Search for displaced photons produced in exotic decays of the Higgs boson
using 13 TeV $pp$ collisions with the ATLAS detector A search is performed for
delayed and nonpointing photons originating from the displaced decay of a
neutral long-lived particle (LLP). The analysis uses the full Run 2 data set
of proton–proton collisions delivered by the LHC at a center-of-mass energy of
$\sqrt{s}=13$ TeV between 2015 and 2018 and recorded by the ATLAS detector,
corresponding to an integrated luminosity of 139 fb-1. The capabilities of the
ATLAS electromagnetic calorimeter are exploited to precisely measure the
arrival times and trajectories of photons. The results are interpreted in a
scenario where the LLPs are pair-produced in exotic decays of the 125 GeV
Higgs boson, and each LLP subsequently decays into a photon and a particle
that escapes direct detection, giving rise to missing transverse momentum. No
significant excess is observed above the expectation due to Standard Model
background processes. The results are used to set upper limits on the
branching ratio of the exotic decay of the Higgs boson. A model-independent
limit is also set on the production of photons with large values of
displacement and time delay. SUSY-2019-14 CERN-EP-2022-096 Phys. Rev. D

###### Contents

  1. 1 Introduction
  2. 2 ATLAS detector
  3. 3 Data and Monte Carlo samples
  4. 4 Photon pointing and timing measurements
    1. 4.1 Pointing measurements
    2. 4.2 Timing measurements
  5. 5 Event selection
    1. 5.1 Object selection
    2. 5.2 Event selection
    3. 5.3 Analysis strategy and optimization
  6. 6 Background estimation
  7. 7 Systematic uncertainties
    1. 7.1 Signal normalization uncertainties
    2. 7.2 Signal shape systematic uncertainties
    3. 7.3 Background shape systematic uncertainties
  8. 8 Statistical analysis
  9. 9 Results
  10. 10 Conclusion

##  1 Introduction

The Standard Model (SM) provides an accurate description of the fundamental
particles and interactions at energy scales up to a few hundred GeV. However,
the SM fails to explain several key phenomena, such as gravity, dark matter,
and the matter–antimatter asymmetry in the universe. Such shortcomings suggest
the presence of new physics, motivating the search for beyond-the-SM (BSM)
physics.

Supersymmetry (SUSY) [1, 2, 3, 4, 5, 6, 7] is a theoretically well-motivated
extension of the SM that proposes the existence of new SUSY partners
(sparticles) of the SM particles. The sparticle has identical quantum numbers
to its SM partner, differing only by half a unit of spin. A new quantum
number, dubbed R-parity, distinguishes between supersymmetric and SM
particles. In R-parity-conserving SUSY models [8, 9, 10, 11, 12], the lightest
SUSY particle (LSP) is stable, and SUSY production in proton–proton ($pp$)
collisions at the LHC would produce sparticles in pairs. The sparticles would
then decay in cascades involving other sparticles and SM particles until the
LSP is produced. A neutral, weakly interacting LSP would usually escape the
detector without interacting, so LHC searches for SUSY often require a large
amount of missing transverse momentum ($E_{\text{T}}^{\text{miss}}$) in the
event.

In gauge-mediated SUSY breaking (GMSB) models [13, 14, 15, 16, 17, 18], the
gravitino ($\tilde{G}$) is the LSP for typical model parameter values. GMSB
phenomenology is largely determined by the properties of the next-to-lightest
supersymmetric particle (NLSP), since the decay chains of the sparticles with
higher mass would typically lead to the NLSP, which would subsequently decay
to produce the LSP. The weak coupling of the NLSP to the gravitino LSP could
generate a non-neglible lifetime of the NSLP, leading to displaced NLSP decays
[17].

An electrically neutral long-lived NLSP would avoid direct detection as it
traveled partway (or entirely) through the ATLAS detector before decaying,
with a prominent decay mode being to a photon plus the LSP. The photons
resulting from displaced NLSP decays would be produced with some delay
compared to prompt photons, due mostly to the time-of-flight of the massive
NLSP before its decay. In addition, due to the opening angle between the
photon and LSP produced in the NLSP decay, the photons would tend to be
“nonpointing”, meaning that their flight path would not be consistent with
originating from the primary vertex (PV) of the hard scatter of the event. The
capabilities of the ATLAS liquid-argon (LAr) electromagnetic (EM) calorimeter
to make precise “pointing” measurements of the flight direction and the
arrival time of photons ($t_{\gamma}$) can be exploited to search for such
photons. Given the size of the ATLAS detector, requiring LAr calorimeter
measurements of the delayed photons restricts the sensitivity of the analysis
to NLSP lifetimes of $\mathcal{O}$(ns).

Previous ATLAS analyses searched for nonpointing and delayed photons produced
in long-lived NLSP decays in the data sets of $pp$ collisions collected at
center-of-mass energies of 7 TeV [19] and 8 TeV [20] during Run 1 of the LHC.
Neither search found an excess above the SM background expectation, and set
limits in the context of a particular set of GMSB SUSY models. A recent Run 2
CMS search for long-lived particles in such models also found that their data
agreed with the SM prediction [21].

The discovery in 2012 of a particle with a mass near 125 GeV that is
consistent with expectations for the SM Higgs boson opens new avenues for
searches with BSM sensitivity. Specifically, the coupling of the Higgs boson
to mass leads to search prospects for pairs of sparticles produced in Higgs
boson decays, provided the sparticles are sufficiently light. Additional
motivation for this production mechanism arises from current uncertainties in
the branching ratios of the Higgs boson. Current experimental constraints from
the Higgs boson property measurements allow the branching ratio for Higgs
boson decays into invisible BSM states to be as large as 13% at 95% CL [22].

This analysis utilizes the full Run 2 ATLAS data set of 13 TeV $pp$
collisions, and searches for evidence of Higgs boson decays into a pair of
long-lived electrically neutral NLSP particles, each of which subsequently
decays into an LSP, which escapes the detector, and a photon. Given the value
of the Higgs boson mass, the energies of the final-state photons, as well as
the value of $E_{\text{T}}^{\text{miss}}$, are not sufficiently high to
provide a trigger for the events. Instead, the analysis considers SM Higgs
boson production in association with either a $Z$ boson, $W$ boson or
$t\bar{t}$ system. Leptonic decays of the associated $W/Z$ boson or $t\bar{t}$
system are exploited to provide a single-lepton (electron or muon) trigger. An
example leading-order Feynman diagram, for the case of $H+Z$ production, is
shown in Figure 1. To reduce the model-dependence of the results, the analysis
considers a simplified model where the masses of the NLSP and LSP and the
lifetime of the NLSP are treated as independent parameters.

Figure 1: Example Feynman diagram of the signal model, with a Higgs boson
produced in association with a $Z$ boson. The Higgs boson decays into a pair
of electrically neutral NLSP particles which are long-lived, each of which
then decays into a photon and an LSP. The $Z$ boson decays into a charged-
lepton pair (electrons or muons).

In addition to the requirement of the lepton trigger, selected events must
have at least one photon candidate. Events are divided into two orthogonal
final states, with the $1\gamma$ final state requiring exactly one photon and
the $\geq 2\gamma$ final state requiring two or more photons. As detailed in
Section 5, the analysis strategy exploits measures of photon pointing and
timing. The pointing information is used to divide the selected events into
five nonoverlapping categories with different signal-to-background ratios,
while the timing information provides the distribution for a likelihood fit
used to search for the possible presence of signal events.

The analysis exploits the expectation of significant
$E_{\text{T}}^{\text{miss}}$ in signal events, due to the escaping LSPs, as
well as any neutrinos produced in the leptonic decays of the associated
system. In addition, the signal would produce delayed photons, which would
have a positive value for their time of arrival at the EM calorimeter, whereas
prompt backgrounds have time distributions centered at zero. The analysis
therefore performs the signal search utilizing a signal region (SR) defined by
requiring high $E_{\text{T}}^{\text{miss}}$ and $t_{\gamma}>0$. The data-
driven background estimate is determined using a control region (CR) with low
values of $E_{\text{T}}^{\text{miss}}$. The background estimate is validated
using two different validation regions (VRs): the first requires intermediate
values of $E_{\text{T}}^{\text{miss}}$ and is denoted by
VR($E_{\text{T}}^{\text{miss}}$), while the second, denoted by VR($t$),
imposes the same $E_{\text{T}}^{\text{miss}}$ requirement as the SR but
reverses the timing requirement by requiring $t_{\gamma}<0$. By construction,
the various regions are mutually exclusive, and any signal contamination of
the CR, VR($E_{\text{T}}^{\text{miss}}$) and VR($t$) data samples can be
safely neglected. The development of the background modeling using the CR, and
its validation using the VRs, were finalized before the data in the SR was
unblinded and the final results obtained.

