††thanks: Corresponding author

# Probing Light Dark Matter with positron beams at NA64

Yu. M. Andreev  0000-0002-7397-9665 Authors affiliated with an institute
covered by a cooperation agreement with CERN  A. Antonov  0000-0003-1238-5158
INFN, Sezione di Genova, 16147 Genova, Italia  D. Banerjee
0000-0003-0531-1679 CERN, European Organization for Nuclear Research, CH-1211
Geneva, Switzerland  B. Banto Oberhauser  0009-0006-4795-1008 ETH Zürich,
Institute for Particle Physics and Astrophysics, CH-8093 Zürich, Switzerland
J. Bernhard  0000-0001-9256-971X CERN, European Organization for Nuclear
Research, CH-1211 Geneva, Switzerland  P. Bisio  /0009-0006-8677-7495
<EMAIL_ADDRESS>INFN, Sezione di Genova, 16147 Genova, Italia
Università degli Studi di Genova, 16126 Genova, Italia  M. Bondí
0000-0001-8297-9184 INFN, Sezione di Catania, 95125 Catania, Italia  A.
Celentano  0000-0002-7104-2983 INFN, Sezione di Genova, 16147 Genova, Italia
N. Charitonidis  0000-0001-9506-1022 CERN, European Organization for Nuclear
Research, CH-1211 Geneva, Switzerland  D. Cooke  UCL Departement of Physics
and Astronomy, University College London, Gower St. London WC1E 6BT, United
Kingdom  P. Crivelli  0000-0001-5430-9394 ETH Zürich, Institute for Particle
Physics and Astrophysics, CH-8093 Zürich, Switzerland  E. Depero
0000-0003-2239-1746 ETH Zürich, Institute for Particle Physics and
Astrophysics, CH-8093 Zürich, Switzerland  A. V. Dermenev  0000-0001-5619-376X
Authors affiliated with an institute covered by a cooperation agreement with
CERN  S. V. Donskov  0000-0002-3988-7687 Authors affiliated with an institute
covered by a cooperation agreement with CERN  R. R. Dusaev
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affiliated with an international laboratory covered by a cooperation agreement
with CERN  V. N. Frolov  Authors affiliated with an international laboratory
covered by a cooperation agreement with CERN  A. Gardikiotis
0000-0002-4435-2695 Physics Department, University of Patras, 265 04 Patras,
Greece  S. G. Gerassimov  0000-0001-7780-8735 Authors affiliated with an
institute covered by a cooperation agreement with CERN  Technische Universität
München, Physik Department, 85748 Garching, Germany  S. N. Gninenko
0000-0001-6495-7619 Authors affiliated with an institute covered by a
cooperation agreement with CERN  M. Hösgen  Universität Bonn, Helmholtz-
Institut für Strahlen-und Kernphysik, 53115 Bonn, Germany  M. Jeckel  CERN,
European Organization for Nuclear Research, CH-1211 Geneva, Switzerland  V. A.
Kachanov  0000-0002-3062-010X Authors affiliated with an institute covered by
a cooperation agreement with CERN  Y. Kambar  0009-0000-9185-2353 Authors
affiliated with an international laboratory covered by a cooperation agreement
with CERN  A. E. Karneyeu  0000-0001-9983-1004 Authors affiliated with an
institute covered by a cooperation agreement with CERN  G. Kekelidze
0000-0002-5393-9199 Authors affiliated with an international laboratory
covered by a cooperation agreement with CERN  B. Ketzer  0000-0002-3493-3891
Universität Bonn, Helmholtz-Institut für Strahlen-und Kernphysik, 53115 Bonn,
Germany  D. V. Kirpichnikov  0000-0002-7177-077X Authors affiliated with an
institute covered by a cooperation agreement with CERN  M. M. Kirsanov
0000-0002-8879-6538 Authors affiliated with an institute covered by a
cooperation agreement with CERN  V. N. Kolosov  Authors affiliated with an
institute covered by a cooperation agreement with CERN  I. V. Konorov
0000-0002-9013-5456 Technische Universität München, Physik Department, 85748
Garching, Germany  S. V. Gertsenberger  0009-0006-1640-9443 Authors affiliated
with an international laboratory covered by a cooperation agreement with CERN
E. A. Kasianova  Authors affiliated with an international laboratory covered
by a cooperation agreement with CERN  V. A. Kramarenko  0000-0002-8625-5586
Authors affiliated with an institute covered by a cooperation agreement with
CERN  Authors affiliated with an international laboratory covered by a
cooperation agreement with CERN  L. V. Kravchuk  0000-0001-8631-4200 Authors
affiliated with an institute covered by a cooperation agreement with CERN  N.
V. Krasnikov  0000-0002-8717-6492 Authors affiliated with an institute covered
by a cooperation agreement with CERN  Authors affiliated with an international
laboratory covered by a cooperation agreement with CERN  S. V. Kuleshov
