Search for light long-lived neutral particles that decay to collimated pairs
of leptons or light hadrons in $pp$ collisions at $\sqrt{s}=13$ TeV with the
ATLAS detector A search for light long-lived neutral particles with masses in
the O(MeV–GeV) range is presented. The analysis targets the production of
long-lived dark photons in the decay of a Higgs boson produced via gluon–gluon
fusion or in association with a $W$ boson. Events that contain displaced
collimated Standard Model fermions reconstructed in the calorimeter or muon
spectrometer are selected in 139 fb-1 of $\sqrt{s}=13$ TeV $pp$ collision data
collected by the ATLAS detector at the LHC. Background estimates for
contributions from Standard Model processes and instrumental effects are
extracted from data. The observed event yields are consistent with the
expected background. Exclusion limits are reported on the production cross-
section times branching fraction as a function of the mean proper decay length
$c\tau$ of the dark photon, or as a function of the dark-photon mass and
kinetic mixing parameter that quantifies the coupling between the Standard
Model and potential hidden (dark) sectors. A Higgs boson branching fraction
above 1% is excluded at 95% CL for a Higgs boson decaying into two dark
photons for dark-photon mean proper decay lengths between 10 mm and 250 mm and
dark photons with masses between 0.4 GeV and 2 GeV. EXOT-2019-05 CERN-
EP-2022-054 JHEP

##  1 Introduction

Dark sectors weakly coupled to the Standard Model (SM) are predicted in
several theories of physics beyond the Standard Model (BSM) [1, 2, 3, 4]. Such
theories often include a complex spectrum of new particles that form the dark
sector and can couple to the SM only via specific portals. Depending on the
structure of the dark sector and its coupling to the SM, unstable dark states
may be produced at colliders, and could decay into SM particles with sizeable
branching fractions.

This paper investigates the case where the SM and dark sectors communicate
through the Higgs portal for the production of the BSM states and through a
vector portal for the decay of the produced particles. In particular, a dark
photon ($\gamma_{\mathrm{d}}$) is assumed to mix kinetically with the SM
photon and decay into leptons and light quarks [5, 6, 7]. The kinetic mixing
parameter ($\epsilon$), which is theoretically allowed to vary over a wide
range of values, determines the lifetime of the dark photon. This analysis
focuses on small values of the kinetic mixing parameter, $\epsilon<10^{-5}$,
corresponding to $\gamma_{\mathrm{d}}$ decays happening at a macroscopic
distance from their production point, and a dark-photon mass in the O(MeV–GeV)
range. The decay branching fractions of a light dark photon that mixes
kinetically with the SM photon depend on its mass [6, 8, 9]. Due to their
small mass compared to the energy scale of the hard-scattering process, the
dark photons are expected to be produced with large Lorentz boosts, resulting
in collimated groups of fermions in a jet-like structure, referred to
hereafter as dark-photon jets (DPJs).

Two mutually exclusive search categories, referred to as gluon–gluon fusion
selection and WH associated production selection, are designed to target the
production of dark photons in these two Higgs boson production modes. Two
models are used for the optimisation of the event selections and the
interpretation of the final results: the Falkowski–Ruderman–Volansky–Zupan
(FRVZ) model [6, 7] and the Hidden Abelian Higgs Model (HAHM) [4]. In the FRVZ
model, a pair of dark fermions $f_{\mathrm{d}}$ is produced via a Higgs boson
decay and can lead to final states with either two (Figure 1(a)) or four
(Figure 1(b)) dark photons. In the two dark photon case, $f_{\mathrm{d}}$
decays into a dark photon and a stable dark fermion, which is assumed to be
the hidden lightest stable particle (HLSP). Whereas in the four dark photon
case, $f_{\mathrm{d}}$ decays into an HLSP and a dark scalar $s_{\textrm{d}}$
that in turn decays into a pair of $\gamma_{\mathrm{d}}$. The HAHM model
predicts a direct decay of the Higgs boson into a pair of
$\gamma_{\mathrm{d}}$, as shown in Figure 1(c). The mean proper lifetime
$\tau$ of the $\gamma_{\mathrm{d}}$ is inversely related to the mass of the
dark photon and to the square of the kinetic mixing parameter [10, 9].

((a)) FRVZ 2$\gamma_{\mathrm{d}}$ ((b)) FRVZ 4$\gamma_{\mathrm{d}}$

((c)) HAHM 2$\gamma_{\mathrm{d}}$

Figure 1: (a) FRVZ process with the dark fermion $f_{\mathrm{d}}$ decaying
into a $\gamma_{\mathrm{d}}$ and an HLSP. (b) FRVZ process with the dark
fermion $f_{\mathrm{d}}$ decaying into an HLSP and a dark scalar
$s_{\textrm{d}}$ that in turn decays into a pair of dark photons. (c) HAHM
process with the two $\gamma_{\mathrm{d}}$ directly produced by the Higgs
boson ($H$). The $\gamma_{\mathrm{d}}$ decays into SM fermions, denoted by $f$
and $\bar{f}$.

In general, dark-sector radiation [11] can produce extra dark photons
proportionally to the size of the dark gauge coupling $\alpha_{\textrm{d}}$
[5]. In this work a dark coupling $\alpha_{\textrm{d}}\lesssim 0.01$ is
assumed, leading to no significant additional $\gamma_{\mathrm{d}}$ emission.
For this reason, no specific interpretations are given for these scenarios.

The search for displaced DPJs presented in this paper uses the dataset
collected at the Large Hadron Collider (LHC) by the ATLAS detector during
2015–2018 in proton–proton ($pp$) collisions at a centre-of-mass energy
$\sqrt{s}=13$ TeV, corresponding to an integrated luminosity of 139 fb-1.
Previous searches for displaced DPJs were performed by ATLAS using
proton–proton collision data at lower centre-of-mass energies [12, 13] or a
partial dataset corresponding to 36.1 fb-1 of proton–proton collision data at
a centre-of-mass energy of $\sqrt{s}=13$ TeV [14]. The results are
complementary to those from related ATLAS searches for prompt DPJs [15, 16,
17], which probed higher values of $\epsilon$, and for displaced dimuon
vertices using 13 TeV data [18], which probed higher dark-photon mass values.
Related searches for dark photons were conducted by the CDF and D0
collaborations at the Tevatron [19, 20, 21] and by the CMS [22, 23, 24, 25]
and LHCb [26, 27] collaborations at the LHC. Additional constraints on
scenarios with dark photons are extracted from, e.g., beam-dump and fixed-
target experiments [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], $e^{+}e^{-}$
colliders [39, 40, 41, 42, 43, 44, 45, 46, 47], electron and muon anomalous
magnetic moment measurements [48, 49, 50] and astrophysical observations [51,
52]. Given the various constraints, a displaced dark-photon decay with a
kinetic mixing parameter $\epsilon<10^{-5}$ is allowed for
$\gamma_{\mathrm{d}}$ masses greater than 10 MeV. In this paper, the
sensitivity to displaced DPJs is significantly higher than in the previous
ATLAS search [14], due to the higher integrated luminosity of the dataset and
to new analysis methods. The new analysis methods include updated signal-
region selection criteria and DPJ reconstruction techniques. Multivariate
techniques based on convolutional neural networks, exploiting the three-
dimensional representations of calorimeter energy deposits associated with
jets, and a dense neural network, using muon track information, are employed
to indentify DPJ candidates. The addition of a new set of event selection
criteria, targeting events where the dark photons arise from the decay of a
scalar particle produced in association with a $W$ boson, significantly
increases the sensitivity in scenarios with $\gamma_{\mathrm{d}}$ mean proper
decay lengths $c\tau$ below (above) 3 (40) mm.

##  2 ATLAS detector

The ATLAS detector [53] at the LHC covers nearly the entire solid angle around
the collision point.111 ATLAS uses a right-handed coordinate system with its
origin at the nominal interaction point (IP) in the centre of the detector and
the $z$-axis along the beam pipe. The $x$-axis points from the IP to the
centre of the LHC ring, and the $y$-axis points upwards. Cylindrical
coordinates $(r,\phi)$ are used in the transverse plane, $\phi$ being the
azimuthal angle around the $z$-axis. The pseudorapidity is defined in terms of
the polar angle $\theta$ as $\eta=-\ln\tan(\theta/2)$, and the rapidity is
defined as $y=(1/2)[(E+p_{z})/(E-p_{z})]$. It consists of an inner tracking
detector (ID) surrounded by a thin superconducting solenoid, electromagnetic
(EM) and hadron (HCAL) calorimeters, and a muon spectrometer (MS)
incorporating three large superconducting air-core toroidal magnets.

The inner tracking detector covers the pseudorapidity range $|\eta|<2.5$. It
consists of silicon pixel, silicon microstrip, and transition radiation
tracking detectors.

The calorimeter system covers the pseudorapidity range $|\eta|<4.9$. Within
the region $|\eta|<3.2$, electromagnetic calorimetry is provided by barrel and
endcap high-granularity lead/liquid-argon (LAr) calorimeters, with an
additional thin LAr presampler covering $|\eta|<1.8$ to correct for energy
loss in material upstream of the calorimeters. Hadron calorimetry is provided
by the steel/scintillator-tile calorimeter, segmented into a central barrel
and two extended-barrel cylindrical structures within $|\eta|<1.7$, and two
copper/LAr hadron endcap calorimeters. The solid angle coverage is completed
with forward copper/LAr and tungsten/LAr calorimeter modules optimised for
electromagnetic and hadronic energy measurements respectively. Overall, the
calorimeter system is highly segmented in shower depth, with a total of seven
sampling layers in the barrel region and eight in the endcap regions.

The muon spectrometer comprises separate trigger and high-precision tracking
chambers measuring the deflection of muons in a magnetic field generated by
the superconducting air-core toroidal magnets. The field integral of the
toroids ranges between $2.0$ and $6.0\text{\,}\mathrm{T}\text{\,}\mathrm{m}$
across most of the detector. A set of precision chambers covers the region
$|\eta|<2.7$ with three layers of monitored drift tubes, complemented by
cathode-strip chambers in the forward region, where the background is highest.
The muon trigger system covers the range $|\eta|<2.4$ with resistive-plate
chambers in the barrel, and thin-gap chambers in the endcap regions.