##  2 ATLAS detector

The ATLAS detector [23] 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 center of the detector and
the $z$-axis along the beam pipe. The $x$-axis points from the IP to the
center 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 $\eta=-\ln\tan(\theta/2)$. Angular distance is
measured in units of $\Delta R\equiv\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$.
It consists of an inner tracking detector surrounded by a thin superconducting
solenoid, electromagnetic and hadron calorimeters, and a muon spectrometer
incorporating three large superconducting air-core toroidal magnets.

The inner-detector system (ID) is immersed in a $2\text{\,}\mathrm{T}$ axial
magnetic field and provides charged-particle tracking in the range
$|\eta|<2.5$. The high-granularity silicon pixel detector covers the vertex
region and typically provides four measurements per track, the first hit
normally being in the insertable B-layer installed before Run 2 [24, 25]. It
is followed by the silicon microstrip tracker, which usually provides eight
measurements per track. These silicon detectors are complemented by the
transition radiation tracker (TRT), which enables radially extended track
reconstruction up to $|\eta|=2.0$. The TRT also provides electron
identification information based on the fraction of hits (typically 30 in
total) above a higher energy-deposit threshold corresponding to transition
radiation.

The calorimeter system covers the pseudorapidity range $|\eta|<4.9$. Within
the region $|\eta|<3.2$, electromagnetic (EM) calorimetry is provided by
barrel (EMB) and endcap (EMEC) 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 three barrel 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, optimized for electromagnetic
and hadronic energy measurements respectively.

The muon spectrometer (MS) 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. Three
layers of precision chambers, each consisting of layers of monitored drift
tubes, covers the region $|\eta|<2.7$, 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.

Events of interest are selected by the first-level trigger system implemented
in custom hardware, followed by selections made by algorithms implemented in
software in the high-level trigger [26]. The first-level trigger accepts
events from the $40\text{\,}\mathrm{MHz}$ bunch crossings at a rate below
$100\text{\,}\mathrm{kHz}$, which the high-level trigger further reduces in
order to record events to disk at about $1\text{\,}\mathrm{kHz}$.

An extensive software suite [27] 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 Monte Carlo samples

This analysis uses a data set of $\sqrt{s}=13$ TeV $pp$ collisions recorded by
the ATLAS detector from 2015 to 2018, corresponding to the full Run 2 period
of the LHC. Signal events with a Higgs boson produced in association with a
$Z$ boson were collected using the unprescaled single-electron and single-muon
triggers [28, 29] with the lowest transverse momentum threshold for the
electron or muon. After data quality requirements [30] are applied to ensure
that all detector components were in good working condition, the data set
corresponds to an integrated luminosity of $139.0\pm 2.4$ fb-1 [31, 32]. The
mean number of interactions per bunch crossing (“pileup”) during Run 2 was
$\langle\mu\rangle=33.7$.

No SM background Monte Carlo (MC) simulation samples are needed, since the
analysis uses a fully data-driven background estimation procedure. To study
the signal, MC samples for the $ZH$ production process were generated at
leading order (LO) using MadGraph5_aMC@NLO 2.6.2 [33], with a Higgs boson mass
of 125 GeV, and normalized to the cross sections reported in Ref. [34]. The
generated events were interfaced to Pythia 8 [35] to simulate subsequent
decays of the Higgs boson and supersymmetric particles in the signal model and
also the parton showering. The A14 set of tuned parameters [36] and the
NNPDF2.3lo PDF set [37] were used in the event generation. The signal model MC
events were passed through a Geant4 [38] simulation of the ATLAS detector [39]
and reconstructed with the same software [27] as used for the data. The
generation of the simulated event samples includes the effect of pileup, both
within the same bunch crossing as well as in bunch crossings before or after
the one containing the hard interaction. Events in the simulation are weighted
in order to reproduce the amount of pileup observed in the Run 2 data-taking
period.

The final state and kinematics of the Higgs boson decay depend on the masses
of the NLSP and LSP, in addition to the lifetime of the NLSP. The signal
generation was implemented with the $\textstyle\tilde{\chi}^{0}_{2}$ as the
NLSP and the $\textstyle\tilde{\chi}^{0}_{1}$ as the LSP, but the NLSP and LSP
masses and NLSP lifetime were treated as free parameters. Therefore, GMSB
signal models were generated in a three-dimensional grid. The grid of MC
signal models was constructed with NLSP masses from 30 GeV up to 60 GeV, near
the kinematic limit enforced by the Higgs boson mass. For each value of NLSP
mass, LSP masses from 0.5 GeV up to a value of $m_{\text{NLSP}}-10$ GeV were
considered. Given the typical step size of 10 GeV used for both the NLSP and
LSP masses, a total of 18 different ($m_{\text{NLSP}}$, $m_{\text{LSP}}$)
pairs were simulated for a given value of NLSP lifetime. For each set of NLSP
and LSP mass values, at least two different NLSP lifetimes, typically 2 ns and
10 ns, were simulated. Results can be generalized to other lifetime values by
appropriately reweighting the samples of simulated events.

Some cross-checks of detector performance utilize MC samples of $Z\rightarrow
ee$ events and events with radiative $Z\rightarrow\ell\ell\gamma$ decays, with
$\ell\ell$ designating an opposite-sign same-flavor lepton (electron or muon)
pair. The photon time resolution is not modeled accurately in the MC
simulation, which underestimates the time resolution observed in data.
Therefore, additional smearing is applied to the photon times as simulated in
the MC signal events. In addition to a contribution applied independently to
each photon, the additional smearing includes a correlated event-level
contribution to account for the impact of the spread of the actual $pp$
collision times, which results from the longitudinal profiles of the colliding
proton bunches along the LHC beamline. The combined smearing contributions are
tuned to match the timing performance observed for electrons in data because,
due to their similar EM shower development, electrons and photons have similar
timing performance. The correlated and uncorrelated contributions to the time
measurement are deconvolved by studying the times of electron–positron pairs
in $Z\rightarrow ee$ events.

##  4 Photon pointing and timing measurements

Photons produced in the decays of long-lived NLSPs can be nonpointing,
delayed, or both. This analysis is made possible by the capability of the LAr
EM calorimeters to perform precise measurements of the pointing and timing of
EM objects, and in particular photons, as discussed in more detail below. The
measurements of pointing and timing are almost completely uncorrelated for
prompt backgrounds. The lack of correlation results from the fact that, as
described below, the pointing measurement uses the spread of the EM shower to
precisely measure its centroids in the first two layers in the EM calorimeter,
while the timing measurement uses the time reconstructed from the pulse-shape
of only the second-layer cell with the maximum energy deposit. In contrast,
these measurements would be correlated for a photon that is produced in the
decay of a heavy NLSP. Using both variables therefore provides a powerful way
to separate signal from background.

###  4.1 Pointing measurements

For $|\eta|<2.5$, the LAr EM calorimeter is segmented into three layers in
depth that can be used to measure the longitudinal profile of the shower. The
first layer uses highly granular “strips” segmented in the $\eta$ direction,
with a typical transverse segmentation of
$\Delta\eta\times\Delta\phi=0.003\times 0.1$, designed to allow efficient
discrimination between single-photon showers and two overlapping showers from
the decay of a $\pi^{0}$ meson. The second layer has a typical transverse
segmentation of $\Delta\eta\times\Delta\phi=0.025\times 0.025$, and collects
most of the energy deposited in the calorimeter by EM showers initiated by
electrons or photons. Very high energy EM showers can leave significant energy
deposits in the third layer, which serves as a “tail catcher” and which can
also be used to correct for energy leakage beyond the EM calorimeter.