0000-0002-3065-326X Center for Theoretical and Experimental Particle Physics,
Facultad de Ciencias Exactas, Universidad Andres Bello, Fernandez Concha 700,
Santiago, Chile  Millennium Institute for Subatomic Physics at High-Energy
Frontier (SAPHIR), Fernandez Concha 700, Santiago, Chile  V. E. Lyubovitskij
0000-0001-7467-572X Authors affiliated with an institute covered by a
cooperation agreement with CERN  Millennium Institute for Subatomic Physics at
High-Energy Frontier (SAPHIR), Fernandez Concha 700, Santiago, Chile
Universidad Técnica Federico Santa María and CCTVal, 2390123 Valparaíso, Chile
V. Lysan  0009-0004-1795-1651 Authors affiliated with an international
laboratory covered by a cooperation agreement with CERN  A. Marini
0000-0002-6778-2161 INFN, Sezione di Genova, 16147 Genova, Italia  L.
Marsicano  0000-0002-8931-7498 INFN, Sezione di Genova, 16147 Genova, Italia
V. A. Matveev  0000-0002-2745-5908 Authors affiliated with an international
laboratory covered by a cooperation agreement with CERN  R. Mena Fredes
Millennium Institute for Subatomic Physics at High-Energy Frontier (SAPHIR),
Fernandez Concha 700, Santiago, Chile  R. Mena Yanssen  Millennium Institute
for Subatomic Physics at High-Energy Frontier (SAPHIR), Fernandez Concha 700,
Santiago, Chile  Universidad Técnica Federico Santa María and CCTVal, 2390123
Valparaíso, Chile  Yu. V. Mikhailov  Authors affiliated with an institute
covered by a cooperation agreement with CERN  L. Molina Bueno
0000-0001-9720-9764 Instituto de Fisica Corpuscular (CSIC/UV), Carrer del
Catedratic Jose Beltran Martinez, 2, 46980 Paterna, Valencia, Spain  M.
Mongillo  0009-0000-7331-4076 ETH Zürich, Institute for Particle Physics and
Astrophysics, CH-8093 Zürich, Switzerland  D. V. Peshekhonov
0009-0008-9018-5884 Authors affiliated with an international laboratory
covered by a cooperation agreement with CERN  V. A. Polyakov
0000-0001-5989-0990 Authors affiliated with an institute covered by a
cooperation agreement with CERN  B. Radics  0000-0002-8978-1725 Department of
Physics and Astronomy, York University, Toronto, ON, Canada  K. Salamatin
0000-0001-6287-8685 Authors affiliated with an international laboratory
covered by a cooperation agreement with CERN  V. D. Samoylenko  Authors
affiliated with an institute covered by a cooperation agreement with CERN  H.
Sieber  0000-0003-1476-4258 ETH Zürich, Institute for Particle Physics and
Astrophysics, CH-8093 Zürich, Switzerland  D. Shchukin  0009-0007-5508-3615
Authors affiliated with an institute covered by a cooperation agreement with
CERN  O. Soto  Millennium Institute for Subatomic Physics at High-Energy
Frontier (SAPHIR), Fernandez Concha 700, Santiago, Chile  Departamento de
Fisica, Facultad de Ciencias, Universidad de La Serena, Avenida Cisternas
1200, La Serena, Chile  V. O. Tikhomirov  0000-0002-9634-0581 Authors
affiliated with an institute covered by a cooperation agreement with CERN  I.
Tlisova  0000-0003-1552-2015 Authors affiliated with an institute covered by a
cooperation agreement with CERN  A. N. Toropin  0000-0002-2106-4041 Authors
affiliated with an institute covered by a cooperation agreement with CERN  A.
Yu. Trifonov  Authors affiliated with an institute covered by a cooperation
agreement with CERN  M. Tuzi  0009-0000-6276-1401 Instituto de Fisica
Corpuscular (CSIC/UV), Carrer del Catedratic Jose Beltran Martinez, 2, 46980
Paterna, Valencia, Spain  P. Ulloa  0000-0002-0789-7581 Center for Theoretical
and Experimental Particle Physics, Facultad de Ciencias Exactas, Universidad
Andres Bello, Fernandez Concha 700, Santiago, Chile  P. V. Volkov
0000-0002-7668-3691 Authors affiliated with an international laboratory
covered by a cooperation agreement with CERN  V. Yu. Volkov
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cooperation agreement with CERN  I. V. Voronchikhin  0000-0003-3037-636X
Authors affiliated with an institute covered by a cooperation agreement with
CERN  J. Zamora-Saá  0000-0002-5030-7516 Center for Theoretical and
Experimental Particle Physics, Facultad de Ciencias Exactas, Universidad
Andres Bello, Fernandez Concha 700, Santiago, Chile  Millennium Institute for
Subatomic Physics at High-Energy Frontier (SAPHIR), Fernandez Concha 700,
Santiago, Chile  A. S. Zhevlakov  0000-0002-7775-5917 Authors affiliated with
an international laboratory covered by a cooperation agreement with CERN