A two-level trigger system is used to select events [54]. The first-level (L1)
trigger is implemented in hardware and uses a subset of the detector
information to accept events at a rate of at most $100\text{\,}\mathrm{kHz}$.
This is followed by a software-based high-level trigger (HLT) that reduces the
accepted event rate to $1\text{\,}\mathrm{kHz}$ on average, depending on the
data-taking conditions.

An extensive software suite [55] is used in the reconstruction and analysis of
real and simulated data, in detector operations, and in the trigger and data
acquisition systems of the experiment.

##  3 Data and simulated event samples

The data were collected by the ATLAS detector during Run 2 of the LHC
(2015–2018) with a peak instantaneous luminosity of $2.1\times 10^{34}$
cm-2s-1, resulting in a mean number of $pp$ interactions per bunch crossing of
$\langle\mu\rangle=34$. Data quality requirements [56] are applied to ensure
that all subdetectors were operating normally, and that the LHC beams were in
stable-collision mode. The integrated luminosity of the resulting data sample
is 139 fb-1. The data were collected using a set of triggers specifically
designed to target the displaced $\gamma_{\mathrm{d}}$ decays, described in
detail in Section 5.1, or single-lepton triggers, with requirements on the
identification, isolation, and $p_{\text{T}}$ of the leptons to maintain
efficiency across the full momentum range while controlling the trigger rates
[57, 58].

During $pp$ collisions data-taking the LHC circulates two counter-rotating
proton beams constructed from bunches of protons. However, following LHC
injection not all bunch slots are filled with protons, with the number of
unfilled bunches depending on the LHC filling scheme [59]. An empty bunch
crossing takes place when neither beam is filled with protons and each empty
bunch is separated from filled bunches by at least five empty bunches on each
side. A dataset enriched in cosmic-ray muon background is collected during
empty bunch crossings (the ‘cosmic dataset’) and used for the background
estimation.

Monte Carlo (MC) simulated event samples are used to model the BSM signals.
Signal samples modelling the production of dark photons via a 125 GeV Higgs
portal were generated with MadGraph5_aMC@NLO 2.2.3 [60] interfaced to Pythia
8.186 [61] for the parton showering (PS) and hadronisation. The matrix-element
calculation was performed at tree level. The parton distribution function
(PDF) set used for the generation is NNPDF2.3lo [62]. Higgs boson production
via gluon–gluon fusion and in association with a $W$ boson is included. The
predicted Standard Model cross-sections for these two processes, assuming
$m_{H}=$ 125.09 GeV, are respectively 48.61 pb [63, 64] and 1.369 pb [65, 66].
A second set of signal samples was generated, modelling the production via
gluon–gluon fusion of a high-mass (800 GeV) Higgs-like scalar mediator with
the same decay modes as in the 125 GeV mass case. Effects of higher-order QCD
corrections on the $p_{\text{T}}$ of the Higgs boson, evaluated using a
reweighting procedure [67], change the signal selection efficiency by less
than the MC statistical accuracy and are therefore neglected. A dark photon
with a mass $m_{\gamma_{\textrm{d}}}$ up to a few GeV that mixes kinetically
with the SM photon will decay into leptons or light quarks, with branching
fractions that depend on its mass [6, 8, 9]. Taking as an example a dark-
photon mass of 0.4 GeV, the $\gamma_{\mathrm{d}}$ decay branching ratios are
expected to be 45$\%\leavevmode\nobreak\ e^{+}e^{-}$,
45$\%\leavevmode\nobreak\ \mu^{+}\mu^{-}$, and 10$\%\leavevmode\nobreak\
q\bar{q}$ [6]. The mean proper decay length $c\tau$ of the
$\gamma_{\mathrm{d}}$ is a free parameter of the model. In the generated
samples, $c\tau$ was chosen such that, accounting for the boost of the
$\gamma_{\mathrm{d}}$, a large fraction of the decays occur inside the
sensitive ATLAS detector volume (i.e. up to 7 m in radius and 13 m along the
$z$-axis from the centre of the detector). The decays of the Higgs boson into
dark photons through dark fermions or directly into two dark photons were
simulated at matrix-element level during the generation. In the FRVZ model,
the mass of $f_{\mathrm{d}}$ was chosen to be small relative to the Higgs
boson mass, and far from the kinematic threshold at
$m_{\mathrm{HLSP}}+m_{\gamma_{\mathrm{d}}}=m_{f_{\mathrm{d}}}$. In the HAHM,
the Higgs boson decays directly into two dark photons.

The SM background is estimated using data-driven techniques, with MC simulated
events used to aid in the validation, the evaluation of uncertainties, and the
training of dedicated multivariate classifiers. SM processes that could be
potential sources of background include multi-jet, W+jets, Z+jets, $t\bar{t}$,
single-top-quark, WW, WZ, and ZZ production. The multi-jet samples were
generated with Pythia 8.210 [68] using the same set of tuned parameter values
(tune) and PDF as for the signal samples. Samples of W+jets, Z+jets, WW, WZ,
and ZZ events were generated using Sherpa 2.2.1 [69] with the NNPDF3.0nnlo
[70] PDF set. Single-top and $t\bar{t}$ MC samples were generated using Powheg
Box v2 [71] and Pythia 8.230 with the A14 tune [72] for parton showering and
hadronisation, and the NNPDF2.3lo set of PDFs.

Finally, MC samples of $J/\psi\to\mu\mu$ events were generated and used to
evaluate systematic uncertainties for muon trigger and reconstruction
efficiencies. The MC samples were generated using Pythia8+Photos++ [73] with
the A14 tune for parton showering and hadronisation, and the CTEQ6L1 [74, 75]
PDF set.

All generated Monte Carlo events were processed through a full simulation of
the ATLAS detector geometry and response [76] using the Geant4 [77] toolkit.
The simulation includes multiple $pp$ interactions per bunch crossing (pile-
up), as well as the detector response to interactions in bunch crossings
before and after the one producing the hard interaction. In order to model the
effects of pile-up, simulated inclusive $pp$ events were overlaid on each
generated hard-scatter event and reweighted to match the conditions of the
2015–2018 data sample. The inclusive $pp$ events were simulated with Pythia
8.210 with the A3 tune [78] and the NNPDF2.3lo set of PDFs.

##  4 Event reconstruction

Candidate events are required to have a reconstructed vertex [79] with at
least two associated tracks with transverse momentum ($p_{\text{T}}$) larger
than 500 MeV that are consistent with originating from the beam collision
region in the $x$–$y$ plane. The primary vertex in the event is the vertex
with the highest sum of squared transverse momenta of associated tracks.

Electron candidates are reconstructed from isolated electromagnetic
calorimeter energy deposits matched to ID tracks. They are required to have
$|\eta|<2.47$, a transverse momentum $p_{\text{T}}>$20\text{\,}\mathrm{GeV}$$,
and to satisfy the ‘TightLHElectron’ requirement defined in Ref. [80], which
is based on a likelihood evaluated using measurements of shower shapes in the
calorimeter and track properties in the ID as input variables. Candidates
within the transition region between the barrel and endcap electromagnetic
calorimeters, $1.37<|\eta|<1.52$, are not considered.

Muon candidates are reconstructed in the region $|\eta|<2.5$ from MS tracks
matching ID tracks. They are required to have
$p_{\text{T}}>$20\text{\,}\mathrm{GeV}$$ and satisfy the ‘medium’
identification requirements defined in Ref. [81]. These requirements are based
on the number of hits in the different ID and MS subsystems, and on the ratio
of the charge and momentum ($q/p$) measured in the ID and MS divided by the
sum in quadrature of their corresponding uncertainties.

The tracks associated with the electron and muon candidates are required to
have a significance of the transverse impact parameter,222The transverse
impact parameter, $d_{0}$, is defined as the distance of closest approach of a
track to the beam-line, measured in the transverse plane. The longitudinal
impact parameter, $z_{0}$, corresponds to the $z$-coordinate distance between
the point along the track at which the transverse impact parameter is defined
and the primary vertex. $d_{0}$, of $|d_{0}|/\sigma(d_{0})<5$ for electrons
and $|d_{0}|/\sigma(d_{0})<3$ for muons, and a longitudinal impact parameter,
$z_{0}$, satisfying $|z_{0}\sin\theta|<0.5$ mm.

Isolation criteria are applied to electrons and muons. The scalar sum of the
$p_{\text{T}}$ of tracks within a variable-size cone around the electron
(muon), must be less than 15% of the lepton $p_{\text{T}}$, excluding tracks
associated with the lepton. The track isolation cone size for electrons
(muons) $\Delta R=\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$ is given by the
smaller of $\Delta R=$10\text{\,}\leavevmode\nobreak\
\mathrm{G}\mathrm{e}\mathrm{V}$/p_{\text{T}}$ and $\Delta R=0.2\,(0.3)$. In
addition, the sum of the transverse energy of clusters of calorimeter cells
[82] in a cone of $\Delta R=0.2$ around the electron (muon) must be less than
20% (30%) of the lepton $p_{\text{T}}$, excluding clusters associated with the
lepton.

Jets are reconstructed from three-dimensional energy clusters in the
calorimeter [83] using the anti-$k_{t}$ jet clustering algorithm [84, 85] with
a radius parameter $R=0.4$. Only jet candidates with
$p_{\text{T}}>$20\text{\,}\mathrm{GeV}$$ and $|\eta|<4.9$ are considered. Jets
are calibrated using MC simulation, with corrections obtained from in situ
techniques [86]. To reduce the effects of pile-up, jets with
$p_{\text{T}}<$120\text{\,}\mathrm{GeV}$$ and $|\eta|<2.5$ are required to
have a significant fraction of their associated tracks compatible with
originating from the primary vertex, as defined by the jet vertex tagger [87]
(JVT). This requirement reduces the fraction of jets from pile-up to 1%, with
an efficiency for hard-scatter jets of about 90%. Jets not satisfying basic
quality criteria designed to reject detector noise and non-collision
backgrounds [88] are then discarded.

Jets containing $b$-hadrons ($b$-tagging) are identified with a multivariate
discriminant that makes use of track properties [89, 90]. Jets are considered
to be $b$-tagged if they fulfil a requirement that has 70% average efficiency
for jets containing $b$-hadrons in simulated $t\bar{t}$ events. The rejection
factors for light-quark and gluon jets, jets containing $c$-hadrons, and
hadronically decaying $\tau$-leptons in simulated $t\bar{t}$ events are
approximately 301, 38, and 8, respectively.