By precisely measuring the centroids of the EM shower in the first and second
EM calorimeter layers, the flight direction of photons can be determined, from
which one can calculate the value of $z_{\gamma}$, defined as the
$z$-coordinate of the extrapolation of the line through these centroids in the
$z$–$r$ plane back to the beamline ($x=y=0$). The angular resolution of the EM
calorimeter’s measurement of the flight direction of prompt photons is about
60 mrad/$\sqrt{E}$, where $E$ is the photon energy in GeV. This angular
resolution corresponds to a resolution in $z_{\gamma}$ of about 15 mm for
prompt photons with energies in the range of 50–100 GeV in the EM barrel
calorimeter. Given the geometry, the $z_{\gamma}$ resolution is worse for
photons reconstructed in the endcap calorimeters.

This analysis uses the measurement of the photon flight direction to search
for photons that do not point back to the PV. The pointing variable used in
the analysis is $\Delta z_{\gamma}=z_{\gamma}-z_{\mathrm{PV}}$, with
$z_{\mathrm{PV}}$ being the $z$-coordinate of the PV of the event, selected as
described in Section 5.2. Given that $z_{\mathrm{PV}}$ is measured with high
precision using the tracker, the $\Delta z_{\gamma}$ resolution is essentially
equivalent to the resolution in $z_{\gamma}$.

While the geometry of the EM calorimeter is optimized for detecting particles
that point back to near the nominal interaction point at the center of the
detector (i.e. $x=y=z=0$), the fine segmentation allows good pointing
performance to be achieved over a wide range of photon impact angles. Using a
sample of electrons from $Z\rightarrow ee$ events, the calorimeter pointing
performance is validated in data by using the finite spread of the $pp$
collision region along the $z$-axis. The pointing resolution degrades
gradually with increasing $|\Delta z_{\gamma}|$ (see Figure 1 in Ref. [20]).
The Geant4-based simulation of the ATLAS detector shows good agreement with
the resolution observed in data over the range of $|\Delta z_{\gamma}|$ values
(up to about 200 mm) that can be accessed in data. This gives confidence in
the ability of the MC simulation to estimate the resolution at the larger
$|\Delta z_{\gamma}|$ values characteristic of signal photons. The resolution
remains much smaller than $|\Delta z_{\gamma}|$ in the full region where the
signal is expected. The studies also show similar pointing performance for
photons and electrons, as expected given their similar EM shower development.

###  4.2 Timing measurements

Photons from long-lived NLSP decays would reach the LAr calorimeter with a
slight delay compared to prompt photons produced directly in the hard
scattering. This delay results mostly from the flight time of the heavy NLSP,
which would have a distribution of relativistic speed ($\beta=v/c$) that
typically peaks near 0.9 and has a tail to much lower values. In addition, the
opening angle in the NLSP decay, which causes the photon to be nonpointing,
results in the geometrical path to the calorimeter being longer than that for
a prompt photon from the PV.

The EM calorimeter, with its novel “accordion” design, and its readout, which
incorporates fast shaping, has excellent time resolution. Test-beam
measurements [40] of individual EMB calorimeter modules demonstrated a time
resolution of $\approx 100$ ps in response to high-energy electrons. In situ
calibration tests in the ATLAS cavern show a time resolution of $\approx 70$
ps [41], limited not by the calorimeter but by the jitter of the calibration
pulse injection system. To avoid that the electronic readout significantly
degrades the intrinsic timing performance of the calorimeter, quality-control
tests during production of the electronics required the clock jitter on the
LAr readout boards to be less than 20 ps, with typical values of 10 ps [42].

During the Run 2 period from 2015 to 2018, the hardware configuration of the
readout of the various LAr channels was adjusted such that the channels were
synchronized online within a uniformity of order 1 ns. While this level of
precision is sufficient for the trigger system to correctly identify the bunch
crossing corresponding to a particular event of interest, a dedicated
calibration for each channel is applied offline to improve the time resolution
significantly for EM clusters. The time calibration is based on a large sample
of $W\rightarrow e\nu$ events and is performed with zero defined as the
expected value for a prompt electron from the PV of the hard collision. The
time calibration includes corrections for the dependence of the time
measurement on energy, crosstalk, flight path from the PV, collision
conditions, and the behavior of individual channels.

The LAr energy and time for each calorimeter cell are reconstructed by
applying the optimal filtering algorithm [43] to the set of four samples of
the signal shape read out for each calorimeter channel, with successive
samples of the waveform separated by 25 ns. More specifically, the deposited
energy per cell and the time of the deposition are calculated using
appropriately weighted linear combinations of the set of samples of the
waveform:

$E=\sum_{i=0}^{3}a_{i}S_{i}\ \ \ \ \mathrm{and}\ \ \ \
t=\frac{1}{E}\sum_{i=0}^{3}b_{i}S_{i}\ \ ,$

where $S_{i}$ denotes the four samples of the signal waveform. The parameters
$a_{i}$ and $b_{i}$ are the optimal filter coefficients (OFC), the values of
which are calculated, knowing the pulse shape and noise autocorrelation
matrix, to deliver the best energy and time resolutions.

For this analysis, the arrival time of an EM object is measured using the
second-layer EM calorimeter cell with the largest energy deposit among cells
in the associated EM cluster ($E_{\mathrm{cell}}$). For the EM shower of an
electron or photon with an energy in the range of interest, this cell
typically contains about 20%–50% of the total energy deposited in the EM
shower.

To cover the full dynamic range of physics signals of interest, the ATLAS LAr
calorimeter readout boards [42] employ three overlapping linear gain scales,
dubbed High, Medium and Low, where the relative gain is reduced by a factor of
about ten for each successive scale. For a given event, any individual LAr
readout channel is digitized using the gain scale that provides optimal energy
resolution, given the energy deposited in that calorimeter cell. The
calibration of the time is determined separately for High and Medium gain for
each channel. The number of electron candidates from the $W\rightarrow e\nu$
sample digitized using Low gain, typically those with energies above about 200
GeV, is insufficient to obtain statistically precise results for the
calibration constants. Therefore, the analysis requires that selected photons
be digitized using either High or Medium gain, which has a negligible impact
on the signal efficiency for this analysis.

An independent sample of $Z\rightarrow ee$ events is used to validate the time
calibration and determine the resolution that is obtained by performing
Gaussian fits to the time distributions in bins of cell energy. Figure 2 shows
the time resolution for High and Medium gain cells with $|\eta|<0.4$, as a
function of $E_{\mathrm{cell}}$, the energy in the second-layer calorimeter
cell used to calculate the time for the sample of $Z\rightarrow ee$ events.
Similar results are obtained over the full coverage of the EM calorimeter.

The time resolution, $\sigma(t)$, is expected to follow the form
$\sigma(t)=p_{0}/E_{\mathrm{cell}}\oplus p_{1}$, where $E_{\mathrm{cell}}$ is
the cell energy, $\oplus$ denotes addition in quadrature, and the fit
parameters $p_{0}$ and $p_{1}$ are the coefficients of the so-called noise
term and constant term, respectively. Superimposed on Figure 2 are the results
of fits to this expected form of the time resolution function. The fits yield
values of $p_{1}$, which gives the time resolution in the limit of large
energy deposits, of 208 ps (217 ps) for High (Medium) gain. The slightly
poorer resolution for Medium gain is due to limited event yields in the
$W\rightarrow e\nu$ sample used to determine the time calibration constants.
By examining the timing distributions of the two electrons in $Z\rightarrow
ee$ candidate events, the time measurements are observed to include a
correlated contribution of $\approx 190$ ps, which, for a given PV position,
agrees well with the expected distribution of collision times due to the
longitudinal spread of the proton bunches along the LHC beamline. Assuming
this contribution to the time resolution to be uncorrelated with the LAr
contributions implies the LAr contributions are $\sim$ 50–150 ps, depending on
$|\eta|$, demonstrating that the offline calibration achieves an in situ time
resolution comparable to that achieved in the more constrained testbeam
meaurements [40].