###### Abstract

We present the results of a missing-energy search for Light Dark Matter which
has a new interaction with ordinary matter transmitted by a vector boson,
called dark photon ${A^{\prime}}$. For the first time, this search is
performed with a positron beam by using the significantly enhanced production
of ${A^{\prime}}$ in the resonant annihilation of positrons with atomic
electrons of the target nuclei, followed by the invisible decay of
${A^{\prime}}$ into dark matter. No events were found in the signal region
with $(10.1\pm 0.1)~{}\times~{}10^{9}$ positrons on target with 100 GeV
energy. This allowed us to set new exclusion limits that, relative to the
collected statistics, prove the power of this experimental technique. This
measurement is a crucial first step toward a future exploration program with
positron beams, whose estimated sensitivity is here presented.

Numerous cosmological and astrophysical observations unequivocally indicate
that 85$\%$ of the matter of our Universe is made by a new form of matter,
called “Dark Matter” (DM), gravitationally interacting with the ordinary
matter described by the Standard Model (SM), but not directly emitting or
absorbing light [1, 2, 3]. At present, the particle content of DM is unknown.
Among the different theories that have been postulated to explain the DM
microscopic properties, the Light Dark Matter (LDM) hypothesis assumes that DM
is made by sub-GeV particles, interacting with the SM through a new force. LDM
particles (here denoted as $\chi$) can be the lightest stable states of a new
“Dark Sector” (DS), with its own particles and fields [4, 5, 6]. A
representative LDM model involves the existence of a new $U(1)_{D}$ hidden
symmetry in Nature, associated to a massive gauge boson, also called “dark
photon” (${A^{\prime}}$). The dark photon can kinetically mix with the
ordinary photon, thus acting as a “portal mediator” between the DS and the SM
[7, 8]. In this framework, the new Lagrangian term extending the SM, omitting
the LDM mass term, reads:

$\mathcal{L}\supset-\frac{1}{4}F^{\prime}_{\mu\nu}F^{\prime\mu\nu}+\frac{1}{2}m^{2}_{A^{\prime}}{A^{\prime}}_{\mu}{A^{\prime}}^{\mu}-\frac{\varepsilon}{2}F_{\mu\nu}F^{\prime\mu\nu}-g_{D}{A^{\prime}}_{\mu}J^{\mu}_{D}$
(1)

where $m_{A^{\prime}}$ is the dark photon mass,
$F^{\prime}_{\mu\nu}\equiv\partial_{\mu}A^{\prime}_{\nu}-\partial_{\nu}A^{\prime}_{\mu}$
is the dark photon field strength tensor, $F_{\mu\nu}$ is the SM
electromagnetic field strength, $g_{D}\equiv\sqrt{4\pi\alpha_{D}}$ is the dark
gauge coupling, $J^{\mu}_{D}$ is the LDM current under $U(1)_{D}$, and
$\varepsilon$ parametrizes the mixing strength. While it is reasonable to
assume that at tree level $g_{D}\sim 1$, the range $\sim 10^{-4}-10^{-2}$
($\sim 10^{-6}-10^{-3}$) is predicted for $\varepsilon$, if the kinetic mixing
is generated at the one (two)-loop level [9, 10, 11]. Cosmological arguments
connected to the DM thermal origin in the early Universe provide a relation
between the measured DM relic density and the model parameters. Specifically,
by introducing the dimensionless parameter
$y\equiv\alpha_{D}\varepsilon^{2}\left(\frac{m_{\chi}}{m_{A^{\prime}}}\right)^{4}$,
with $m_{\chi}$ being the LDM mass, the following relation can be derived
[12]:

$y_{relic}\simeq f\cdot 2\cdot
10^{-14}\left(\frac{m_{\chi}}{1\,\mathrm{MeV}}\right)^{2}\;\;,$ (2)

where $f\sim 1$ is a dimensionless quantity that depends on model specific
details such as the LDM quantum numbers and the $m_{A^{\prime}}/m_{\chi}$
ratio. For a given value of $m_{\chi}$, it follows from Eq. 2 that there is a
target value of $y$ that experiments should probe, resulting in a clear,
predictive target to confirm or rule out the LDM theory [13].