Simulated events are corrected for differences from collision data in
$b$-tagging efficiencies and $b$-tagging mis-tag rates [90, 91, 92].
Corrections are also applied to account for minor differences between data and
MC simulation in the single-lepton trigger, reconstruction, identification and
isolation efficiencies [93, 94].

Jet candidates within an angular distance $\Delta R^{\prime}=\sqrt{(\Delta
y)^{2}+(\Delta\phi)^{2}}=0.2$ of a lepton candidate are discarded. Remaining
lepton candidates within $\Delta
R^{\prime}=\operatorname{min}\\{0.4,0.04+p_{\text{T}}(\mu)/($10\text{\,}\mathrm{GeV}$)\\}$
of a jet are then discarded to suppress bottom and charm hadron decays. When
considering muons, if the jet has fewer than three associated tracks, the muon
is retained and the jet is discarded instead to avoid inefficiencies for high-
energy muons undergoing significant energy loss in the calorimeter. Finally,
any electron candidate sharing an ID track with a remaining muon candidate is
also removed.

The missing transverse momentum vector
$\mathbf{p_{\mathrm{T}}^{\mathrm{miss}}}$, whose magnitude is denoted by
$E_{\text{T}}^{\text{miss}}$, is defined as the negative vector sum of the
transverse momenta of all identified electrons, muons and jets, plus an
additional soft term. The soft term is constructed from all tracks that
originate from the primary vertex but are not associated with any identified
lepton or jet. In this way, the $E_{\text{T}}^{\text{miss}}$ is adjusted for
the best calibration of leptons and jets, while contributions from pile-up
interactions are suppressed through their exclusion from the soft term [80,
81].

A displaced $\gamma_{\mathrm{d}}$ decay producing a pair of muons is expected
to leave two or more collimated stand-alone MS tracks, referred to as a muonic
dark-photon jet $\mu$DPJ. Stand-alone MS tracks [95] are reconstructed in the
pseudorapidity region $|\eta|\geq 0.1$ and formed by requiring at least two
matched segments in the MS, and are fit with a primary vertex constraint.
Candidates with pseudorapidity in the range $1.0\leq|\eta|\leq 1.1$ are
rejected to avoid the transition region of the MS between barrel and endcap.
Moreover, only stand-alone MS tracks in the pseudorapidity interval
$|\eta|<2.4$, corresponding to the ID coverage, are selected to allow an
isolation variable based on ID tracks to be computed. Stand-alone MS tracks
are required to not match any prompt muon candidate, in order to discard muons
coming from the main interaction vertex.

A $\gamma_{\mathrm{d}}$ decaying into a displaced electron or quark pair leads
to energy deposits in the calorimeters reconstructed as a single jet with low
EM fraction (EMF), defined as the ratio of the energy deposited in the EM
calorimeter to the total jet energy. Jets with EMF below 0.4 are referred to
as calorimeter dark-photon jet caloDPJ. MC simulations show that DPJs
containing two dark photons both decaying into an electron or quark pair are
reconstructed as a single jet. Low-EM-fraction jets are reconstructed and
calibrated with the same algorithms as described previously. They are,
however, only considered if they have $p_{\text{T}}>$20\text{\,}\mathrm{GeV}$$
and lie within $|\eta|<2.5$. Furthermore, to retain high efficiency for the
targeted signals, they are required to satisfy quality criteria looser than
those in the main jet selection and no JVT selection is applied. To avoid
selecting events where most of the energy associated with a jet could have
been produced by localised noise, events are rejected if the leading jet has
${>}90$% of its energy associated with a single constituent cluster or layer
within the LAr calorimeter. Potential background from noise bursts [96] in the
LAr calorimeter is rejected via a veto on events where the leading jet’s
largest energy deposit is located in the EM calorimeter endcap, where most
noise bursts occur. Noise-induced jets in the hadron calorimeters are removed
by imposing the BadLoose cleaning selection, described in Ref. [88], without
the cuts on the fraction of jet energy deposited within the electromagnetic
calorimeter and the jet charged-fraction. These cleaning requirements reject
approximately 0.8% of low-EM-fraction jets in the signal samples used in this
analysis. Finally, the time $t_{\mathrm{caloDPJ}}$ associated with a caloDPJ,
measured as the energy-weighted average of the timing for each calorimeter
cell related to the jet and corrected by the corresponding time-of-flight from
the interaction point, is required to fall within a window of 4 ns around
zero. CaloDPJs from cosmic-ray muons and beam-induced backgrounds (BIB) would
have a different jet-timing distribution than collision products.

Dedicated algorithms for the identification of DPJs have been developed to
target $\gamma_{\mathrm{d}}$ decays into $\mu$DPJs or caloDPJs. These
algorithms are described in Sections 4.1 and 4.2 respectively.

###  4.1 Muonic dark-photon jets

Muonic dark-photon jets are reconstructed using the Cambridge–Aachen
clustering algorithm [97] that combines all the selected stand-alone MS tracks
lying within a cone of fixed size in ($\eta$, $\phi$) space. The algorithm
starts from the highest-$p_{\text{T}}$ stand-alone MS track and searches for
additional stand-alone MS tracks within a $\Delta R$ = 0.4 cone around the
initial track’s momentum vector. If a second stand-alone MS track is found in
the cone, the axis of the cone is rotated to the vector sum of the momenta of
the two tracks, and the search is repeated until no additional tracks are
found in the cone. The $\mu$DPJs are required to have at least two MS tracks
and are discarded if a jet is found within $\Delta R$ = 0.4 of a $\mu$DPJ, to
ensure orthogonality between reconstructed DPJ types.

Cosmic-ray muons that cross the detector in time coincidence with a $pp$
interaction constitute the main source of background to the $\mu$DPJ. The
cosmic dataset is used to study this background and to train a dense neural
network (DNN), referred to as the cosmic-ray tagger, to discriminate signal
$\mu$DPJs from the $\mu$DPJ candidates that originate from the cosmic-ray
background. For optimal training, a balanced mixture of all available MC
signal samples is used. The DNN is implemented using Keras with the Tensorflow
backend [98] and it is trained to classify each stand-alone MS track
constituting a $\mu$DPJ using the following quantities: the longitudinal
impact parameter $z_{0}$, the track angular direction (in $\eta$ and $\phi$),
and the timing measurements from the MS. The time measurements in different
stations of the MS, when available, allow the muon’s direction of flight to be
identified. A parametric training method is used to only consider the timing
measurements when they are available. The neural network has three dense
hidden layers, with 32, 64 and 128 neurons respectively. An output layer with
a sigmoidal activation function returns a binary classification score between
0 and 1, which can be interpreted as the probability for a $\mu$DPJ
constituent track to originate from a $\gamma_{\mathrm{d}}$ decay, and example
distributions are shown in Figure 2. The figure also illustrates the small
dependency on the specific choice of signal model. A $\mu$DPJ is accepted if
all its stand-alone MS track constituents have a cosmic-ray tagger output
score > 0.5. This selection was optimised to retain a high signal efficiency:
signal $\mu$DPJs are selected with an efficiency above 95% for transverse
decay lengths $L_{xy}$ up to 5 m and for $\gamma_{\mathrm{d}}$ $p_{\text{T}}$
larger than 20 GeV and with a background rejection of 90%.

Figure 2: The distribution of the cosmic-ray tagger output score is shown for
a few signal scenarios (lines) and for data from the cosmic dataset (shaded
area).

The $\mu$DPJ reconstruction efficiency is shown in Figure 3, for $\mu$DPJ
objects that satisfy the cosmic-ray tagger selection, as a function of the
$L_{xy}$ and transverse momentum of the dark photon in a few benchmark signal
scenarios. A drop in efficiency is expected for $\gamma_{\mathrm{d}}$ decays
which occur after the middle layer of the MS (6 m radius in the barrel
region), where muons can no longer be reconstructed.

((a))  ((b))

Figure 3: The reconstruction efficiency for $\mu$DPJ objects satisfying the
cosmic-ray tagger selection produced in the decay of a $\gamma_{\mathrm{d}}$
into a muon pair. Figure (a) shows the reconstruction efficiency for
$\gamma_{\mathrm{d}}$ with $0<|\eta|<1$ as a function of the transverse decay
length $L_{xy}$. Figure (b) shows the reconstruction efficiency for
$\gamma_{\mathrm{d}}$ with $0<|\eta|<1$ as a function of the
$\gamma_{\mathrm{d}}$ transverse momentum in events where the
$\gamma_{\mathrm{d}}$ $L_{xy}$ is below 6 m.

###  4.2 Calorimeter dark-photon jets

All the low-EM-fraction jets satisfying the selection described in Section 4
are considered as calorimeter dark-photon-jet candidates. Candidates in the
transition region between the barrel calorimeters and the endcap cryostat are
removed by requiring the fraction of energy in the Tile Gap scintillators to
be less than 10% of the total jet energy.

The main source of background for caloDPJs is jet production. In order to
reduce this background, a dedicated discriminator (QCD tagger), based on a
convolutional neural network implemented using Keras with the Tensorflow
backend, is used to assign a score to each caloDPJ in the event. The training
of the neural network is based on simulated events and exploits caloDPJs
reconstructed in signal and multi-jet MC events. The QCD tagger inputs are
three-dimensional representations of energy deposits associated with the jet.
The energy deposits are defined by collections of calorimeter cell clusters
used in jet reconstruction [82]. Each collection has $(\eta,\phi)$ coordinates
and holds information about the total amount of energy deposited in each of
the calorimeter samplings.

The $(\eta,\phi)$ space around the jet axis is divided into a $15\times 15$
grid, centred on the jet axis and corresponding to an
$\eta\times\phi=0.9\times 0.9$ area, so that each cell cluster within this 2D
grid corresponds to a $0.06\times 0.06$ portion of the $(\eta,\phi)$ space. A
third axis is added to this grid, to take into account the EM and HCAL
calorimeters’ samplings as an additional coordinate. The resulting 3D grids
are composed of cell clusters which contain the total energy released at the
corresponding $(\eta,\phi)$ coordinates and calorimeter sampling. In order to
exploit the full calorimeter volume, three 3D grids are produced: one
accounting for the barrel samplings, one for the tile calorimeter’s extended
barrel and one for the endcap. These are then used as input to a convolutional
neural network. The output layer of the neural network has a sigmoidal
activation function resulting in a binary classification in the [0, 1] range,
as shown in Figure 4(a).