Figure 2: Time resolution obtained with the LAr EM calorimeter as a function
of $E_{\mathrm{cell}}$, the energy in the second-layer cell with the largest
energy deposit. The solid lines show the results of fitting the expected form,
namely $\sigma(t)=p_{0}/E_{\mathrm{cell}}\oplus p_{1}$, to the data points.
The results shown are obtained using $Z\rightarrow ee$ events recorded in
2018, for electrons in the EMB calorimeter with $|\eta|<0.4$, and for both the
High and Medium gain readout scales. Similar results are obtained over the
full Run 2 data set and for the full coverage of the EM calorimeter.

The signal timing templates are constructed using the timing variable of the
simulated signal samples. Additional resolution uncertainties are assigned to
cover the observed discrepancy between radiative $Z\rightarrow\ell\ell\gamma$
events in data and those from MC simulation. The photon timing resolution
uncertainty is implemented in the likelihood function (see Section 7 for
details) such that the effects in the timing distribution are correlated
between pointing categories. Uncertainties accounting for fluctuations in the
statistically limited signal MC samples are also included in the likelihood
function.

##  5 Event selection

The requirements applied in the selection of various physics objects in the
event are described below, followed by the selection of events and the
optimization of these selections.

###  5.1 Object selection

The reconstruction and identification of electrons and photons are described
in Ref. [44]. Shape variables computed from the lateral and longitudinal
energy profiles of the EM showers in the calorimeter are used to identify
photons and electrons and to distinguish them from backgrounds.

Photons are required to satisfy $p_{\text{T}}>10$ GeV and $|\eta|<2.37$,
excluding the region $1.37<|\eta|<1.52$, which corresponds to the transition
region between the EMB and EMEC calorimeters. In addition, to reduce the
probability of jets being misidentified as photons, photons must satisfy both
calorimeter-based and track-based isolation requirements. The calorimeter
isolation variable is defined as the energy of EM calorimeter-cell clusters in
a cone of size $\Delta R=0.2$ around the photon candidate, excluding a fixed
window containing most of the photon shower. Corrections accounting for photon
energy leakage from this window into the surrounding cone, for pileup, and for
the underlying event are applied [44]. The calorimeter isolation is required
to be less than 6.5% of the photon’s transverse energy. The track-based
isolation variable is defined as the scalar sum of the $p_{\text{T}}$ of all
tracks with $p_{\text{T}}>1$ GeV within $\Delta R=0.2$ of the photon
candidate. The track isolation variable is required to be less than 5% of the
photon’s transverse energy.

A set of photon selection criteria, designed for high efficiency and modest
background rejection, defines the so-called Loose photon identification used
in this analysis. The Loose photon identification does not impose requirements
on the EM shower shape in the first (strips) layer of the EM calorimeter, and
uses only variables that describe the shower shape in the second layer of the
EM calorimeter and leakage into the hadronic calorimeter. The Loose photon
identification thereby minimizes the bias against identification of
nonpointing photons. Figure 3 shows the Loose photon identification efficiency
as a function of $|\Delta z_{\gamma}|$, with the simulated results for the
signal photons from this analysis superimposed on data and MC results for SM
$Z\rightarrow\ell\ell\gamma$ events. For the $Z\rightarrow\ell\ell\gamma$
events, $|\Delta z_{\gamma}|$ is measured from the center of the detector
($z=0$), exploiting the spread of $pp$ collisions along the beamline. The
radiative $Z\rightarrow\ell\ell\gamma$ sample is selected by requiring events
to have a photon plus two same-flavor opposite-sign leptons (electrons or
muons), with $m(\ell\ell\gamma)$ being within 10 GeV of the known $Z$ boson
mass, and the dilepton invariant mass being less than 83 GeV. To reduce the
impact of electrons being misidentified as photons, candidate events in the
$Z\rightarrow ee\gamma$ sample are vetoed if the invariant mass of either
$e\gamma$ pair is consistent with the $Z$ boson mass. The
$Z\rightarrow\ell\ell\gamma$ selection results in a very pure sample of
photons, from which the efficiency of applying the Loose photon identification
requirement is measured after removing the remaining background component
using a template fit to the $\ell\ell\gamma$ invariant mass distribution [45].
As seen in Figure 3, the Loose photon identification efficiency is very high
and quite constant within the beam spot ($|\Delta z_{\gamma}|$ less than about
200 mm), and the results for the various samples are in good agreement,
providing confidence in the use of MC simulation to estimate the
identification efficiency for signal photons at higher $|\Delta z_{\gamma}|$
values. As shown, the efficiency decreases gradually for photons produced
further from the center of the detector. To cover potential mis-modeling of
the efficiency drop in the high pointing region, photon identification
uncertainty is introduced and discussed in Section 7.2

Figure 3: The Loose photon identification efficiency, shown as a function of
$|\Delta z_{\gamma}|$, for photons with transverse momentum in the range
10–100 GeV and with $E_{\mathrm{cell}}>7$ GeV. The simulated results for
signal photons from long-lived particle decays in this analysis are overlaid
with data and MC results for SM $Z\rightarrow\ell\ell\gamma$ events. For the
$Z\rightarrow\ell\ell\gamma$ events, $|\Delta z_{\gamma}|$ is measured from
the center of the detector ($z=0$), exploiting the spread of $pp$ collisions
along the beamline.

Some studies in this analysis use the Medium photon working point, which adds
a requirement on one shower shape variable in the strips layer in order to
increase the photon purity at the cost of some decrease in efficiency.
Detailed descriptions of both the Loose and Medium photon identification
criteria can be found in Ref. [44].

Electrons are reconstructed by matching tracks in the ID to topological
clusters of calorimeter cells formed using the same dynamical, topological
cell-clustering algorithm as in the photon reconstruction [44]. Electron
candidates are required to have $p_{\text{T}}>10$ GeV and $|\eta|<2.47$, and
must satisfy a Medium electron identification selection based on a likelihood
discriminant using calorimeter shower shapes and track parameters [44].
Isolation criteria using calorimeter- and track-based information are applied
to electrons. The reconstructed track matched to the electron candidate must
be consistent with the PV position: its longitudinal impact parameter $z_{0}$
relative to the PV is required to satisfy $|z_{0}\sin\theta|<0.5$ mm. In
addition, the electron track’s transverse impact parameter with respect to the
beam axis divided by its uncertainty, $|d_{0}|/\sigma(d_{0})$, must be less
than 5.

Muons are reconstructed by matching tracks from the MS and ID subsystems.
Muons in the range $2.5<|\eta|<2.7$ without an ID track but whose MS track is
compatible with the interaction point are also considered. Muon candidates are
required to have $p_{\text{T}}>10$ GeV and $|\eta|<2.7$, and must satisfy the
Medium muon identification requirements [46]. Muons are required to satisfy
calorimeter- and track-based isolation requirements [46] that are 95%–97%
efficient for muons with $p_{\text{T}}\in[10,60]$ GeV and 99% efficient for
$p_{\text{T}}>60$ GeV. Muon tracks must satisfy $|z_{0}\sin\theta|<0.5$ mm and
$|d_{0}|/\sigma(d_{0})<3$. Finally, $\tau$ leptons are required to have at
least 20 GeV of $p_{\text{T}}$ and to lie within $|\eta|$ < 1.37 or 1.52 <
$|\eta|$ < 2.5

Jets are used only indirectly in the analysis, through an overlap removal
procedure and corrections to the calculation of $E_{\text{T}}^{\text{miss}}$.
Jets are reconstructed by applying a particle-flow algorithm [47] to noise-
suppressed positive-energy topological clusters [48] of calorimeter cells from
the anti-$k_{t}$ algorithm [49, 50] with radius parameter $R=0.4$. Jets must
have $p_{\text{T}}>25\leavevmode\nobreak\ $GeV and $|y|<4.4$. A jet-vertex-
tagger (JVT) multivariate discriminant [51] is applied to jets with
$p_{\text{T}}<60$ GeV and $|\eta|<2.4$ to suppress jets from pileup.