Among the experimental techniques adopted to search for vector-mediated LDM at
accelerators, the missing-energy strategy has proven particularly effective in
the invisible decay scenario ($m_{A^{\prime}}>2m_{\chi}$) [14, 15, 16, 13, 17,
18]. In this approach, the ${A^{\prime}}$ production signature consists in a
large missing energy, i.e. the difference between the nominal beam energy and
the one deposited in the detector, evaluated event-by-event [19]. For an
electron/positron beam setup, two reactions are relevant: the radiative and
the resonant ${A^{\prime}}$ production [20, 21]. The ${A^{\prime}}$ radiative
production by a high-energy $e^{+}/e^{-}$ on a heavy nucleus,
$e^{\pm}Z\rightarrow e^{\pm}Z{A^{\prime}}$, scales as
$\alpha_{EM}^{3}\varepsilon^{2}/m^{2}_{A^{\prime}}$, and it is further
suppressed at large dark photon mass due to the loss of nuclear coherence and
the reduction of the Weizäcker-Williams effective photon flux [22]. For a
narrow ${A^{\prime}}$ ($\alpha_{D}\lesssim 0.1$), the cross section
$\sigma^{P}_{res}$ for the annihilation of the incoming $e^{+}$ with an atomic
$e^{-}$, $e^{+}e^{-}\rightarrow{A^{\prime}}$ at peak reads

$\sigma_{res}^{P}=\frac{1}{\Gamma}\frac{4\pi\alpha_{EM}\varepsilon^{2}}{m_{A^{\prime}}}\;\;,$
(3)

where $\Gamma$ is the total ${A^{\prime}}$ width. As a result, the total
${A^{\prime}}$ yield, obtained by integrating the cross section over the
positrons track-length distribution $T(E_{+})$ [23], scales as
$N_{S}\simeq\Gamma\sigma_{res}^{P}T(E_{R})\propto T(E_{R})/m_{A^{\prime}}$,
where $E_{R}=m^{2}_{A^{\prime}}/(2m_{e})$ is the resonant energy. In a recent
work, we showed how this mechanism leads to a sensitivity improvement for an
electron-beam missing-energy measurement, thanks to the secondary positrons of
the electromagnetic shower developing in the active target [24]; this effect
is maximum for $E_{R}$ close to the missing-energy threshold, and decreases
for $E_{R}\rightarrow E_{0}$ due to the $T(E_{+})$ suppression, where $E_{0}$
is the beam energy. In a positron-beam missing-energy measurement, instead,
thanks to the $T(E_{+})$ enhancement for $E_{+}\rightarrow E_{0}$ associated
to the primary positron, no signal suppression is present, and thus a large
sensitivity to $\varepsilon$ is obtained within the ${A^{\prime}}$ mass range
rigidly delimited by the experimental threshold on the missing energy and the
energy of the beam [20].

Figure 1: Schematic illustration of the NA64 setup to search for invisible
decays of the ${A^{\prime}}$s resonantly produced by the annihilation of the
100 GeV impinging $e^{+}$ with the atomic electrons of the active ECAL target.

NA64 is a missing energy experiment exploiting the $100$ GeV electron beam
from the H4 beamline [25, 24]. The H4 beam is produced by the interaction of
the primary 400 GeV SPS proton beam with a thick Be target; emerging forward-
going photons are pair-converted on a thin lead foil, with a downstream dipole
magnet selecting the charge and the momentum of the particles further
transported downstream. The beam-line can be also operated in “positron mode”,
by appropriately setting the magnets. At 100 GeV/c, the hadronic contamination
in positron (electron) mode is $\simeq 4\%$ ($\simeq 0.3\%$) [26]. Incoming
particles are tagged by a set of three plastic scintillator counters (Sc) and
a veto counter. Their momentum is measured by a magnetic spectrometer,
composed of a set of tracking detectors (GEMs, MicroMegas, and Straw tubes)
installed upstream and downstream a two dipole magnets with total magnetic
strength $\int Bdl\simeq 7\,$T$\,\cdot\,$m [27]. The momentum resolution
$\delta p/p$ is $\simeq 1\%$. Particle identification is achieved by measuring
the synchrotron radiation (SR) emitted by electrons deflected by the magnetic
field through a compact Pb/Sc calorimeter (SRD) [28]. The NA64 active target
is a 40$X_{0}$ Pb/Sc inhomogeneous calorimeter (ECAL), assembled as a $5\times
6$ matrix of $3.82\times 3.82$ cm2 cells with independent PMT readout,
segmented into a $4X_{0}$ pre-shower section (ECAL0) and a main section
(ECAL1). Downstream the ECAL, a hermetic Fe/Sc hadron calorimeter (HCAL), made
by three modules with total length $\simeq 30\lambda_{I}$, is installed, to
detect secondary hadrons and muons produced in the ECAL and upstream detector
elements. A high-efficiency plastic scintillator counter (VETO) is installed
between the ECAL and the HCAL to further suppress backgrounds.