CaloDPJs are defined to be candidates if they have a QCD tagger score larger
than 0.5. This selection was optimised to retain a high signal efficiency and
corresponds to a background rejection of approximately 94%. The QCD tagger has
a signal efficiency above 95% for $L_{xy}$ above 2.5 m, which decreases to
about 60% for $\gamma_{\mathrm{d}}$ decaying within the ID. The efficiency was
also found to be independent of the $\gamma_{\mathrm{d}}$ $p_{\text{T}}$ for a
$p_{\text{T}}$ larger than 20 GeV.

Muons arising from BIB and crossing the detector longitudinally, at radial
distances $r>2\ $\mathrm{m}$$, can deposit energy in the HCAL by radiative
losses, which can be reconstructed as caloDPJs owing to the resulting low EM
fraction. To reduce the residual contamination from misidentified caloDPJs
from BIB, a dedicated per-jet tagger (BIB tagger) was developed. This tagger
uses the same strategy as the QCD tagger, exploiting the topology of the
energy deposits to classify the candidate jets.

((a))  ((b))

Figure 4: The output score for the taggers used on calorimeter dark-photon
jets. Figure (a) shows the QCD tagger score for a few signal scenarios and
simulated multi-jet events. Figure (b) shows the BIB tagger score for the same
signal scenarios and BIB-enriched data.

The BIB tagger shares the network architecture of the QCD tagger. To remove
any possible energy dependence and rely only on geometrical information, the
input tensors are preprocessed by substituting the cluster energies with a
fixed value. The jet $(\eta,\phi)$ coordinates are passed as inputs to the
dense neural network, in addition to the convolution layers’ output. CaloDPJs
from a BIB-enriched data selection and from all available MC signal samples
are used as input for the training. The BIB-enriched data selection uses a
dedicated calorimeter trigger exploiting cell timing and position to select
BIB events. This trigger’s performance was validated on BIB events occurring
during empty bunch crossings. CaloDPJs are accepted if the output score of the
BIB tagger, shown in Figure 4(b), is larger than 0.2, corresponding to a BIB
rejection rate of 68%. This value is chosen in order to keep the signal
efficiency above 80%, irrespective of the signal scenario, for events entering
the analysis selections.

The caloDPJ reconstruction efficiency is shown in Figure 5 for caloDPJ
candidates that satisfy the QCD tagger selection but no BIB tagger
requirement, as a function of the transverse decay length and transverse
momentum, for several benchmark signal samples. A $\gamma_{\mathrm{d}}$
decaying into hadrons or electrons is reconstructed only within the HCAL
volume (at radii of 2.28–4.25 m in the barrel region).

((a))  ((b))

Figure 5: The reconstruction efficiency for caloDPJs produced by the decay of
$\gamma_{\mathrm{d}}$ into $e^{+}e^{-}$ or $q\bar{q}$. Figure (a) shows the
reconstruction efficiency for $\gamma_{\mathrm{d}}$ with $0<|\eta|<1.1$ as a
function of the transverse decay length $L_{xy}$. The efficiency drop at 2.5 m
corresponds to the end of the first layer of the HCAL. Figure (b) shows the
reconstruction efficiency for $\gamma_{\mathrm{d}}$ with $0<|\eta|<1.1$ as a
function of their transverse momentum in events where the
$\gamma_{\mathrm{d}}$ $L_{xy}$ is between 2 m and 4 m.

##  5 Event selection and background estimation

The events are classified into the two exclusive search categories,
gluon–gluon fusion (ggF) and WH associated production, based on their charged-
lepton multiplicity. The ggF category targets the Higgs boson production mode
where events have no charged-lepton candidates satisfying the selections
described in Section 4. The WH category targets Higgs boson production in
association with a $W$ boson and requires exactly one charged lepton. Two DPJs
satisfying the selection criteria described in Section 4 are required in the
selected events. If more than two DPJs are reconstructed, the one with the
highest transverse momentum, called the ‘leading DPJ’, and the one farthest in
$\Delta\phi$ from the leading one, called the ‘far DPJ’, are used to classify
the event. Each search category further separates the events into different
orthogonal search channels based on the numbers of $\mu$DPJs and caloDPJs,
resulting in a total of six signal regions (SRs) that were optimised for the
best discovery sensitivity.

The main sources of background for the DPJ signals are cosmic-ray muons for
the $\mu$DPJs and jet production for both the caloDPJs and $\mu$DPJs. Cosmic-
ray muon background for the $\mu$DPJs is strongly reduced after the event
selection, and the main residual background is found to consist of hadronic or
electromagnetic showers which reach the MS after not being fully contained in
the calorimeter (punch-through jets) and originate from multi-jet events. A
subdominant background for both DPJ types is BIB. All background contributions
are estimated from data. For the different decay modes considered, dedicated
sets of the following discriminating variables are used to separate the BSM
signal from the backgrounds or in the definition of regions used to aid the
estimation of the background contributions:

  * •

$m_{\textrm{jj}}$: the invariant mass of the two leading jets.

  * •

$|\Delta t_{\textrm{caloDPJs}}|$: the absolute time difference between a pair
of caloDPJs. This quantity is useful for further rejecting contributions from
cosmic-ray muons and BIB, as these caloDPJ candidates do not originate from a
single interaction vertex.

  * •

JVT: the JVT score of a caloDPJ can be used to reject candidates that are
likely to originate from the primary proton–proton interaction vertex.

  * •

$\Delta\phi_{\textrm{DPJ}}$: the azimuthal angle between the leading DPJ and
far DPJ. Signal events are expected to contain anti-aligned DPJs.

  * •

$\sum_{\Delta R=0.5}p_{\mathrm{T}}$: the scalar sum of the transverse momenta
of all tracks within a $\Delta R=0.5$ cone around the direction of the DPJ
momentum vector. Displaced DPJs are expected to have very little nearby track
activity in the ID.

  * •

$\mathrm{max}(\sum p_{\mathrm{T}})$: the largest of the $\sum_{\Delta
R=0.5}p_{\mathrm{T}}$ values for the two DPJs in an event.

  * •

$\prod\mathrm{QCD\ tagger}$: the product of the QCD tagger scores of the two
DPJs, or the single QCD tagger score when only one caloDPJ is available.

  * •

$m_{\textrm{T}}$: the event transverse mass, computed as
$m_{\textrm{T}}=\sqrt{2p_{\textrm{T}}^{\ell}E_{\text{T}}^{\text{miss}}(1-\cos\Delta\phi)}$
where $\Delta\phi$ is the azimuthal angle between the missing transverse
momentum vector and the lepton, is exploited to reduce possible background
contributions from multi-jet processes with a misidentified or non-prompt
lepton.

  * •

caloDPJ width: the $p_{\text{T}}$-weighted sum of the $\Delta R$ between each
energy cluster of the jet and the jet axis. Jets from DPJs are expected to be
narrower, on average, than ordinary jets since they are produced just before
or inside the calorimeters.

  * •

$\mathrm{min}(\Delta\phi)$: smallest azimuthal angular separation between a
selected DPJ and the $\mathbf{p_{\mathrm{T}}^{\mathrm{miss}}}$ vector.

  * •

$\mathrm{min}(\mathrm{QCD\ tagger})$: the minimum QCD tagger score, computed
for up to two caloDPJs in an event.

The detailed selection requirements, together with the dedicated methods for
the estimation of the residual background contributions, are described in
Section 5.1 for the ggF selection and in Section 5.2 for the WH selection.

###  5.1 Gluon–gluon fusion selection

In this selection, events are accepted if they satisfy one or more of three
dedicated triggers targeting displaced objects. The trigger strategy is
analogous to that described in Ref. [14]. It comprises two MS-based triggers
and one calorimeter-based trigger.

The first MS-based trigger, referred to as tri-muon MS-only [57], accepts
events with at least three muons with $p_{\text{T}}\geq 6$ GeV reconstructed
using only MS information. The second MS-based trigger, referred to as muon
narrow-scan [99], requires a muon candidate from the L1 trigger with
$p_{\text{T}}\geq 20$ GeV to be confirmed by the HLT using only MS
information. At the HLT a ‘scan’ is then performed in a cone of $\Delta R=0.5$
around this muon, looking for a second muon reconstructed using only MS
information. The $p_{\text{T}}$ requirement on the second muon was increased
from 6 GeV to 15 GeV during the course of the 2015–2016 data taking and kept
constant afterwards. Both muons were required to be not matched to any track
in the ID, and track isolation was required for the leading muon. During the
2017–2018 data taking, the narrow-scan trigger was extended to take advantage
of the L1 topological trigger [100] by requiring partially matched topological
items. The leading L1 muon candidate was then combined with an HLT object:
either an unmatched muon, a jet, or transverse momentum imbalance. The last
two were added to target events with one $\mu$DPJ and one caloDPJ. The narrow-
scan triggers have constant efficiencies of 75% and 40% for the 2015–2016 and
2017–2018 versions, respectively, when considering events with
$\gamma_{\mathrm{d}}$ decays into muon pairs with $|\eta|<1.0$ and $L_{xy}$
below 6 m.

A L1 calorimeter trigger called the CalRatio trigger [99] is designed to
select narrow jets produced in the hadronic calorimeter. During 2015–2016, the
CalRatio L1 trigger selected narrow jets with transverse energy
$E_{\mathrm{T}}>60$ GeV in a $0.2\times 0.2$ $(\Delta\eta\times\Delta\phi)$
region of the EM and HCAL. During 2017–2018 data taking, the CalRatio trigger
instead used a L1 topological trigger [100] to select jets in the HCAL that
have $E_{\mathrm{T}}>30$ GeV and are isolated from energy deposits in the EM
calorimeter with $E_{\text{T}}>3$ GeV within $\Delta R=0.2$ around the most
energetic HCAL energy deposit. At the HLT, jets from both of these L1
selections were required to have $E_{\mathrm{T}}\geq 30$ GeV, $|\eta|<2.4$ and
EM fraction $<0.06$. These jets were also required to have no tracks with
$p_{\mathrm{T}}\geq 2$ GeV within $\Delta R=0.2$ of the jet axis, and to
satisfy BIB-suppression requirements on calorimeter cell timing and position.
The CalRatio triggers used in 2015–2016 and 2017–2018 are found to have
constant efficiencies of approximately 5% and 20%, respectively, for FRVZ
signal events where the $\gamma_{\mathrm{d}}$ is produced in the decay of a
125 GeV Higgs boson and decays into $e^{+}e^{-}$ or $q\bar{q}$ within
$|\eta|<1.0$. These efficiencies rise to 25% and 30%, on average, for signal
events in HAHM models, or to 55% and 45%, on average, when assuming
$\gamma_{\mathrm{d}}$ production via an 800 GeV scalar mediator.