An overlap removal procedure is performed in order to avoid double-counting of
objects. First, electrons overlapping with photons within $\Delta R<0.4$ are
removed. Jets overlapping with photons ($\Delta R<0.4$) and electrons ($\Delta
R<0.2$) are removed. Electrons overlapping with the remaining jets ($\Delta
R<0.4$) are removed, to match the requirements imposed when measuring isolated
electron efficiencies. Finally, muons overlapping with photons or jets
($\Delta R<0.4$) are removed.

The value of $E_{\text{T}}^{\text{miss}}$ is defined as the magnitude of the
negative vector sum of the transverse momenta of the selected photons,
electrons, muons and jets, as well as the transverse momenta of remaining
low-$p_{\text{T}}$ particles, estimated using tracks matched to the PV but not
to any of the selected objects [52].

###  5.2 Event selection

Selected events are required to have a candidate PV for the hard scatter,
reconstructed from at least two charged tracks, each with $p_{\text{T}}>500$
MeV. If multiple vertices are reconstructed, the PV is selected as the one
with the largest sum of the $p_{\text{T}}^{2}$ values of the associated
tracks.

Events are required to pass the lowest unprescaled single-lepton (electron or
muon) trigger. For data taken in 2015, the electron (muon) trigger required
the lepton to satisfy $p_{\text{T}}>24\ (20)$ GeV. For 2016–2018, these
kinematic requirements on the trigger leptons were both increased to
$p_{\text{T}}>26$ GeV. The offline event selection required at least one
electron or muon, with $p_{\text{T}}>27$ GeV, satisfying Medium
identification, and matching the trigger lepton.

Triggered events are subject to a common set of preselection requirements. In
addition to the lepton requirement above, selected events are required to have
at least one photon candidate with $p_{\text{T}}>10$ GeV and satisfying Loose
photon identification.

To reject events with an electron misreconstructed as a photon, events with an
electron are rejected if the invariant mass of the electron–photon pair
satisfies $|m_{e\gamma}-m_{Z}|<15$ GeV.

To suppress photons from earlier or later bunch crossings, photons are
required to have measured arrival times at the LAr calorimeter of
$|t_{\gamma}|<12$ ns.

###  5.3 Analysis strategy and optimization

Due to their different signal-to-background ratios, the $1\gamma$ and $\geq
2\gamma$ final states are treated separately in the analysis. The $\geq
2\gamma$ final state has lower background, and higher sensitivity for smaller
values of the NLSP lifetime, for which there is a significant probability for
both NLSPs to decay before reaching the EM calorimeter. The use of the
$1\gamma$ final state recovers sensitivity for events where one photon is
missed, or where one NSLP decays beyond the boundaries of the ID.

For all selected events, the final statistical analysis uses the pointing and
timing measurements of only one photon. Since the $\Delta z_{\gamma}$
distribution should be symmetric for both signal and background, the pointing
distribution is folded by taking $|\Delta z_{\gamma}|$ as the variable of
interest instead of $\Delta z_{\gamma}$ itself. The inputs to the statistical
analysis are therefore the values of $|\Delta z_{\gamma}|$ and $t_{\gamma}$
measured for the photon selected in each event. Given the fact that, due to
geometry, the pointing resolution along the $z$-axis is better for EMB photons
than for EMEC photons, the photon selected for the statistical analysis must
be detected in the EMB. Thus, $1\gamma$ events must have their single photon
in the EMB. For $\geq 2\gamma$ events, at least one photon must be in the EMB;
for events with more than one EMB photon, the highest-$E_{\text{T}}$ EMB
photon is selected for the statistical analysis.

A study was performed in order to optimize the SR definition across the two-
dimensional signal grid of NLSP and LSP masses. The kinematic distributions of
the signal have a significant dependence on the value of the mass splitting,
$\Delta m=m_{\text{NLSP}}-m_{\text{LSP}}$. Higher values of $\Delta m$
typically lead to production of higher-energy photons and to more modest
values of $E_{\text{T}}^{\text{miss}}$. In contrast, lower $\Delta m$ leads to
lower photon energies and higher values of $E_{\text{T}}^{\text{miss}}$. The
optimization study revealed that close-to-optimal results can be achieved
across the entire signal grid by the definition of two different SR
selections: the low-$\Delta m$ SR selection is applied to signal models with
$\Delta m=10$ GeV

and the high-$\Delta m$ SR selection is applied to signal models with $\Delta
m>10$ GeV. The definitions of the two SR selections, which include different
requirements on $E_{\mathrm{cell}}$ and $E_{\text{T}}^{\text{miss}}$, are
summarized in Table 1, along with definitions of the corresponding CRs and
VRs.

While exploring the optimal SR selections, the optimization procedure also
examined how to best choose the number of bins and locations of the
corresponding bin edges for both the photon pointing and timing variables.
Considerations included maximizing the sensitivity while avoiding any empty
bins in the required background templates constructed from CR data. The
binning optimization was performed separately for the $1\gamma$ and $\geq
2\gamma$ final states. The results led to the choice of a common pointing
binning, with a total of five pointing categories. Optimization of the timing
binning led to seven (six) bins for the $1\gamma$ ($\geq 2\gamma$) final
state, the difference being mostly due to the lower CR event yields for the
$\geq 2\gamma$ channel. The bin definitions are included in Table 1.

Table 1: Optimized requirements defining the low-$\Delta m$ and high-$\Delta m$ SR selections, as well as the corresponding CRs and VRs. Also included are the optimized binnings for photon pointing and timing for both the $1\gamma$ and $\geq 2\gamma$ channels. | Low-$\Delta m$ selection | High-$\Delta m$ selection
---|---|---
| ($\Delta m=10$ GeV) | ($\Delta m>10$ GeV)
Parameter |  CR | VR($E_{\text{T}}^{\text{miss}}$) | VR($t$) |  SR |  CR | VR($E_{\text{T}}^{\text{miss}}$) | VR($t$) |  SR
$E_{\mathrm{cell}}$ [GeV] | > 7 | > 10
$E_{\text{T}}^{\text{miss}}$ [GeV] | < 30 | 30–50 | > 80 | > 80 | < 30 | 30–50 | > 50 | > 50
$t_{\gamma}$ [ns] |  > 0 |  > 0 |  < 0 |  > 0 |  > 0 |  > 0 |  < 0 |  > 0
$|\Delta z_{\gamma}|$ bins [mm] | [0, 50, 100, 200, 300, 2000]
$|t_{\gamma}|$ bins [ns] |
$1\gamma$ channel | [0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.5, 12.0]
$\geq 2\gamma$ channel | [0, 0.2, 0.4, 0.6, 0.8, 1.0, 12.0]

The signal acceptance is about 22% (10%) for signal models with an NLSP
lifetime of 2 ns (10 ns), where the acceptance is defined as the fraction of
simulated signal events passing the trigger and having at least one lepton and
at least one photon reconstructed in the final state. The lower acceptance for
higher NLSP lifetimes is expected, as more NLSP decays occur outside the ATLAS
ID and their decay photons are not captured by the EM calorimeter. The signal
efficiency is around 40% for all the signal models, defined as the fraction of
simulated signal events within the signal acceptance that pass the SR
selection requirements.

##  6 Background estimation

The background is expected to be completely dominated by $pp$ collision
events, with possible backgrounds due to cosmic rays, beam-halo events, or
other noncollision processes being negligible. The source of the Loose photons
in background events contributing to the selected sample is expected to be
either a prompt photon, an electron misidentified as a photon, or a jet
misidentified as a photon. In each case, the object providing the Loose photon
signature is prompt and originates from the PV.

The pointing and timing distributions expected for the background sources are
determined entirely using control samples in data. This avoids a reliance on
the precise MC simulation of the pointing and timing performance for the
backgrounds, and especially of the tails of their $|\Delta z_{\gamma}|$ and
$t_{\gamma}$ distributions. Additionally, using data samples naturally
accounts for the influence of pileup, the possibility of selecting the wrong
PV, and any instrumental or other effects that might influence the background
measurements.