In this work, we present the results of the first dedicated positron-beam
missing-energy measurement for the search of LDM. This effort paves the way
for a future experimental program with positrons, whose sensitivity projection
- evaluated in light of this first measurement - is here presented. The
analysis is based on a total statistics of about $N_{e^{+}OT}=(10.1\pm
0.1)\times 10^{9}$ 100 GeV positrons on target ($e^{+}OT$). A blind-analysis
approach was adopted; all selection cuts were optimized by maximizing the
signal efficiency for the resonant ${A^{\prime}}$ production. The final cuts
configuration required the presence of a well-identified impinging track with
momentum in the range $[97~{}\mathrm{GeV},\mathrm{103}~{}\mathrm{GeV}]$, in
time with a total energy deposition in each SRD cell of at least 2.5 MeV. The
VETO energy was required to be less than 17 MeV for each panel. A 500 MeV
threshold on the ECAL0 energy deposition was applied, and the shape of the
electromagnetic shower in the ECAL was required to be compatible with that
expected for the ${A^{\prime}}$ signal [29]. Finally, the $E_{ECAL}<50~{}$GeV,
$E_{HCAL}<1$ GeV signal region conditions were applied.

The LDM yield in the signal window was estimated through a full Geant4-based
simulation of the NA64 setup, using the DMG4 package [30]. The signal
efficiency was evaluated using data, by applying the analysis cuts on
tracking, SRD, VETO, and HCAL to measured 100 GeV $e^{+}$ events. The ECAL
shower-shape cut was optimized to obtain a 95$\%$ efficiency on measured
events where a high-energy $\mu^{+}\mu^{-}$ pair is produced in the ECAL
mainly from one of the two following production channels: (I) the radiative
production, i.e. a Bremsstrahlung photon converting in the EM field of a
nucleus, $\gamma N\rightarrow\mu^{+}\mu^{-}N$; (II) the atomic annihilation
production, $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$. These “di-muon” events,
given the MIP nature of high-energy muons, feature a signal-like topology from
the point of view of the energy deposition in the different cells of the ECAL.
An additional correction factor, obtained from Monte Carlo (MC), was
introduced for each $m_{A^{\prime}}$ value to account for residual kinematic
differences between signal events and di-muon events. Di-muon events were also
used to assess the overall data normalization: the ratio of the observed di-
muon absolute yield to the MC-predicted one was $\mathcal{F}_{\mu\mu}=(0.84\pm
0.04)$. In the analysis, we added this as an additional global efficiency
factor for the ${A^{\prime}}$ signal. Overall, the average signal efficiency
was $\xi\simeq(55\pm 1)\%$, with a $\simeq 20\%$ dependency on
$m_{A^{\prime}}$ due to the shower-shape cut. The $\xi$ uncertainty was
evaluated by varying the cuts used to identify the impinging $e^{+}$ and
assessing the variation of the obtained efficiency.

Background source | Background, $n_{b}$
---|---
(i) $\pi,~{}K$ decays | $(0.06\pm 0.03)$
(ii) $e^{+}$ hadronic interactions in the beam line | $(0.011\pm 0.007)$
(iii) di-muons | $\leq 0.017$
(iv) $\mu$ decays | $(1.2\pm 0.2)\times 10^{-3}$
(v) $e^{+}$ hadronic interactions in the target | $\ll 10^{-3}$
(vi) hadrons interactions in the target | $\ll 10^{-3}$
Total $n_{b}$ (conservatively) | $(0.09\pm 0.03)$

Table 1: Expected backgrounds for $N_{e^{+}OT}=(10.1\pm 0.1)\times 10^{9}$
$e^{+}OT$.

Background events can originate from different sources, summarized in Tab. 1.
The dominant contribution is from the $K^{+}\rightarrow e^{+}\pi^{0}\nu_{e}$
decay of a kaon contaminant upstream w.r.t. the ECAL, if the neutrino energy
is larger than 50 GeV and the $e^{+}\gamma\gamma$ particles produce a single
low-energy EM shower in the calorimeter. The $K^{+}$ can be misidentified by
the SRD cut if a $\delta$-ray is emitted from the $K^{+}$ through the
interaction with upstream beamline materials and then impinges on the SRD
detector, or if the $K^{+}$ is superimposed in time with a low-energy $e^{+}$
from the beam tail, that emits enough SR and is then deflected away by the
magnets. A lesser contribution comes from the branching-ratio suppressed
$\pi^{+}\rightarrow e^{+}\nu_{e}$ decay. These backgrounds, evaluated from
data, are potentially critical for a positron measurement at the H4 line,
since the hadronic contamination is significantly larger when compared to the
electron beam case [26]. The second most intense background channel is the
upstream hadrons production by the $e^{+}$ interaction with beamline materials
[19], when the low-energy positron is measured by the ECAL, and one or more
high-energy neutral hadrons escape detection. We estimated this contribution
from a larger dataset collected by NA64 in 2022 using a 100 GeV electron beam,
with the same detector configuration and reversed beamline optics, following
the method described in Ref. [29]. The remaining sub-dominant background
sources include: (III) the loss of di-muon events, evaluated mainly from data
by accounting for the VETO inefficiency ($\simeq 1.4\%$) and for the
probability of the di-muons deposited HCAL energy to fall below 1 GeV ($\simeq
5\times 10^{-6}$); (IV) the in-flight decay of a muon contaminant, evaluated
from Monte Carlo simulations; and (V - VI) the probability for a positron or a
hadron contaminant to undergo a hard interaction in the ECAL, depositing
therein less than 50 GeV and producing final-state particles all undetected by
the VETO and the HCAL. Summing up all these contributions, a total background
yield of $B=(0.09\pm 0.03)$ events is expected.

