, $M_{\chi}/M_{A^{\prime}}=0.6$

$M_{A^{\prime}}$ | Observed | Expected |  Expected $1\sigma$ |  Expected $2\sigma$
---|---|---|---|---
(MeV) |  |  |  |
10 |  7.34$\cdot 10^{-10}$ |  7.46$\cdot 10^{-10}$ |  (6.08–9.43)$\cdot 10^{-10}$ |  (0.52–1.19)$\cdot 10^{-9}$
20 |  1.72$\cdot 10^{-9}$ |  1.76$\cdot 10^{-9}$ |  (1.44–2.21)$\cdot 10^{-9}$ |  (1.22–2.78)$\cdot 10^{-9}$
30 |  4.07$\cdot 10^{-9}$ |  4.14$\cdot 10^{-9}$ |  (3.38–5.20)$\cdot 10^{-9}$ |  (2.87–6.54)$\cdot 10^{-9}$
40 |  9.07$\cdot 10^{-9}$ |  9.20$\cdot 10^{-9}$ |  (0.75–1.16)$\cdot 10^{-8}$ |  (0.64–1.45)$\cdot 10^{-8}$
50 |  1.67$\cdot 10^{-8}$ |  1.70$\cdot 10^{-8}$ |  (1.39–2.14)$\cdot 10^{-8}$ |  (1.18–2.70)$\cdot 10^{-8}$
60 |  3.15$\cdot 10^{-8}$ |  3.18$\cdot 10^{-8}$ |  (2.60–4.00)$\cdot 10^{-8}$ |  (2.20–5.04)$\cdot 10^{-8}$
70 |  6.01$\cdot 10^{-8}$ |  6.00$\cdot 10^{-8}$ |  (4.90–7.55)$\cdot 10^{-8}$ |  (4.15–9.52)$\cdot 10^{-8}$
80 |  9.41$\cdot 10^{-8}$ |  9.55$\cdot 10^{-8}$ |  (0.78–1.20)$\cdot 10^{-7}$ |  (0.66–1.51)$\cdot 10^{-7}$
90 |  1.62$\cdot 10^{-7}$ |  1.61$\cdot 10^{-7}$ |  (1.32–2.03)$\cdot 10^{-7}$ |  (1.12–2.56)$\cdot 10^{-7}$
100 |  2.31$\cdot 10^{-7}$ |  2.32$\cdot 10^{-7}$ |  (1.89–2.91)$\cdot 10^{-7}$ |  (1.60–3.67)$\cdot 10^{-7}$
200 |  3.55$\cdot 10^{-6}$ |  3.63$\cdot 10^{-6}$ |  (2.96–4.56)$\cdot 10^{-6}$ |  (2.51–5.75)$\cdot 10^{-6}$
300 |  6.46$\cdot 10^{-5}$ |  6.58$\cdot 10^{-5}$ |  (5.37–8.30)$\cdot 10^{-5}$ |  (0.45–1.05)$\cdot 10^{-4}$
400 |  2.92$\cdot 10^{-3}$ |  2.91$\cdot 10^{-3}$ |  (2.37–3.67)$\cdot 10^{-3}$ |  (2.00–4.66)$\cdot 10^{-3}$

Table 4:  The $90\%$ CL observed limits on $\varepsilon^{2}$ for a scalar DM
particle $\chi$ as a function of $M_{A^{\prime}}$ for $\alpha_{D}=0.1$ and
$\alpha_{D}=1$, and $M_{\chi}/M_{A^{\prime}}=0.6$, together with the $1$ and
$2$ standard deviation bands around the median expected limits. The results
are obtained with a data set of $7.2\times 10^{20}$ POT.