It is known [20] that the measured $|\Delta z_{\gamma}|$ and $t_{\gamma}$
distributions of genuine photons are narrower than those of other physics
objects (particularly jets) that are misreconstructed as photons. The $|\Delta
z_{\gamma}|$ distribution is particularly sensitive to the background
composition, since it depends on details of the EM shower development.
Consequently, the statistical analysis (see Section 8) uses the $|\Delta
z_{\gamma}|$ value only to separate the events into mutually independent
pointing categories, and implements the normalization of the background in
each pointing category as an independent, unconstrained nuisance parameter.
Therefore, the fitting procedure eliminates the need to model the pointing
distribution of the background, and reduces the background modeling issue to
one of determining the shape of the background timing template.

Variations of the shape of the background $t_{\gamma}$ distribution are
explored by dividing the CR data into a genuine-photon enriched sample and a
fake-photon enriched sample. The actual purities of the individual samples do
not need to be known; they just need to be different from each other, so that
the total background timing template can then be constructed as a linear
combination of the timing templates from these two samples.

The sample of events from the CR is divided into mutually exclusive subsamples
according to whether or not the photon passes an identification requirement
(typically either the Medium or Tight definitions given in Ref. [44]) that is
more stringent than the Loose identification used to define the overall CR.
The photons that fail the more stringent identification are used to form the
fake-photon enriched sample, while those that pass it are put in the genuine-
photon enriched sample. To increase the event yield, the genuine-photon
enriched control sample also includes photons from the radiative
$Z\rightarrow\ell\ell\gamma$ decay sample.

As discussed earlier, the time resolution depends on $E_{\mathrm{cell}}$, the
energy deposited in the second-layer cell of the EM calorimeter that is used
to measure the arrival time of the photon. To mitigate any potential
discrepancy between the CR and SR timing shapes that arises from kinematic
differences, a reweighting procedure is applied to the control samples so that
the photon $E_{\mathrm{cell}}$ distribution in the CR after reweighting is the
same as that in the SR.

Due to imperfections in calibrations, limited event yields, or other effects,
the photon timing distributions in the data can have small nonzero mean
values, at the level of up to a few tens of picoseconds. Before constructing
the final background timing templates, the means are determined by fitting the
timing distributions, which are then shifted to eliminate any small residual
nonzero mean values. To avoid any bias being introduced by possible signal
contributions, the fits are restricted to the core of the timing
distributions.

Figure 4 shows the unit-normalized shapes of the pointing and timing
distributions for the inclusive genuine-photon enriched and fake-photon
enriched samples. Superimposed are the distributions for selected signal
models.

Figure 4: Photon pointing ($\Delta z_{\gamma}$) and timing ($t_{\gamma}$)
distributions for the genuine-photon enriched and fake-photon enriched
samples. Superimposed are the expected distributions for selected signal
models. The signal models are labeled by the values of their NLSP and LSP
masses ($m_{\mathrm{NLSP}}$ and $m_{\mathrm{LSP}}$ in GeV) and NLSP lifetime
($\tau$ in ns).

The timing templates are derived separately in each of the pointing
categories. In general, the fraction of the genuine-photon enriched template
in the linear combination is a free nuisance parameter in the likelihood
function and can be determined when fitting to data. In two of the pointing
categories, the shapes of the timing distributions for the genuine-photon
enriched and fake-photon enriched samples are found to be statistically
consistent according to a Kolmogorov–Smirnov test, so the fit cannot obtain a
well-determined value for the nuisance parameter. For these two pointing
categories, the genuine-photon enriched and fake-photon enriched template
fractions are each set to a constant value of 0.5, to exploit the statistical
power obtained by combining the two samples.

##  7 Systematic uncertainties

Systematic uncertainties affecting the search are primarily experimental.
Theoretical uncertainties are also relevant, when the search result is
interpreted to yield a constraint on the decay of the Higgs boson into long-
lived particles.

In the statistical analysis, the background normalization for each $|\Delta
z_{\gamma}|$ category is determined using an independent nuisance parameter.
Therefore, it is not necessary to include systematic uncertainties for the
background’s normalization or its shape in the variable $|\Delta z_{\gamma}|$.
The various systematic uncertainties relevant for this analysis can be divided
into two categories: so-called “flat” uncertainties are not a function of
$|\Delta z_{\gamma}|$ or $t_{\gamma}$ and affect only the overall signal
yield, while “shape” uncertainties are those that are related to the shapes of
the unit-normalized $|\Delta z_{\gamma}|$ and $t_{\gamma}$ distributions for
signal or to the shape of the background $t_{\gamma}$ template.

The various systematic uncertainties are outlined below. However, their impact
on the limits is very minor, and the results of the analysis are dominated by
the statistical uncertainties. Implementing the full set of systematic
uncertainties degrades the sensitivity, as measured by the expected 95% CL
limits on the branching ratio $\mathcal{B}$($H\rightarrow{\mathrm{NLSP\
NLSP}}$), by only about 5% compared to the results obtained with only
statistical uncertainties considered.

###  7.1 Signal normalization uncertainties

Theoretical uncertainties of the signal model arise from the fact that the
Higgs boson production processes are calculated at LO in QCD, from
uncertainties in the value of the strong coupling constant
$\alpha_{\text{s}}$, and from the choice of parton distribution functions.
While the cross sections of the Higgs boson production processes are
normalized to the best available theoretical calculations [34], the missing
higher order QCD calculations in the event generation may lead to a potential
mis-modeling of signal efficiency. Such uncertainties are estimated by varying
the factorization and renormalization scales used in the event generation
separately by factors of 0.5 and 2.0 relative to their nominal values, and the
uncertainty is taken as the maximum difference between the yields obtained
with the nominal scales and the different variations. In addition, the
variation in signal selection efficiency due to the choice of PDF sets in the
event generation is also estimated. The total combined theoretical uncertainty
in the signal yield reaches about 17%, and is included as a single nuisance
parameter in the fit.

Examples of instrumental systematic uncertainties which affect the overall
signal yield include uncertainties in photon and lepton reconstruction and
identification efficiencies, reconstruction of $E_{\text{T}}^{\text{miss}}$,
and the integrated luminosity measurement. The uncertainty in the integrated
luminosity is 1.7%, evaluated using the methodology described in Ref. [31].
The photon identification efficiency is studied using a sample of photons from
radiative $Z\rightarrow\ell\ell\gamma$ decays, selected as described in
Section 6. The efficiency of the Loose identification is around 95% for
$p_{\text{T}}(\gamma)\approx 20$ GeV, increasing gradually to about 99% for
$p_{\text{T}}(\gamma)>60$ GeV. The measured efficiency in data is well
reproduced in MC simulations.

Summing the instrumental uncertainties in quadrature yields a small total
uncertainty in the signal yield, up to a few percent. These uncertainties are
incorporated into the likelihood model as a single nuisance parameter scaling
the signal yields uniformly across the ten pointing categories, where the
magnitude of the uncertainty is calculated separately for the $1\gamma$ and
$\geq 2\gamma$ channels.

###  7.2 Signal shape systematic uncertainties

Uncertainties in photon timing and pointing measurements, the identification
efficiency for photons with large pointing values, as well as the timing
templates, affect the distribution of relative event yields between timing
bins in different pointing categories. These uncertainties therefore affect
the shapes of the signal distributions, and are incorporated into the
likelihood model such that varying their corresponding nuisance parameters
introduces correlated shape effects in the timing and pointing distributions.
For example, as shown in Figure 3, the Loose photon identification
requirements provide very high efficiency for identifying photons, but that
efficiency has a modest dependence on $|\Delta z_{\gamma}|$, dropping by about
15% (5%) for the highest $|\Delta z_{\gamma}|$ bin for low (high)
$p_{\text{T}}(\gamma)$ values. To obtain a conservative estimate of the
uncertainty in identifying nonprompt photons, the ratio of efficiency in each
$|\Delta z_{\gamma}|$ category above 150 mm to the efficiency measured in data
for $|\Delta z_{\gamma}|=100$–150 mm is calculated. The deviation of these
ratio values from unity provides a binned relative uncertainty in the photon
identification efficiency that increases as $|\Delta z_{\gamma}|$ increases.

###  7.3 Background shape systematic uncertainties

Several types of systematic uncertainties are considered for the shapes of the
background timing distributions. These include the bin-by-bin statistical
uncertainties for both the genuine-photon enriched and fake-photon enriched
templates to account for fluctuations in data control samples.