Figure 2: The unblinded ECAL vs HCAL energy distribution for events satisfying
all analysis cuts. The unblinded signal region is also reported – for better
visualization, it has been expanded by a factor $\times 3$ along the vertical
axis.

The ECAL vs HCAL energy distribution for the events satisfying all analysis
cuts is reported in Fig. 2. The diagonal band corresponds to events from the
reaction $e^{+}N\rightarrow e^{+}X$ in the ECAL, with final state hadrons
interacting with the HCAL. After unblinding, no events have been observed in
the signal region. Based on this result, we derived the upper limit on the
${A^{\prime}}$ coupling $\varepsilon$ as a function of $m_{A^{\prime}}$. We
adopted a frequentist approach, considering the $90\%$ Confidence Level (CL)
of a one-sided profile-likelihood test statistics [31]. The likelihood model
was built assuming a Poisson PDF for the number of events in the signal
region, with mean value $\mu=S+B=(\varepsilon/\varepsilon_{0})^{2}S_{0}+B$,
where $S_{0}$ is the signal yield for the nominal coupling value
$\varepsilon_{0}$ obtained from Monte Carlo.

Figure 3:  The new exclusion limits from the positron-beam, missing energy
measurement presented in this work in the $\varepsilon$ vs $m_{A^{\prime}}$
space, considering fermionic LDM for $\alpha_{D}=0.1$. The most stringent LDM
exclusion limits from BaBar [32] and NA64 [33] are also shown, as well as the
favored area from the muon $g-2$ anomaly [34, 35] (red lines). The green
dashed lines reports the sensitivity for a future positron-beam effort at
lower energy, as described in the text. For a full review on other exclusion
limits and planned measurements see Ref. [14, 15, 16, 13, 17, 18].

The systematic uncertainties were accounted for in the statistical procedure
by introducing, for each contribution, an additional log-normal PDF term in
the Likelihood, taking the measured value as the observed one, and handling
the expected value as a nuisance parameter [31]. The uncertainty on the number
of expected background events (30$\%$) and that on the signal yield from the
number of $e^{+}OT$ ($0.1\%$), from the tracking, SRD, VETO, and HCAL cuts
($\simeq 1\%$), and from the overall normalization as obtained from the di-
muons analysis ($4.8\%$) are the same for all $m_{A^{\prime}}$ values. In
contrast, the systematic uncertainty arising from the ECAL energy calibration,
and thus affecting the pre-shower and missing energy thresholds, result in a
$m_{A^{\prime}}$-dependent effect. For the missing-energy threshold, we
estimated $\delta_{E}/E\simeq 1.5\%$ by measuring the position of the full-
energy peak in the calorimeter across the different runs. For the pre-shower
threshold, instead, we evaluated $\delta_{E}/E\simeq 1.0\%$ from the analysis
of di-muon events. To evaluate the impact of $\delta_{E}/E$ on $S_{0}$, first
we computed the signal yield for the nominal threshold values. Then, we
randomly sampled these multiple times from two Gaussian distributions with
$\mu=50$ GeV and $\sigma=0.75$ GeV and $\sigma=0.5$ GeV, respectively,
computing $S_{0}$ for each configuration and then taking the RMS of all
results as the systematic uncertainty. The largest systematic uncertainty,
$\simeq 25\%$, is observed for the dark photon resonant production at
$m_{A^{\prime}}\simeq 225$ MeV, since in this case the resonant energy is
close to the missing energy threshold.