Events where the two leading jets have an invariant mass $m_{\textrm{jj}}>1$
TeV and $E_{\text{T}}^{\text{miss}}>225$ GeV are removed to allow for a
statistically independent study of vector-boson fusion (VBF) production modes
in the future.

The events satisfying the requirements listed above are further separated into
orthogonal search channels based on the DPJ types. Events with pairs of
$\mu$DPJs (caloDPJs) are targeted by SR${}_{2\mu}^{\textrm{ggF}}$
(SR${}_{2\textrm{c}}^{\textrm{ggF}}$), while events with one DPJ of each kind
are targeted by SR${}_{\textrm{c}+\mu}^{\textrm{ggF}}$. Each of these SRs
includes additional selection requirements, as summarised in Table 1 together
with the triggers employed. The impact of beam-induced background is
suppressed by requiring caloDPJ candidates to pass the BIB tagger selection
and, only in the SR${}_{2\textrm{c}}^{\textrm{ggF}}$ where two caloDPJs are
reconstructed, to have a time difference $|\Delta t_{\textrm{caloDPJs}}|$
smaller than $2.5$ ns. In SR${}_{2\textrm{c}}^{\textrm{ggF}}$ and
SR${}_{\textrm{c}+\mu}^{\textrm{ggF}}$, the leading and far DPJs are further
required to have an azimuthal angular difference $\Delta\phi_{\textrm{DPJ}}$
larger than 0.5. Multi-jet background events are rejected by requiring the
DPJs to satisfy ID isolation criteria. DPJ candidates are required to have a
$\sum_{\Delta R=0.5}p_{\mathrm{T}}$ below 4.5 GeV.

Table 1:  Definition of the SRs used in the ggF selection. All SRs require at least two DPJs, but only the leading and the far DPJs are considered for the event classification. Dashes indicate cases where a requirement is not applied. Requirement / Region | SR${}_{2\mu}^{\textrm{ggF}}$ | SR${}_{2\textrm{c}}^{\textrm{ggF}}$ | SR${}_{\textrm{c}+\mu}^{\textrm{ggF}}$
---|---|---|---
Number of $\mu$DPJs | 2 | 0 | 1
Number of caloDPJs | 0 | 2 | 1
Tri-muon MS-only trigger | yes | - | -
Muon narrow-scan trigger | yes | - | yes
CalRatio trigger | - | yes | -
$|\Delta t_{\textrm{caloDPJs}}|$ [ns] | - | $<2.5$ | -
caloDPJ JVT | - | $<0.4$ | -
$\Delta\phi_{\textrm{DPJ}}$ | $>\pi/5$ | $>\pi/5$ | $>\pi/5$
BIB tagger score | - | $>0.2$ | $>0.2$
$\mathrm{max}(\sum p_{\mathrm{T}})$ [GeV] | $<4.5$ | $<4.5$ | $<4.5$
$\prod\mathrm{QCD\ tagger}$ | - | $>0.95$ | $>0.9$

The main sources of background after the SR selections are punch-through jets
from rare multi-jet events for SR${}_{2\mu}^{\textrm{ggF}}$ and multi-jet
production for SR${}_{2\textrm{c}}^{\textrm{ggF}}$ and
SR${}_{\textrm{c}+\mu}^{\textrm{ggF}}$. The second leading background for all
SRs is the cosmic-ray muon background.

In order to estimate the background contribution to each SR, a data-driven
‘ABCD’ method is used. This method relies on the assumption that the
distribution of background events can be factorised in the plane of two
uncorrelated variables so that it is subdivided into four regions: A, B, C,
and D. The number of background events in the SR can be evaluated as
$N_{\mathrm{A}}=N_{\mathrm{D}}\times N_{\mathrm{B}}/N_{\mathrm{C}}$. Any
possible signal leakage outside the SR region is accounted for by using a
modified ABCD method that simultaneously fits the signal and background
events, taking into account potential signal contamination in the ABCD
estimate.

The planes used in the estimation of the backgrounds for the three ggF SRs are
formed by $\mathrm{max}(\sum p_{\mathrm{T}})$ and either
$\Delta\phi_{\textrm{DPJ}}$ in SR${}_{2\mu}^{\textrm{ggF}}$, or
$\prod\mathrm{QCD\ tagger}$ in SR${}_{2\textrm{c}}^{\textrm{ggF}}$ and
SR${}_{\textrm{c}+\mu}^{\textrm{ggF}}$. The definition of the three sets of
ABCD control regions are given in Table 2. In the ABCD convention, region A is
the SR and regions B, C and D are respectively the control regions CRB, CRC
and CRD.

Table 2:  Definition of the control regions used in the ggF background estimation. All control regions require at least two DPJs, but only the leading DPJ and the far DPJ are considered for the event classification. All CR requirements are the same as for the respective SR, with the exception of the selections reported in this table. Requirement / Region | CRB${}_{2\mu}^{\textrm{ggF}}$ | CRC${}_{2\mu}^{\textrm{ggF}}$ | CRD${}_{2\mu}^{\textrm{ggF}}$
---|---|---|---
$\Delta\phi_{\textrm{DPJ}}$ | $>\pi/5$ | $(0.1,\pi/5]$ | $(0.1,\pi/5]$
$\mathrm{max}(\sum p_{\mathrm{T}})$ [GeV] | $[4.5,20)$ | $[4.5,20)$ | $<4.5$
Requirement / Region | CRB${}_{2\textrm{c}}^{\textrm{ggF}}$ | CRC${}_{2\textrm{c}}^{\textrm{ggF}}$ | CRD${}_{2\textrm{c}}^{\textrm{ggF}}$
$\prod\mathrm{QCD\ tagger}$ | $>0.95$ | $(0.8,0.95]$ | $(0.8,0.95]$
$\mathrm{max}(\sum p_{\mathrm{T}})$ [GeV] | $[4.5,20)$ | $[4.5,20)$ | $<4.5$
Requirement / Region | CRB${}_{\textrm{c}+\mu}^{\textrm{ggF}}$ | CRC${}_{\textrm{c}+\mu}^{\textrm{ggF}}$ | CRD${}_{\textrm{c}+\mu}^{\textrm{ggF}}$
$\prod\mathrm{QCD\ tagger}$ | $>0.9$ | $(0.75,0.9]$ | $(0.75,0.9]$
$\mathrm{max}(\sum p_{\mathrm{T}})$ [GeV] | $[4.5,20)$ | $[4.5,20)$ | $<4.5$

The cosmic-ray muon background contribution is estimated first, using the
cosmic dataset. The ratio of the number of filled bunch crossings to the
number of empty bunch crossings is used to scale the number of events in the
cosmic dataset to that in the $pp$ collision data. The value of this
correction depends on the trigger used in the SR selection and ranges from a
minimum of 2.2 for the CalRatio trigger in 2015–2016 to a maximum of 3.9 for
the tri-muon MS-only trigger in 2018. Only three regions were found to have a
non-negligible cosmic-ray muon background contribution: $7\pm 5$ events in
SR${}_{2\mu}^{\textrm{ggF}}$, $7\pm 4$ events in
SR${}_{2\textrm{c}}^{\textrm{ggF}}$ and $14\pm 7$ events in
CRD${}_{2\textrm{c}}^{\textrm{ggF}}$. Since the events forming the cosmic
dataset are collected in empty bunch crossings, there is no significant
activity in the ID, which implies that the $\sum p_{\mathrm{T}}$ is 0 for all
reconstructed DPJs. For this reason, events populate only the SRs and CRD.
These estimated cosmic-ray muon backgrounds are subtracted from the observed
data in the main dataset before performing the ABCD procedure.

Events with BIB energy deposits are very likely to have two caloDPJs which can
contribute to the background. The possible contribution from misidentified
caloDPJs was studied using the BIB-enriched dataset. This selection is thus
orthogonal to the signal regions. The efficiency of the SR and CR selection
requirements when applied to BIB-enriched events was found to be less than
$2.3\cdot 10^{-4}$, reducing the BIB contamination to a negligible
contribution. The BIB background is hence neglected.

Figure 6 shows the distribution of data events in the ABCD plane for the three
search channels, together with the expected distribution for a benchmark FRVZ
model assuming a 125 GeV Higgs boson and a decay branching fraction to the
dark sector of 10%. The acceptance times efficiency for the FRVZ signal
processes after applying all selection criteria is 0.74%, 0.02% and 0.15%, in
the case of a $\gamma_{\mathrm{d}}$ mass of 400 MeV and a generated $c\tau$ of
50 mm, for the signal regions SR${}_{2\mu}^{\textrm{ggF}}$,
SR${}_{2\textrm{c}}^{\textrm{ggF}}$ and SR${}_{\textrm{c}+\mu}^{\textrm{ggF}}$
respectively.

((a)) SR${}_{2\mu}^{\textrm{ggF}}$ ((b))
SR${}_{\textrm{c}+\mu}^{\textrm{ggF}}$ ((c))
SR${}_{2\textrm{c}}^{\textrm{ggF}}$

((d)) SR${}_{2\mu}^{\textrm{ggF}}$ ((e))
SR${}_{\textrm{c}+\mu}^{\textrm{ggF}}$ ((f))
SR${}_{2\textrm{c}}^{\textrm{ggF}}$

Figure 6: The per-event distributions in the ABCD planes defined for the ggF
search channels. Figures (a, b, c) show data events, while Figures (d, e, f)
show simulated signal events. FRVZ signal samples assuming a 125 GeV Higgs
boson with a decay branching fraction of 10% to the dark sector and a
$\gamma_{\mathrm{d}}$ mass of 400 MeV are shown.