In addition, a shape uncertainty correlated between timing bins within the
same pointing category is introduced to account for uncertainties arising from
the choice of CR samples for determining the genuine-photon enriched and fake-
photon enriched timing distributions. The uncertainties are derived by
comparing different choices for the division of events into those two
subsamples, namely by requiring photons to pass the Medium or the Tight
identification requirements in order to be included in the genuine-photon
enriched sample.

Finally, a background-only fit of the likelihood function to the VR($t$) data
is performed. The VR($t$) data has the same requirements as the SR but with
$t_{\gamma}<0$. This selection ensures negligible signal contamination, and
can therefore be used to test the robustness of the background model. The
residual difference between data in the VR($t$) region and the fitted
background is considered as a nonclosure uncertainty for the background
templates. While the nonclosure uncertainties are zero for most of the time
bins, a few have values up to about 40% of the background uncertainty in that
bin.

##  8 Statistical analysis

Binned likelihood fits to data in the SR are performed to test the background-
only and signal-plus-background hypotheses. The likelihood function is
constructed by simultaneously fitting the ten timing distributions to their
pdfs. The systematic uncertainties described in Section 7 are incorporated
into the likelihood function as nuisance parameters with Gaussian constraint
pdfs. The background normalization and the background template admixture in
each pointing category are floating nuisance parameters constrained by data.

A profile likelihood ratio is used to perform frequentist hypothesis tests
[53]. The $p_{0}$-value, defined as the probability of statistical
fluctuations making the background distributions appear to contain at least as
much signal as the data distributions, is calculated for different signal
models.

Before applying the statistical analysis to the SR data, the method and
background model were validated by fitting to the
VR($E_{\text{T}}^{\text{miss}}$) data. Since the background estimation method
essentially models the photon object property of timing and does not depend on
the event selection, the fits to the VR($E_{\text{T}}^{\text{miss}}$) photon
sample, which is statistically independent from the SR, provide a validation
of the method. As shown in Figure 5, the VR($E_{\text{T}}^{\text{miss}}$) data
are well described by a background-only fit, for both the high-$\Delta m$ and
low-$\Delta m$ selections. Further validation of the statistical analysis
included signal injection tests, in which varying signal models and strengths
were injected into the VR($E_{\text{T}}^{\text{miss}}$) data; in all cases,
the fit correctly extracted the strength of the injected signal.

Figure 5: Timing distributions for VR($E_{\text{T}}^{\text{miss}}$) data and
the estimated background, as determined by a background-only fit. The upper
(lower) row shows the results for the high-$\Delta m$ (low-$\Delta m$)
selection. The fit for each selection is performed simultaneously across ten
timing distributions, namely the five pointing categories for each of the two
final states ($1\gamma$ and $\geq 2\gamma$). For conciseness, the figures show
the sum of the data across all five pointing categories, compared with the sum
of the background-only fits, with the left (right) figure displaying the
results for the $1\gamma$ ($\geq 2\gamma$) final state.

##  9 Results

The post-fit plots comparing the timing distributions of the SR data and the
estimated background in each pointing category are shown in Figures 6 and 7
for the high-$\Delta m$ and low-$\Delta m$ selections, respectively.
Superimposed on the figures are the expected signal distributions for a few
example signal models. The data are in good agreement with the fitted
background-only hypothesis in all categories for both the high-$\Delta m$ and
low-$\Delta m$ selections. No significant excesses are observed, with the
largest excess for any signal model considered having a significance of about
1.2 standard deviations. Therefore, a modified frequentist CLs method [54] is
used to set 95% CL upper limits on $\mathcal{B}(H\rightarrow{\mathrm{NLSP\
NLSP}})$, assuming that
$\mathcal{B}(\mathrm{NLSP}\rightarrow\mathrm{LSP}+\gamma)$ is equal to unity.

Figure 6: Timing distributions for SR data and the estimated background, as
determined by the background-only fit, for the high-$\Delta m$ selection. The
plots on the left (right) show the time distributions for each of the five
pointing categories for the $1\gamma$ ($\geq 2\gamma$) final state. For
comparison, the expected timing shapes for a few different signal points are
shown superimposed, with a signal normalization corresponding to a value of
$\mathcal{B}(H\rightarrow{\mathrm{NLSP\ NLSP}})=20$%. The signal models are
labeled by their values of the NLSP and LSP masses (in GeV) and NLSP lifetime
(in ns).

Figure 7: Timing distributions for SR data and the estimated background, as
determined by the background-only fit, for the low-$\Delta m$ selection. The
plots on the left (right) show the time distributions for each of the five
pointing categories for the $1\gamma$ ($\geq 2\gamma$) final state. For
comparison, the expected timing shapes for a few different signal points are
shown superimposed, with a signal normalization corresponding to a value of
$\mathcal{B}(H\rightarrow{\mathrm{NLSP\ NLSP}})=20$%. The signal models are
labeled by their values of the NLSP and LSP masses (in GeV) and NLSP lifetime
(in ns).

For a fixed NLSP lifetime, the limits are directly obtained for the 18
simulated signal models, covering a two-dimensional grid of NLSP masses
ranging from 30 GeV to 60 GeV, and LSP masses from 0.5 GeV to
$m_{\mathrm{NSLP}}-10$ GeV. For visualization purposes, a bilinear
interpolation between the discrete signal points simulated for each fixed
lifetime is performed in order to obtain the 95% CL limits across the full
available range of NLSP and LSP masses. In the high-$\Delta m$ case, the
interpolation is performed on a two-dimensional grid of $m_{\text{LSP}}$
versus $m_{\text{NLSP}}$. The results for fixed NLSP lifetimes of 2 ns and 10
ns are shown in Figure 8. A selection of contours corresponding to fixed
values of the expected and observed 95% CL upper limits on
$\mathcal{B}(H\rightarrow{\mathrm{NLSP\ NLSP}})$ is superimposed on the
$(m_{\text{NLSP}},m_{\text{LSP}})$ plane.

Figure 8: Interpolated 95% CL limits on
$\mathcal{B}$($H\rightarrow{\mathrm{NLSP\ NLSP}}$) in the plane of LSP mass
versus NLSP mass for the high-$\Delta m$ selection. The plot on the left
(right) corresponds to the results with the NLSP lifetime equal to 2 ns (10
ns). A smoothing function is used to continuously fill the plane,
interpolating between the results for the discrete signal models which were
simulated on a grid with 10 GeV spacing, and with a minimum value of $\Delta
m=20$ GeV for the signal models covered by the high-$\Delta m$ selection. For
visualization purposes, contours are superimposed indicating where the
expected and observed 95% CL limits on
$\mathcal{B}$($H\rightarrow{\mathrm{NLSP\ NLSP}}$) reach values of 2%, 5%, and
10%.

In the low-$\Delta m$ case, the interpolation can only be performed in one
dimension, providing the 95% CL limit versus the NLSP mass, since the LSP mass
is fixed according to the relation $m_{\text{NLSP}}-m_{\text{LSP}}=10$ GeV.
The observed and expected 95% CL $\mathcal{B}$($H\rightarrow{\mathrm{NLSP\
NLSP}}$) upper limit curves for the low-$\Delta m$ selection are shown in
Figure 9 for NLSP lifetimes of 2 ns and 10 ns.

Figure 9:  Interpolated 95% CL limits on
$\mathcal{B}$($H\rightarrow{\mathrm{NLSP\ NLSP}}$) as a function of NLSP mass
for signal points in the the low-$\Delta m$ selection. Straight lines are
drawn between the results for the discrete signal models considered by the
low-$\Delta m$ selection, which were simulated with $\Delta m=10$ GeV. The
plot on the left (right) corresponds to the results with the NLSP lifetime
equal to 2 ns (10 ns).