The obtained upper limit on $\varepsilon$ as a function of $m_{A^{\prime}}$ is
shown in Fig. 3 for fermionic LDM, with $\alpha_{D}=0.1$ and
$m_{A^{\prime}}=3m_{\chi}$. In the same figure, the combined projection for a
future measurement program at two different beam energies - 60 GeV and 40 GeV
- is presented. For both measurements, a statistics of $10^{11}\,\,e^{+}$OT
was considered, assuming a missing energy threshold corresponding to half the
energy of the beam and a $50\%$ signal efficiency, in analogy with the
strategy adopted for the 100 GeV data analysis. Limits were evaluated
considering no expected background events; this assumption is corroborated by
the fact that the main background source for such measurements comes from the
hadronic contaminant fraction in the beam, whose intensity wanes at decreasing
beam energy [26]. The same results are reported in Fig. 4 as a function of
$y$. The black lines correspond to the aforementioned “targets” predicted by
cosmology, in the hypothesis of a thermal LDM origin. To assess the variation
of our result as a function of the LDM model parameters, we repeated the upper
limit calculation considering also a scalar LDM model and the two $\alpha_{D}$
values, 0.1 and 0.5.

Figure 4: The new NA64 positron-beam exclusion limit in the $(m_{\chi},y)$
plane, for $\alpha_{D}=0.1$ (left) and $\alpha_{D}=0.5$ (right). The other
curves and shaded areas report already-existing limits in the same parameter
space from NA64 in electron-beam mode [33], COHERENT [36], and BaBar [32]. The
black lines show the favoured parameter combinations for the observed DM relic
density for different model variations [12]. In each panel, the green dashed
line shows the sensitivity for a future positron-beam effort at lower energy.

In conclusion, we performed the first positron-beam missing-energy measurement
searching for LDM. The resulting limits touch our latest electron-beam
results, corresponding to $\sim$2 orders of magnitude larger electron
statistics [26]. This proves the outstanding potential of NA64 to probe dark
sectors using the unique H4 positron beam at the CERN SPS, and motivates a
multi-energy measurement program to fully exploit the peculiarities of the
resonant ${A^{\prime}}$ production to “scan” the LDM parameter space. Such
experimental program would allow to explore a significant region of the LDM
parameter space, in synergy with our well established electron-beam
measurements.

###### Acknowledgements.

We gratefully acknowledge the support of the CERN management and staff, and
the technical staffs of the participating institutions for their vital
contributions. This result is part of a project that has received funding from
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme, Grant agreement No. 947715 (POKER). This
work was supported by the HISKP, University of Bonn (Germany), ETH Zurich and
SNSF Grant No. 169133, 186181, 186158, 197346 (Switzerland), and ANID -
Millenium Science Initiative Program - ICN2019 044 (Chile), and RyC-030551-I
and PID2021-123955NA-100 funded by MCIN/AEI/ 10.13039/501100011033/FEDER, UE
(Spain).