The background estimation procedure was validated by applying the ABCD method
in subregions constructed in combinations of the regions CRB, CRC and CRD, for
varying choices of the subregion boundaries. For each validation test, a new
set of (ABCD)′ test regions was defined within the union of either CRB and CRC
or CRC and CRD. The test region boundaries were varied in discrete steps after
dividing the allowed range into ten equal-size intervals. The validation was
performed for each of the SR${}_{2\mu}^{\textrm{ggF}}$,
SR${}_{2\textrm{c}}^{\textrm{ggF}}$ and SR${}_{\textrm{c}+\mu}^{\textrm{ggF}}$
selections. The expected background yields in the resulting test SRs range
between 15 and 800 events. An additional validation of the method was
performed by checking the closure on orthogonal selections, which were
obtained either by inverting the cosmic-ray tagger cut or by selecting
different ranges of QCD tagger score. These selections were chosen in order to
have negligible signal contributions. In all the channels, the observed yields
were found to agree within one standard deviation with the expected yields.
The Pearson linear correlation coefficient between the two variables defining
the ABCD plane in all subregions and in the validation regions was found to be
less than 3%. The signal leakage in regions CRB, CRC and CRD, was found to be
less than 10% of the total signal in the ABCD selection for all signal
scenarios considered in the analysis.

###  5.2 WH associated production selection

In the WH selection, candidate events are required to pass single-electron or
single-muon triggers. The offline requirements on the $p_{\text{T}}$,
identification and isolation of the lepton are tighter than those applied
online, so as to be on the trigger efficiency plateau [54].

The selection requires one electron or muon with $p_{\text{T}}$ $\geq$ 27 GeV,
$E_{\text{T}}^{\text{miss}}$ $\geq$ 30 GeV,
$m_{\mathrm{T}}\leavevmode\nobreak\ \geq$ 40 GeV, and no additional lepton
with $p_{\text{T}}$ $\geq$ 10 GeV. Events where the two leading jets have an
invariant mass $m_{\textrm{jj}}\geq 1$ TeV are removed to allow for a future
statistically independent study of VBF production modes. Contributions from
single-top and $t\bar{t}$ processes are reduced by requiring three or fewer
jets with $p_{\text{T}}$ $\geq$ 30 GeV and no $b$-tagged jets in each event.
After these selections the background is dominated by W+jets events.

Events are then separated into orthogonal search channels based on the numbers
of $\mu$DPJs and caloDPJs. Events with exactly one caloDPJ are targeted by
SR${}_{\textrm{c}}^{\textit{WH}}$, and events with two or more caloDPJs by
SR${}_{2\textrm{c}}^{\textit{WH}}$. The mixed channel with one DPJ of each
type is targeted by SR${}_{\textrm{c}+\mu}^{\textit{WH}}$. Channels with only
$\mu$DPJs are not considered for the WH selection as the constraints they
provide are not competitive with those from other signal production modes. The
SR requirements are summarised in Table 3. In all SRs, caloDPJ candidates are
required to have a JVT score below 0.6 to suppress jets originating from the
interaction point. Events belonging to SR${}_{\textrm{c}}^{\textit{WH}}$,
where one $\gamma_{\mathrm{d}}$ is reconstructed, are required to have
$m_{\mathrm{T}}\leavevmode\nobreak\ \geq$ 120 GeV and a caloDPJ with
$p_{\text{T}}$ $\geq$ 30 GeV to further reduce the initially overwhelming
W+jets background. In this SR, the second $\gamma_{\mathrm{d}}$ is expected to
decay outside the detector volume and yield a higher
$E_{\text{T}}^{\text{miss}}$. CaloDPJ candidates are required to have a width
$w\leq$ 0.08 in SR${}_{\textrm{c}}^{\textit{WH}}$ and $w\leq$ 0.1 in
SR${}_{\textrm{c}+\mu}^{\textit{WH}}$. In SR${}_{2\textrm{c}}^{\textit{WH}}$
the leading (far) caloDPJ candidate is required to have $w\leq$ 0.1 (0.15).
These selections are optimised to reject hadronic jets and maximise the
discovery significance.

Table 3:  Definition of the signal regions used in the WH selection. In signal regions requiring at least two DPJs, only the leading DPJ and the far DPJ are considered for the event classification. Each DPJ SR is exclusive in the number of DPJs in the event. Dashes indicate cases where a requirement is not applied. Requirement / Region | SR${}_{\textrm{c}}^{\textit{WH}}$ | SR${}_{2\textrm{c}}^{\textit{WH}}$ | SR${}_{\textrm{c}+\mu}^{\textit{WH}}$
---|---|---|---
Number of $\mu$DPJs | 0 | 0 | 1
Number of caloDPJs | 1 | 2 | 1
Single-lepton trigger ($\mu$, $e$) | yes | yes | yes
$m_{\mathrm{T}}$ [GeV] | $>120$ | - | -
$|t_{\mathrm{caloDPJ}}|$ [ns] | $<4$ | $<4$ | $<4$
Leading (far) caloDPJ width | $<0.08$ | $<0.10\ (0.15)$ | $<0.1$
caloDPJ $p_{\mathrm{T}}$ [GeV] | $>30$ | - | -
JVT | $<0.6$ | $<0.6$ | $<0.6$
$\mathrm{min}(\Delta\phi)$ | $<3\pi/5$ | $<3\pi/10$ | $<7\pi/20$
$\mathrm{min}(\mathrm{QCD\ tagger})$ | $>0.99$ | $>0.91$ | $>0.9$

After the selection the main background is due to W+jets events where rare QCD
phenomena give rise to in-flight decays and punch-through jets that mimic
signal DPJs. The residual W+jets background in the signal region is estimated
with a data-driven ABCD method, the same as used for the ggF selection and
explained in detail in Section 5.1. The non-collision background from cosmic-
ray muons and BIB was found to be negligible for this selection.

The ABCD planes used in the estimation of the backgrounds for the three WH SRs
are based on $\mathrm{min}(\Delta\phi)$ and $\mathrm{min}(\mathrm{QCD\
tagger})$. These variables provide good discrimination between signal and
background, and are highly uncorrelated in W+jets events. The definitions of
the three sets of ABCD regions are given in Table 4.

Table 4:  Definition of the control regions used in the WH background estimation. The requirements for all regions are the same as those for the respective SRs, except for the selections reported in this table. Requirement / Region | CRB${}_{\textrm{c}}^{\textit{WH}}$ | CRC${}_{\textrm{c}}^{\textit{WH}}$ | CRD${}_{\textrm{c}}^{\textit{WH}}$
---|---|---|---
$\mathrm{min}(\Delta\phi)$ | $<3\pi/5$ | $>3\pi/5$ | $>3\pi/5$
$\mathrm{min}(\mathrm{QCD\ tagger})$ | $[0.9,0.99)$ | $[0.9,0.99)$ | $>0.99$
Requirement / Region | CRB${}_{2\textrm{c}}^{\textit{WH}}$ | CRC${}_{2\textrm{c}}^{\textit{WH}}$ | CRD${}_{2\textrm{c}}^{\textit{WH}}$
$\mathrm{min}(\Delta\phi)$ | $<3\pi/10$ | $>3\pi/10$ | $>3\pi/10$
$\mathrm{min}(\mathrm{QCD\ tagger})$ | $[0.8,0.91)$ | $[0.8,0.91)$ | $>0.91$
Requirement / Region | CRB${}_{\textrm{c}+\mu}^{\textit{WH}}$ | CRC${}_{\textrm{c}+\mu}^{\textit{WH}}$ | CRD${}_{\textrm{c}+\mu}^{\textit{WH}}$
$\mathrm{min}(\Delta\phi)$ | $<7\pi/20$ | $>7\pi/20$ | $>7\pi/20$
$\mathrm{min}(\mathrm{QCD\ tagger})$ | $[0.8,0.9)$ | $[0.8,0.9)$ | $>0.9$

All the SRs lie at high QCD tagger score and small azimuthal angular
separation between the $\mathbf{p_{\mathrm{T}}^{\mathrm{miss}}}$ and the
nearest DPJ, where most of the signal events are expected.

The background estimation procedure was validated by applying the ABCD method
in subregions constructed in combinations of the regions CRB, CRC and CRD, for
various choices of the subregion boundaries, as done for the ggF selection.
The expected background yields in the resulting test SRs range between 80 and
1550 events. Excellent agreement, within one standard deviation, between data
and the predicted yields was observed in most regions, with discrepancies of
up to two standard deviations appearing in the validation of
SR${}_{\textrm{c}}^{\textit{WH}}$ for values of $\mathrm{min}(\mathrm{QCD\
tagger})$ close to 0.9. An additional validation of the method was performed
by checking the closure of the ABCD procedure on orthogonal selections, which
were obtained by selecting different ranges of QCD tagger score. The
validation ranges extend to a minimum of 0.6. In all the channels, the
observed yields were found to agree within one standard deviation with the
expected yields. The Pearson linear correlation coefficient between the two
variables defining the ABCD plane in all subregions and in the validation
regions was found to be less than 2%.

Figure 7 shows the distribution of data events in the ABCD plane for the three
search channels, together with the expected distribution for a benchmark FRVZ
model assuming a 125 GeV Higgs boson. The acceptance times efficiency for the
FRVZ signal processes after applying all selection criteria is 0.12% in the
case of a $\gamma_{\mathrm{d}}$ mass of 400 MeV and $c\tau$ of 50 mm in
SR${}_{\textrm{c}+\mu}^{\textit{WH}}$, and 0.05% and 0.26% in
SR${}_{2\textrm{c}}^{\textit{WH}}$ and SR${}_{\textrm{c}}^{\textit{WH}}$,
respectively, in the case of a $\gamma_{\mathrm{d}}$ mass of 100 MeV and
$c\tau$ of 15 mm. The signal leakage in the CRs can be as high as 25% for CRB,
and up to 15% for CRD and CRC. However, the stability of the fit is not
affected by the potential signal leakage because the expected signal yields
are negligible in comparison with the SM background in those regions.

((a)) SR${}_{\textrm{c}}^{\textit{WH}}$ ((b))
SR${}_{2\textrm{c}}^{\textit{WH}}$ ((c)) SR${}_{\textrm{c}+\mu}^{\textit{WH}}$

((d)) SR${}_{\textrm{c}}^{\textit{WH}}$ ((e))
SR${}_{2\textrm{c}}^{\textit{WH}}$ ((f)) SR${}_{\textrm{c}+\mu}^{\textit{WH}}$

Figure 7: The per-event $\mathrm{min}(\Delta\phi)$ vs
$\mathrm{min}(\mathrm{QCD\ tagger})$ distributions for the WH search channels.
Figures (a, b, c) show data, while Figures (d, e, f) show simulated signal
events. FRVZ signal samples with a 125 GeV Higgs boson and a
$\gamma_{\mathrm{d}}$ mass of 100 MeV are shown for
SR${}_{\textrm{c}}^{\textit{WH}}$ and SR${}_{2\textrm{c}}^{\textit{WH}}$,
while a $\gamma_{\mathrm{d}}$ mass of 400 MeV is shown for
SR${}_{\textrm{c}+\mu}^{\textit{WH}}$.