Fermion Dark Matter, $\alpha_{D}=0.1$, $M_{\chi}/M_{A^{\prime}}=0.6$

$M_{A^{\prime}}$ | Observed | Expected |  Expected $1\sigma$ |  Expected $2\sigma$
---|---|---|---|---
(MeV) |  |  |  |
10 |  7.36$\cdot 10^{-9}$ |  6.86$\cdot 10^{-9}$ |  (5.36–9.34)$\cdot 10^{-9}$ |  (0.44–1.32)$\cdot 10^{-8}$
20 |  1.58$\cdot 10^{-8}$ |  1.61$\cdot 10^{-8}$ |  (1.31–2.05)$\cdot 10^{-8}$ |  (1.10–2.64)$\cdot 10^{-8}$
30 |  3.45$\cdot 10^{-8}$ |  3.51$\cdot 10^{-8}$ |  (2.86–4.42)$\cdot 10^{-8}$ |  (2.42–5.59)$\cdot 10^{-8}$
40 |  6.64$\cdot 10^{-8}$ |  6.74$\cdot 10^{-8}$ |  (5.50–8.48)$\cdot 10^{-8}$ |  (0.46–1.07)$\cdot 10^{-7}$
50 |  1.17$\cdot 10^{-7}$ |  1.19$\cdot 10^{-7}$ |  (0.97–1.50)$\cdot 10^{-7}$ |  (0.82–1.89)$\cdot 10^{-7}$
60 |  2.17$\cdot 10^{-7}$ |  2.19$\cdot 10^{-7}$ |  (1.78–2.75)$\cdot 10^{-7}$ |  (1.51–3.46)$\cdot 10^{-7}$
70 |  4.12$\cdot 10^{-7}$ |  4.12$\cdot 10^{-7}$ |  (3.36–5.18)$\cdot 10^{-7}$ |  (2.84–6.53)$\cdot 10^{-7}$
80 |  6.61$\cdot 10^{-7}$ |  6.72$\cdot 10^{-7}$ |  (5.48–8.46)$\cdot 10^{-7}$ |  (0.46–1.07)$\cdot 10^{-6}$
90 |  1.18$\cdot 10^{-6}$ |  1.18$\cdot 10^{-6}$ |  (0.96–1.48)$\cdot 10^{-6}$ |  (0.81–1.87)$\cdot 10^{-6}$
100 |  1.81$\cdot 10^{-6}$ |  1.81$\cdot 10^{-6}$ |  (1.48–2.28)$\cdot 10^{-6}$ |  (1.25–2.87)$\cdot 10^{-6}$
200 |  2.30$\cdot 10^{-5}$ |  2.35$\cdot 10^{-5}$ |  (1.91–2.95)$\cdot 10^{-5}$ |  (1.62–3.73)$\cdot 10^{-5}$
300 |  2.75$\cdot 10^{-4}$ |  2.80$\cdot 10^{-4}$ |  (2.28–3.53)$\cdot 10^{-4}$ |  (1.93–4.46)$\cdot 10^{-4}$
400 |  8.13$\cdot 10^{-3}$ |  8.09$\cdot 10^{-3}$ |  (0.66–1.02)$\cdot 10^{-2}$ |  (0.56–1.30)$\cdot 10^{-2}$

Fermion Dark Matter, $\alpha_{D}=1.0$, $M_{\chi}/M_{A^{\prime}}=0.6$

$M_{A^{\prime}}$ | Observed | Expected |  Expected $1\sigma$ |  Expected $2\sigma$
---|---|---|---|---
(MeV) |  |  |  |
10 |  2.33$\cdot 10^{-10}$ |  2.17$\cdot 10^{-10}$ |  (1.70–2.95)$\cdot 10^{-10}$ |  (1.40–4.16)$\cdot 10^{-10}$
20 |  5.01$\cdot 10^{-10}$ |  5.10$\cdot 10^{-10}$ |  (4.13–6.49)$\cdot 10^{-10}$ |  (3.49–8.33)$\cdot 10^{-10}$
30 |  1.09$\cdot 10^{-9}$ |  1.11$\cdot 10^{-9}$ |  (0.90–1.40)$\cdot 10^{-9}$ |  (0.77–1.77)$\cdot 10^{-9}$
40 |  2.10$\cdot 10^{-9}$ |  2.13$\cdot 10^{-9}$ |  (1.74–2.68)$\cdot 10^{-9}$ |  (1.47–3.38)$\cdot 10^{-9}$
50 |  3.69$\cdot 10^{-9}$ |  3.77$\cdot 10^{-9}$ |  (3.08–4.74)$\cdot 10^{-9}$ |  (2.61–5.97)$\cdot 10^{-9}$
60 |  6.86$\cdot 10^{-9}$ |  6.92$\cdot 10^{-9}$ |  (5.64–8.70)$\cdot 10^{-9}$ |  (0.48–1.09)$\cdot 10^{-8}$
70 |  1.30$\cdot 10^{-8}$ |  1.30$\cdot 10^{-8}$ |  (1.06–1.64)$\cdot 10^{-8}$ |  (0.90–2.06)$\cdot 10^{-8}$
80 |  2.09$\cdot 10^{-8}$ |  2.13$\cdot 10^{-8}$ |  (1.73–2.67)$\cdot 10^{-8}$ |  (1.47–3.37)$\cdot 10^{-8}$
90 |  3.72$\cdot 10^{-8}$ |  3.72$\cdot 10^{-8}$ |  (3.04–4.68)$\cdot 10^{-8}$ |  (2.58–5.90)$\cdot 10^{-8}$
100 |  5.71$\cdot 10^{-8}$ |  5.73$\cdot 10^{-8}$ |  (4.68–7.21)$\cdot 10^{-8}$ |  (3.96–9.08)$\cdot 10^{-8}$
200 |  7.26$\cdot 10^{-7}$ |  7.42$\cdot 10^{-7}$ |  (6.06–9.34)$\cdot 10^{-7}$ |  (0.51–1.18)$\cdot 10^{-6}$
300 |  8.69$\cdot 10^{-6}$ |  8.85$\cdot 10^{-6}$ |  (0.72–1.12)$\cdot 10^{-5}$ |  (0.61–1.41)$\cdot 10^{-5}$
400 |  2.57$\cdot 10^{-4}$ |  2.56$\cdot 10^{-4}$ |  (2.08–3.23)$\cdot 10^{-4}$ |  (1.76–4.10)$\cdot 10^{-4}$

Table 5:  The $90\%$ CL observed limits on $\varepsilon^{2}$ for a fermion DM
particle $\chi$ as a function of $M_{A^{\prime}}$ for $\alpha_{D}=0.1$ and
$\alpha_{D}=1$, and $M_{\chi}/M_{A^{\prime}}=0.6$, together with the $1$ and
$2$ standard deviation bands around the median expected limits. The results
are obtained with a data set of $7.2\times 10^{20}$ POT.