## References

  * Bertone and Hooper [2018] G. Bertone and D. Hooper, Rev. Mod. Phys. 90, 045002 (2018), arXiv:1605.04909 [astro-ph.CO] .
  * Liddle [1998] A. R. Liddle,  _An introduction to modern cosmology_ (1998).
  * Arcadi  _et al._ [2018] G. Arcadi, M. Dutra, P. Ghosh, M. Lindner, Y. Mambrini, M. Pierre, S. Profumo, and F. S. Queiroz, Eur. Phys. J. C 78, 203 (2018), arXiv:1703.07364 [hep-ph] .
  * Arkani-Hamed  _et al._ [2009] N. Arkani-Hamed, D. P. Finkbeiner, T. R. Slatyer, and N. Weiner, Phys. Rev. D 79, 015014 (2009), arXiv:0810.0713 [hep-ph] .
  * Strassler and Zurek [2007] M. J. Strassler and K. M. Zurek, Phys. Lett. B 651, 374 (2007), arXiv:hep-ph/0604261 .
  * Andreas  _et al._ [2013] S. Andreas, M. D. Goodsell, and A. Ringwald, Phys. Rev. D 87, 025007 (2013), arXiv:1109.2869 [hep-ph] .
  * Holdom [1986] B. Holdom, Phys. Lett. B 166, 196 (1986).
  * Knapen  _et al._ [2017] S. Knapen, T. Lin, and K. M. Zurek, Phys. Rev. D 96, 115021 (2017), arXiv:1709.07882 [hep-ph] .
  * Essig  _et al._ [2010] R. Essig, J. Kaplan, P. Schuster, and N. Toro, arXiv:1004.0691 [hep-ph] (2010).
  * del Aguila  _et al._ [1988] F. del Aguila, G. D. Coughlan, and M. Quiros, Nucl. Phys. B 307, 633 (1988), [Erratum: Nucl. Phys. B 312, 751(E) (1989)].
  * Arkani-Hamed and Weiner [2008] N. Arkani-Hamed and N. Weiner, JHEP 12, 104, arXiv:0810.0714 [hep-ph] .
  * Berlin  _et al._ [2019] A. Berlin, N. Blinov, G. Krnjaic, P. Schuster, and N. Toro, Phys. Rev. D 99, 075001 (2019), arXiv:1807.01730 [hep-ph] .
  * Battaglieri  _et al._ [2017] M. Battaglieri _et al._ , in  _U.S. Cosmic Visions: New Ideas in Dark Matter_ (2017) arXiv:1707.04591 [hep-ph] .
  * Fabbrichesi  _et al._ [2020] M. Fabbrichesi, E. Gabrielli, and G. Lanfranchi (2020) SpringerBrief in Physics, arXiv:2005.01515 [hep-ph] .
  * Filippi and De Napoli [2020] A. Filippi and M. De Napoli, Rev. Phys. 5, 100042 (2020), arXiv:2006.04640 [hep-ph] .
  * Beacham  _et al._ [2020] J. Beacham _et al._ , J. Phys. G 47, 010501 (2020), arXiv:1901.09966 [hep-ex] .
  * Ilten  _et al._ [2022] P. Ilten _et al._ , in  _Snowmass 2021_ (2022) arXiv:2206.04220 [hep-ex] .
  * Graham  _et al._ [2021] M. Graham, C. Hearty, and M. Williams (2021) arXiv:2104.10280 [hep-ph] .
  * Gninenko  _et al._ [2016] S. N. Gninenko, N. V. Krasnikov, M. M. Kirsanov, and D. V. Kirpichnikov, Phys. Rev. D 94, 095025 (2016), arXiv:1604.08432 [hep-ph] .
  * Marsicano  _et al._ [2018a] L. Marsicano, M. Battaglieri, M. Bondí, C. D. R. Carvajal, A. Celentano, M. De Napoli, R. De Vita, E. Nardi, M. Raggi, and P. Valente, Phys. Rev. Lett. 121, 041802 (2018a), arXiv:1807.05884 [hep-ex] .
  * Marsicano  _et al._ [2018b] L. Marsicano, M. Battaglieri, M. Bondi’, C. D. R. Carvajal, A. Celentano, M. De Napoli, R. De Vita, E. Nardi, M. Raggi, and P. Valente, Phys. Rev. D 98, 015031 (2018b), arXiv:1802.03794 [hep-ex] .
  * Izaguirre  _et al._ [2013] E. Izaguirre, G. Krnjaic, P. Schuster, and N. Toro, Phys. Rev. D 88, 114015 (2013), arXiv:1307.6554 [hep-ph] .
  * Chilton [1978] A. Chilton, Health Physics 34, 715 (1978).
  * Andreev  _et al._ [2021] Y. M. Andreev _et al._ , Phys. Rev. D 104, L091701 (2021), arXiv:2108.04195 [hep-ex] .
  * Banerjee  _et al._ [2019] D. Banerjee _et al._ , Phys. Rev. Lett. 123, 121801 (2019), arXiv:1906.00176 [hep-ex] .
  * Andreev  _et al._ [2023a] Y. M. Andreev _et al._ , (2023a), arXiv:2305.19411 [hep-ex] .
  * Banerjee  _et al._ [2018a] D. Banerjee _et al._ , Nucl. Instrum. Meth. A 881, 72 (2018a), arXiv:1708.04087 [physics.ins-det] .
  * Depero  _et al._ [2017] E. Depero _et al._ , Nucl. Instrum. Meth. A 866, 196 (2017), arXiv:1703.05993 [physics.ins-det] .
  * Banerjee  _et al._ [2018b] D. Banerjee _et al._ (NA64), Phys. Rev. D 97, 072002 (2018b), arXiv:1710.00971 [hep-ex] .
  * Bondi  _et al._ [2021] M. Bondi, A. Celentano, R. R. Dusaev, D. V. Kirpichnikov, M. M. Kirsanov, N. V. Krasnikov, L. Marsicano, and D. Shchukin, Comput. Phys. Commun. 269, 108129 (2021), arXiv:2101.12192 [hep-ph] .
  * Gross [2007] E. Gross, in  _PHYSTAT-LHC Workshop on Statistical Issues for LHC Physics_ (2007).
  * Lees  _et al._ [2017] J. P. Lees _et al._ (BaBar), Phys. Rev. Lett. 119, 131804 (2017), arXiv:1702.03327 [hep-ex] .
  * Andreev  _et al._ [2023b] Y. M. Andreev _et al._ , (2023b), arXiv:2307.02404 [hep-ex] .
  * Pospelov [2009] M. Pospelov, Phys. Rev. D 80, 095002 (2009), arXiv:0811.1030 [hep-ph] .
  * Abi  _et al._ [2021] B. Abi _et al._ (Muon g-2), Phys. Rev. Lett. 126, 141801 (2021), arXiv:2104.03281 [hep-ex] .
  * Akimov  _et al._ [2023] D. Akimov _et al._ (COHERENT), Phys. Rev. Lett. 130, 051803 (2023), arXiv:2110.11453 [hep-ex] .