##  6 Systematic uncertainties

The overall uncertainty in the SR yields is dominated by the statistical
uncertainty. Nevertheless, potential sources of experimental uncertainty are
considered for the background estimates and the simulated signal yields.

The statistical uncertainties of the observed yields in regions CRB, CRC, and
CRD are propagated to the background expectation obtained from the ABCD
method. An additional uncertainty is assigned to account for the size of the
cosmic dataset and scaled by the ratio of the number of empty bunch crossings
to the number of filled bunch crossings while the triggers were active. This
uncertainty is estimated to be as large as 80% of the expected cosmic-ray muon
background, but has no impact on the final results because of the limited
contribution from this specific background.

Experimental uncertainties in the reconstruction and simulation of the signal
events are considered. For jets, these include jet energy resolution (JER) and
jet energy scale (JES) uncertainties from the standard calibration scheme
[86], which amount to up to 3% of the expected signal yields. An extra JES
uncertainty is applied to take into account a possible dependence on the low-
EM-fraction selection. It is estimated following the same procedure as used in
the 2015–2016 dark-photon jet search [14] and is about 3%.

For muons, a systematic uncertainty in the single-$\gamma_{\mathrm{d}}$
reconstruction efficiency is evaluated using a tag-and-probe method applied to
$J/\psi\to\mu\mu$ events in data and simulation, as $J/\psi\to\mu\mu$ decays
were found to behave similarly to $\gamma_{\mathrm{d}}$ decays in simulated
events. The $J/\psi$ reconstruction efficiency is evaluated in both data and
simulation as a function of the opening angle $\Delta R$ between the two
muons. For low $\Delta R$ values, the efficiency decreases due to the
difficulty of reconstructing two tracks with small angular separation in the
MS. The difference in $J/\psi\to\mu\mu$ reconstruction efficiency between
simulation and data in the $\Delta R$ interval between 0 and 0.06, where the
DPJ samples are concentrated, amounts to 9.6$\%$ and is taken as the
uncertainty.

The systematic uncertainties in the efficiency of the two dedicated muon
triggers are evaluated using a tag-and-probe method applied to
$J/\psi\to\mu\mu$ events in data and simulation. The difference between the
trigger efficiency in data and that in simulation is evaluated as a function
of the opening angle between the two muons. The difference in the region
$\Delta R$ < 0.05, corresponding to the $\Delta R$ expected for signal, is
taken as the uncertainty and is found to be 6$\%$ for the muon narrow-scan
trigger and 5.8$\%$ for the tri-muon MS-only trigger.

The systematic uncertainty in the CalRatio trigger efficiency is taken from
Ref. [101] and recomputed for each signal sample. It is estimated to be up to
4% for the heavy (800 GeV) scalar-mediator signal samples, and up to 30% for
the 125 GeV Higgs signal samples.

The uncertainty associated with the MC modelling of the input variables used
for the cosmic-ray tagger training is evaluated by comparing the distributions
of the tagger’s output scores for $Z\rightarrow\mu\mu$ events in data and MC
samples. Due to the high boost of low-mass dark photons and the high
$p_{\mathrm{T}}$ of signal muons, the muon $z_{0}$ and timing distributions
are similar to those of prompt muons from $Z\rightarrow\mu\mu$ decays. Muons
from $Z$ boson decays were reconstructed using only MS information. The ratio
of the two score distributions is computed and applied as a binned scale
factor to the muons used for $\mu$DPJ reconstruction. The resulting relative
variation of the final signal yield in the SR is taken as the systematic
uncertainty. The same procedure is used for the QCD and BIB taggers, where the
ratios of data to simulated distributions are computed from data and MC
samples of multi-jet events after selecting signal-like events with
reconstructed caloDPJs but failing the analysis trigger selections. In-depth
modelling studies of jet calorimeter-cell clusters, upon which the taggers’
training images are built, show no significant mis-modelling.

A pile-up modelling uncertainty is assigned to account for the difference
between the predicted and measured inelastic cross-sections [102].

Finally, an uncertainty in the computed total integrated luminosity used to
rescale the expected number of signal events is considered. The uncertainty in
the combined 2015–2018 integrated luminosity is 1.7% [103], obtained using the
LUCID-2 detector [104] for the primary luminosity measurements.

A summary of the experimental systematic uncertainties taken into account for
the signal samples in this analysis is shown in Figure 8, assuming signal
production via a 125 GeV Higgs boson. The average uncertainties are
representative for the bulk of the signal samples, with small variations of
the order of a few percent being observed as a function of the
$\gamma_{\mathrm{d}}$ mass. The uncertainty related to the efficiency of the
CalRatio trigger was found to have the largest variation, ranging between 15%
and 26% with a linear dependence on the $\gamma_{\mathrm{d}}$ mass.

Figure 8:  Comparison of the relative uncertainty in the signal yield in each
SR, showing the contributions from the different sources of uncertainty. The
quoted values are averaged over different $\gamma_{\mathrm{d}}$ masses, in
scenarios assuming signal production via a 125 GeV Higgs boson. The ‘Muon
uncertainties’ category contains all muon-related systematic uncertainties and
is dominated by the uncertainty in the single-$\gamma_{\mathrm{d}}$
reconstruction efficiency. The ‘NN taggers’ category contains the three
taggers adopted in the analysis and is dominated by the BIB tagger
uncertainties for the hadronic channels and by the cosmic-ray tagger in the
muonic channels. The ‘Triggers’ category contains all trigger systematic
uncertainties, which are relevant only for the dedicated triggers in the ggF
selection. The ‘Jet uncertainties’ category contains the JES, JER and low-EM-
fraction JES systematic uncertainties. Some sets of systematic uncertainties
apply to only a subset of the SRs.

##  7 Results and interpretations

The data-driven background estimate in each SR is obtained by performing a
maximum-likelihood fit to the yields in the four (i.e. ABCD) regions in data.
The fitted likelihood function is formed from a product of Poisson functions,
one for each of the SR, CRB, CRC, and CRD regions, describing signal and
background expectations. The ABCD ansatz is introduced as nuisance parameters
in the background component of the expected yield in each region. The
likelihood-based ABCD fit is robust against control regions with a small
number of events and takes into account possible signal contamination in the
control regions. All systematic uncertainties described in Section 6 are
included in the fit as nuisance parameters, parameterised with Gaussian
probability density functions that multiply the fit likelihood. They are
assumed to be uncorrelated across regions. An alternative correlation model,
where the uncertainties are assumed to be fully correlated across regions, has
a negligible impact on the results. The mean value of the Gaussian probability
distribution function is constrained by the nominal value of the parameter and
the variance is defined by the 68% confidence interval of the systematic
uncertainty associated with the parameter.

The observed and expected numbers of events in the signal regions are
summarised in Table 5. The reported yields are extrapolated by the fit
assuming no signal, and with unblinded data in all ABCD regions in the fit.
When comparing the results with a likelihood fit using blinded data in the SR,
the background yields are found to be consistent at the percent level except
for SR${}_{2\mu}^{\textrm{ggF}}$, where an 11% change is observed. For the ggF
selection results, the estimated cosmic-ray muon yields are subtracted from
each of the ABCD regions before using the ABCD method to estimate the multi-
jet background yield.

Table 5: Observed and expected yields in the ABCD regions. The total uncertainty in the background expectation is computed by the ABCD fit to unblinded data. In the ggF selection regions the estimated contribution from cosmic-ray muons is subtracted from each of the ABCD regions before the ABCD fit, and added back post-fit (as shown in this table). Selection | Search channel | CRB | CRC | CRD | SR expected | SR observed
---|---|---|---|---|---|---
| $2\mu$ |  55 |  61 |  389 |  $317\pm 47$ |  269
ggF | c+$\mu$ |  169 |  471 |  301 |  $108\pm 13$ |  110
| 2c |  97 | 1113 | 12146 | $1055\pm 82$ | 1045
| c | 1850 | 3011 |  155 |  $93\pm 12$ |  103
WH | c+$\mu$ |  30 |  49 |  31 |  $19\pm 8$ |  20
| 2c |  79 |  155 |  27 |  $14\pm 5$ |  15

The results of the search are used to set upper limits on the production
cross-section times branching fraction, $\sigma\times B$, of the process
$pp\rightarrow H\rightarrow\leavevmode\nobreak\ 2\gamma_{\textrm{d}}+X$, with
$X$ assumed to be undetected, as a function of the $\gamma_{\mathrm{d}}$ mean
proper decay length $c\tau$. The upper limit on the signal strength is
obtained with the CL${}_{\text{s}}$ method [105] with the asymptotic
calculator [106], considering the background and the predicted signal yields
from simulation in the four ABCD regions. The validity of the asymptotic
approximation is checked against a full calculation using pseudo-experiments,
and the CL${}_{\text{s}}$ values of the two methods typically agree within 1%,
and maximally within 5%. Model-dependent limits are computed for the various
signal scenarios considered in the analysis. The hypothesis tests take account
of the expected signal yield and its uncertainties in the CRs and SRs and are
performed with a likelihood fit to the observed data.

A summary of the observed limits for scenarios assuming the production of a
125 GeV Higgs boson that can decay into hidden-sector particles, resulting in
two $\gamma_{\mathrm{d}}$ in an FRVZ model, is shown in Figures 9 and 10 for
the ggF and WH selections, respectively. The sensitivity of each search
channel is reported in a separate subfigure. The sensitivity of the $2\mu$ and
c+$\mu$ selections to $\gamma_{\mathrm{d}}$ masses less than twice the muon
mass is due to signal decays into a displaced electron or quark pair within
the muon spectrometer volume which are reconstructed as $\mu$DPJs. For pairs
of dark photons decaying into $\mu$DPJs, branching fractions larger than 1%
are excluded at 95% confidence level (CL) if the dark photons have a mean
proper decay length $c\tau$ between 10 mm and 250 mm and a mass between 0.4
GeV and 2 GeV. When considering dark-photon decays into pairs of caloDPJs,
branching fractions larger than 10% are excluded at 95% CL for mean proper
decay lengths between 2 mm and 3 mm and masses between 17 MeV and 50 MeV.