Examining the results in Figures 8 and 9, there is a clear dependence of the
achieved limit on the signal mass splitting, $\Delta m$. The most stringent
constraints are achieved toward the bottom right corner of Figure 8, where 95%
CL upper limits on $\mathcal{B}$($H\rightarrow{\mathrm{NLSP\ NLSP}}$) of order
1% are obtained. In that region of the signal mass grid, the mass splitting
between the NLSP and LSP is large, giving rise to higher-energy photons and,
therefore, higher photon acceptance and lower backgrounds. Moving from the
bottom right to upper left in Figure 8, the limits smoothly become less
stringent as the mass splitting decreases, a trend which continues into the
low-$\Delta m$ results in Figure 9. However, it should be noted that the 95%
CL upper limit on $\mathcal{B}$($H\rightarrow{\mathrm{NLSP\ NLSP}}$) is below
10% over most of the two-dimensional mass plane for the high-$\Delta m$ case
shown in Figure 8.

The observed BSM limits also depend on the NLSP lifetime. Comparing the left
and right plots of Figures 8 and 9 shows that, for a given set of NLSP and LSP
mass values, the upper limits on $\mathcal{B}$($H\rightarrow{\mathrm{NLSP\
NLSP}}$) for an NLSP lifetime of 2 ns are tighter than those for a 10 ns NLSP
lifetime. This reflects the fact that the probability of at least one NLSP
decaying before reaching the EM calorimeter is higher for low-lifetime signal
points, resulting in higher photon (and therefore signal) acceptance. The
lifetime dependence of the results is further explored by appropriately
reweighting the simulated MC samples, typically generated with fixed NLSP
lifetimes of either 2 ns or 10 ns, in order to produce reweighted samples
corresponding to other lifetime values, ranging from 250 ps to 100 ns.

Using this lifetime reweighting method, Figure 10 presents the 95% CL limits
on $\mathcal{B}$($H\rightarrow{\mathrm{NLSP\ NLSP}}$) in the two-dimensional
plane of LSP mass versus NLSP lifetime. An interpolation is again used to
provide smooth results throughout the plane, connecting the discrete mass
values that were simulated. The results are shown separately in four plots,
with the NLSP mass fixed within a given plot, and with the four fixed NLSP
mass values considered being 30, 40, 50, and 60 GeV.

Figure 10: Interpolated 95% CL limits on
$\mathcal{B}$($H\rightarrow{\mathrm{NLSP\ NLSP}}$) in the plane of LSP mass
versus NLSP lifetime. Within each plot, the NLSP mass is held fixed, with
values of 60 GeV (top left), 50 GeV (top right), 40 GeV (bottom left), and 30
GeV (bottom right). A smoothing function is used to continuously fill the
plane, interpolating between the results for the discrete signal models, which
were simulated on a grid with 10 GeV spacing. For visualization purposes,
contours are superimposed to indicate where the expected and observed 95% CL
limits on $\mathcal{B}$($H\rightarrow{\mathrm{NLSP\ NLSP}}$) reach values of
1%, 2%, 5%, 10% or 20%, depending on the plot.

The results in Figure 10 demonstrate that the limits are most stringent for
NLSP lifetimes of the order of a few ns, as expected given the size of the ID.
The limits become weaker for both shorter and longer lifetimes. The
sensitivity decreases for longer lifetimes because more NLSP decays occur
outside the ID volume and therefore fewer decay photons are reconstructed in
the EM calorimeter, with a resultant loss of signal acceptance. For very short
lifetimes, the signal pointing and timing distributions become more difficult
to separate from the prompt background, reducing the sensitivity of the
analysis. These trends can be illustrated more clearly by examining the
evolution of the limits as a function of NLSP lifetime for fixed NLSP and LSP
masses, as shown in Figure 11.

Figure 11: The 95% CL limits on $\mathcal{B}$($H\rightarrow{\mathrm{NLSP\
NLSP}}$) from the high-$\Delta m$ selection as a function of NLSP lifetime for
fixed values of the NLSP and LSP masses. Within each plot, the NLSP mass is
held fixed, with values of 60 GeV (top left), 50 GeV (top right), 40 GeV
(bottom left), and 30 GeV (bottom right). Each plot includes multiple curves,
each with the LSP mass fixed, as shown in the legend. The 95% CL observed
limits are shown as solid lines, while the 95% CL expected limits and their
$\pm 1\sigma$ uncertainties are shown as the dotted lines and shaded bands,
respectively. The crossings of the observed and expected limits are a result
of changing relative importance of the $1\gamma$ and $\geq 2\gamma$ channels
as a function of signal lifetimes.

In addition to the results reported above for LLPs produced in Higgs boson
decays, a more model-independent test is performed to provide a limit on the
production of displaced photons with large values of both pointing and timing,
using the selection requirements of the high-$\Delta m$ analysis. The test is
performed separately for the $1\gamma$ and $\geq 2\gamma$ channels, as well as
their combination. The model-independent test region is defined for each
channel by the lower bound for the single bin with the highest pointing and
timing values. The pointing requirement is thus $|\Delta z_{\gamma}|>300$ mm
for both channels, while the timing requirement is $t_{\gamma}>1.5$ ns ($1$
ns) for the $1\gamma$ ($\geq 2\gamma$) channel. The same likelihood fit
procedure described in Section 8 is applied, but eliminating the model-
independent test regions from the fit. The background prediction and
uncertainty in the model-independent test regions are then determined by
extrapolating the results of this background-only fit to the model-independent
region with higher pointing and timing values.

The model-independent limits are summarized in Table 2. For the two channels
combined, four events are observed in the data, whereas the background
prediction is $4.1\pm 1.7$ events. Since the observed numbers of events are
consistent with the background expectation in all regions, the results are
used to set 95% CL upper limits on the visible cross sections of BSM signals
in the model-independent test regions. The reconstruction efficiency for
signal events, after taking into account the acceptance of the selection, is
typically around 5% for signal samples considered in this analysis.

Table 2: Expected background event yield and observed event yield in the model-independent test regions, as defined for the $1\gamma$ and $\geq 2\gamma$ channels by the listed requirements, as well as for the combination of the two channels. Also shown are the values of $\sigma^{95}_{\mathrm{vis}}$, the corresponding 95% CL limit on the visible cross section of BSM signals. Channel | Test region requirements | Expected | Observed |  $\sigma^{95}_{\mathrm{vis}}$ [fb]
---|---|---|---|---
$1\gamma$ |  $1.5<t_{\gamma}<12$ ns, |$\Delta z_{\gamma}$ | $>$ 300 mm | 3.8 $\pm$ 1.6 | 4 | 0.042
$\geq 2\gamma$ |  $1.0<t_{\gamma}<12$ ns, |$\Delta z_{\gamma}$ | $>$ 300 mm | 0.28 $\pm$ 0.04 | 0 | 0.022
$\geq 1\gamma$ |  | 4.1 $\pm$ 1.7 | 4 | 0.041

##  10 Conclusion

A search is performed for delayed and nonpointing photons produced from exotic
decays of the 125 GeV Higgs boson into a pair of BSM LLPs, such as the NLSP
present in GMSB models. The data set used was recorded by the ATLAS detector
at the LHC and corresponds to an integrated luminosity of 139 fb-1 of $pp$
collisions at a center-of-mass energy of $\sqrt{s}=13$ TeV.

Using measurements of the trajectories and arrival times of photons in the
ATLAS LAr EM calorimeter to search for possible displaced photons, no excess
is observed above the expectation from prompt backgrounds, which are modeled
entirely using data control samples. The results are used to set limits on
$\mathcal{B}$($H\rightarrow{\mathrm{NLSP\ NLSP}}$) for NLSP lifetimes ranging
from 0.25 ns to 100 ns, assuming that
$\mathcal{B}(\mathrm{NLSP}\rightarrow\mathrm{LSP}+\gamma)=100$%. The limits
are determined in a two-dimensional grid of NLSP mass values from 30 GeV to 60
GeV and LSP masses from 0.5 GeV up to $m_{\text{NLSP}}-10$ GeV. The most
stringent constraints limit $\mathcal{B}$($H\rightarrow{\mathrm{NLSP\ NLSP}}$)
to less than about 1% for intermediate NLSP lifetime values and for the
largest NLSP mass and mass-splitting between the NLSP and LSP. A model-
independent limit is also set on the production of photons with large values
of displacement and time delay.

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