((a)) SR${}_{2\mu}^{\textrm{ggF}}$ ((b))
SR${}_{\textrm{c}+\mu}^{\textrm{ggF}}$

((c)) SR${}_{2\textrm{c}}^{\textrm{ggF}}$

Figure 9: Observed 95$\%$ CL upper limits on the branching ratio ($B$) for the
decay $H\rightarrow\leavevmode\nobreak\ 2\gamma_{\textrm{d}}+X$ and ggF
selection, for different $\gamma_{\mathrm{d}}$ masses and a 125 GeV Higgs
boson. The limits are shown separately for the (a) $2\mu$, (b) c$+\mu$ and (c)
2c search channels, assuming an FRVZ signal model. The hatched band denotes
the region in which the branching ratio of $H\rightarrow\leavevmode\nobreak\
2\gamma_{\textrm{d}}+X$ is larger than unity.

((a)) SR${}_{\textrm{c}}^{\textit{WH}}$ ((b))
SR${}_{\textrm{c}+\mu}^{\textit{WH}}$

((c)) SR${}_{2\textrm{c}}^{\textit{WH}}$

Figure 10: Observed 95$\%$ CL upper limits on the branching ratio ($B$) for
the decay $H\rightarrow\leavevmode\nobreak\ 2\gamma_{\textrm{d}}+X$ and WH
selection, for different $\gamma_{\mathrm{d}}$ masses and a 125 GeV Higgs
boson. The limits are shown separately for the three search channels: (a) c,
(b) c$+\mu$ and (c) 2c, assuming an FRVZ signal model. The hatched band
denotes the region in which the branching ratio of
$H\rightarrow\leavevmode\nobreak\ 2\gamma_{\textrm{d}}+X$ is larger than
unity.

Figure 11 reports the observed limits for scenarios assuming the production of
a 125 GeV Higgs boson that can decay into two $\gamma_{\mathrm{d}}$ in the
HAHM model. The sensitivity of each search channel in the ggF selection is
reported in a separate subfigure. The sensitivities in the WH selection are
significantly lower because the CalRatio trigger is less efficient for those
events. For pairs of dark photons decaying into $\mu$DPJs, branching fractions
larger than 1% are excluded at 95% CL if the dark photons have a mean proper
decay length $c\tau$ between 4 mm and 200 mm and a mass between 0.4 GeV and 2
GeV. When considering dark-photon decays into pairs of caloDPJs, branching
fractions larger than 10% are excluded at 95% CL for mean proper decay lengths
between 1.5 mm and 8 mm and masses between 17 MeV and 100 MeV.

((a)) SR${}_{2\mu}^{\textrm{ggF}}$ ((b))
SR${}_{\textrm{c}+\mu}^{\textrm{ggF}}$

((c)) SR${}_{2\textrm{c}}^{\textrm{ggF}}$

Figure 11: Observed 95$\%$ CL upper limits on the branching ratio ($B$) for
the decay $H\rightarrow\leavevmode\nobreak\ 2\gamma_{\textrm{d}}$ and ggF
selection, for different $\gamma_{\mathrm{d}}$ masses and a 125 GeV Higgs
boson. The limits are shown separately for the (a) $2\mu$, (b) c$+\mu$ and (c)
2c search channels, assuming a HAHM signal model. The hatched band denotes the
region in which the branching ratio of $H\rightarrow\leavevmode\nobreak\
2\gamma_{\textrm{d}}$ is larger than unity.

Alternative signal scenarios were also considered in the context of FRVZ
models. The sensitivity to scenarios where a pair of dark photons is produced
in the decay of an 800 GeV scalar mediator is driven by the selection
efficiency being higher than for the decay of the 125 GeV Higgs boson because
of the harder dark-photon energy spectrum. The signal region targeting decays
into one caloDPJ and one $\mu$DPJ in the ggF selection was found to be the
most sensitive: branching fractions above 10% are excluded at 95% CL if the
dark photons have a mean proper decay length between 6 mm and 30 mm and a mass
between 400 MeV and 2 GeV. In the case of models leading to a total of four
dark photons in the final state, the sensitivity is lower than for the
$H\rightarrow\leavevmode\nobreak\ 2\gamma_{\textrm{d}}+X$ process because the
large boost of dark-photon pairs from the $s_{\textrm{d}}$ decays results in
caloDPJs overlapping with $\mu$DPJs and failing the object-level selections.
The signal region targeting decays into one caloDPJ and one $\mu$DPJ in the
ggF selection was found to be the most sensitive: branching fractions above
10% are excluded at 95% CL for dark photons with a mean proper decay length
between 20 mm and 50 mm and a mass between 400 MeV and 1 GeV. In order to set
limits on the $pp\rightarrow H\rightarrow\leavevmode\nobreak\
2\gamma_{\textrm{d}}+X$ process as a function of the kinetic mixing parameter
$\epsilon$ and $\gamma_{\mathrm{d}}$ mass, the best-performing search
channels, based on their expected sensitivity, are taken into account. In the
case of FRVZ models, the mutually exclusive search channels targeting dark-
photon decays into caloDPJs are statistically combined to maximise the
exclusion power. The combined fit considers a product of the likelihood
functions of the individual search channels with independent parameters, but
with a common signal normalisation. Figure 12 considers the case of an FRVZ
model with a $\gamma_{\mathrm{d}}$ mass of 100 MeV and presents the results of
the statistical combination of the SR${}_{2\textrm{c}}^{\textrm{ggF}}$,
SR${}_{2\textrm{c}}^{\textit{WH}}$ and SR${}_{\textrm{c}}^{\textit{WH}}$, as
95% CL upper limits on the branching fraction of a 125 GeV Higgs boson
decaying into hidden-sector particles and resulting in two dark photons. For
simplicity, the SR${}_{\textrm{c}+\mu}^{\textrm{ggF}}$ is not used in the
combination although it could offer an increase in sensitivity at the percent
level. The figure illustrates the complementary sensitivity of the WH
selection, in this case providing most of the sensitivity for dark-photon mean
proper decay lengths below 2 mm or above 50 mm.

Figure 12: Exclusion limits at the 95$\%$ CL on the branching ratio ($B$) for
the process $H\rightarrow\leavevmode\nobreak\ 2\gamma_{\textrm{d}}+X$, for a
$\gamma_{\mathrm{d}}$ mass of 100 MeV and a 125 GeV Higgs boson in an FRVZ
model, as a function of the dark-photon mean proper decay length $c\tau$. The
three dashed curves, taken from Figures 9(c), 10(a) and 10(c), show the
individual sensitivities of the search channels targeting caloDPJs in the ggF
and WH selections. The solid black curve shows the observed exclusion limit
from their statistical combination, and is almost identical to the expected
limit. The green and yellow bands represent $\pm 1\sigma$ and $\pm 2\sigma$
uncertainty in the expected limit. The hatched band denotes the region in
which the branching ratio of $H\rightarrow\leavevmode\nobreak\
2\gamma_{\textrm{d}}+X$ is larger than unity.

Upper limits at 90% CL333The limits are quoted at 90% CL in this
interpretation of the results to ease the comparison with other experiments
that conventionally use this CL threshold. are also set, in the context of the
FRVZ model and the HAHM vector-portal model, in terms of kinetic mixing
parameter $\epsilon$ and $\gamma_{\mathrm{d}}$ mass and presented in Figure 13
for different Higgs decay branching fractions into $\gamma_{\mathrm{d}}$,
ranging from 0.1% to 10%. The limits are interpolated between different masses
by branching fraction variations [9] as a function of the
$\gamma_{\mathrm{d}}$ mass, corrected by a linear interpolation of the signal
efficiency between adjacent available MC signal samples. For
$\gamma_{\mathrm{d}}$ mass below twice the muon mass, no coverage from
$\mu$DPJ signal regions is expected, motivating a significant drop in
sensitivity. The combination of the caloDPJ signal regions provides the most
stringent limit in this mass range. In the mass range above twice the muon
mass, only the best-performing channel for each $\gamma_{\mathrm{d}}$ mass
probed is considered. The structures observed in this mass range depend on the
expected $\gamma_{\mathrm{d}}$ branching ratio, where decays to QCD resonances
can significantly alter the sensitivities of the different search channels.

Exclusion regions from previous ATLAS searches for dark-photon jets are shown
for the displaced-decay search [14] and for the complementary prompt-decay
search [17]. The contributions from the caloDPJ channels allow ATLAS to set
limits for $m_{\gamma_{d}}<0.1\,\text{GeV}$ for the first time.

((a)) FRVZ ((b)) HAHM

Figure 13: The 90$\%$ CL exclusion contours of the branching ratio ($B$) for
the decay $H\rightarrow\leavevmode\nobreak\ 2\gamma_{\textrm{d}}+X$ as a
function of the $\gamma_{\mathrm{d}}$ mass and the kinetic mixing parameter
$\epsilon$ for (a) the FRVZ model and (b) the HAHM model. These limits are
obtained assuming branching fractions between 0.1% and 10% for Higgs boson
decays resulting in dark photons. For $\gamma_{\mathrm{d}}$ mass below twice
the muon mass, the combination of the caloDPJ signal regions is used. The
figure also shows regions excluded by the previous ATLAS searches for jets
from displaced [14] (orange line) and prompt [17] (red line) decays of dark
photons.

This search has significantly higher sensitivity than the previous ATLAS
search for light long-lived neutral particles [14], reaching a 0.1% branching
ratio for a 125 GeV Higgs boson decaying into dark photons.

##  8 Conclusion

This paper presents a search for light long-lived neutral particles which
decay into collimated pairs of fermions in the ATLAS detector at the LHC. Data
collected from 139 fb-1 of $\sqrt{s}=13$ TeV $pp$ collisions during 2015–2018
were analysed for evidence of long-lived dark photons from Higgs boson decays
and were found to be consistent with the background prediction. Upper limits
on the Higgs boson branching fraction to dark photons as a function of their
mass and mean proper decay length $c\tau$ are reported, assuming SM Higgs
boson production cross-sections. Branching fractions above 0.1% are excluded
at 95% CL for Higgs boson decays to two dark photons with mean proper decay
length between 10 mm and 250 mm and mass between 0.4 and 2 GeV. This analysis
has significantly higher sensitivity than previous ATLAS searches for light
long-lived neutral particles that decay to collimated pairs of fermions. The
higher sensitivity is due to the addition of an event selection targeting
long-lived dark photons from decays of Higgs bosons produced in association
with a $W$ boson and to the use of a larger dataset and new methodologies for
reconstructing dark-photon candidates and rejecting non-collision backgrounds.

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