11institutetext: Université Clermont Auvergne, CNRS/IN2P3, LPCA, 63000
Clermont-Ferrand, France22institutetext: Boston University, 590 Commonwealth
Avenue, Boston, Massachusetts 02215, USA

# Probing exotic long-lived particles from the prompt side using the CONTUR
method

L. Corpe 1 A. Goudelis 1 S. Jeannot 2 S. Jeon<EMAIL_ADDRESS>
###### Abstract

A method to derive constraints on new physics models featuring exotic long-
lived particles using detector-corrected measurements of prompt states is
presented. The Contur workflow is modified to account for the fraction of
long-lived particles which decay early enough to be reconstructed as prompt,
making it possible to determine how many of signal events would be selected in
the Rivet routines which encapsulate the fiducial regions of dozens of
measurements of Standard Model processes by the ATLAS and CMS collaborations.
New constraints are set on several popular exotic long-lived particle models
in the early-decay regime, which is often poorly covered by direct searches.
The probed models include feebly-interacting dark matter, hidden sector models
mediated by a heavy neutral scalar, dark photon models and a model featuring
photo-phobic axion-like particles.

##  1 Introduction

The discovery of the Higgs boson in 2012 closed an important chapter of the
history of particle physics. But despite all the triumph of the discovery, the
Standard Model (SM) is not the end of the story of fundamental physics.
Indeed, the lack of a convincing candidate for Dark Matter (DM), the issue of
the hierarchy problem, the matter-antimatter asymmetry of the universe, and
the established fact of neutrino oscillations, all point towards the existence
of new, undiscovered particles. Nevertheless, now that the first decade of LHC
data has been scrutinized, no long-standing signs of new particles or
interactions have come to light. This prompts the question: “are there
assumptions in the LHC research programme that could be hampering a
discovery?” Indeed, one possible blind spot is the existence of semi-stable
new particles, which could travel far enough into the detector to leave
neither missing transverse momentum signatures, nor be reconstructed as
standard objects. Since the standard ATLAS and CMS reconstruction algorithms
assume that activity started at the interaction point, many of these new
signals could have been thrown away as noise. Besides the experimental
motivation, these exotic Long-Lived Particles (LLPs) also have good
theoretical credentials. Long lifetimes occur when a particle’s width becomes
small, typically because of a small matrix element (for example, small
couplings or heavy virtual particles mediating the decay) or phase space (for
example, due to small mass differences between parent and daughter particles).
These mechanisms occur frequently in the SM, and it should be expected that
they would occur in a wide variety of new physics models that resolve the
fundamental questions facing particle physics, from supersymmetry
ArkaniHamed:2012gw ; Giudice:1998bp ; Barbier:2004ez ; Csaki:2015fea ;
Fan:2011yu ; Fan:2012jf  to hidden sector (HS) Strassler:2006im ;
Strassler:2006ri ; Chan:2011aa  and axion-like particle models Brivio:2017ije
.

Over the last decade, there has been an upsurge in interest in exotic LLP
searches at the LHC Alimena:2019zri , encompassing an impressive variety of
signatures from displaced vertices and jets to anomalous ionization (see Refs
ATLAS:2023cjw ; CMS-EXO-22-020-new ; ATLAS:2024ocv ; CMS:2024trg ;
ATLAS:2023zxo ; ATLAS:2024qoo  for recent examples). These searches have had
to re-invent the way that particles are reconstructed in multipurpose
experiments like ATLAS or CMS, to bypass trigger constraints, re-purpose
standard reconstruction algorithms and control unusual backgrounds (such as
cosmic rays or beam-induced activity). These dedicated searches can be
correspondingly complex and time-consuming. Nevertheless, much progress has
been made in recent years, and constraints on the production cross-sections
(times branching fraction) of LLP production against the lifetime are set.
Summary plots that give an overview of the experimental situation, for example
for HS models can be found in Ref. ATL-PHYS-PUB-2022-007  for ATLAS and Ref.
CMSsummar  for CMS.

However, despite this progress, the low-lifetime (or equivalently, early-
decay) regime remains ill-covered. Many models have no constraints set by the
multipurpose experiments in the region of $c\tau$ (the speed of light times
mean proper lifetime) that straddles prompt and displaced decays. This is
ostensibly because of the difficulty in reconstructing secondary vertices and
the presence of sometimes overwhelming SM backgrounds such as multijet
production. In this paper, we argue that existing measurements of prompt
processes at the LHC can be used to constrain LLP production: whatever the
mean proper lifetime of an exotic LLP, the decay distribution is a falling
exponential, and therefore, some fraction of LLPs will behave like prompt
particles. Hence, some LLP signatures might already have been visible in well-
studied final states for which precision measurements exist at ATLAS and CMS,
and can therefore be excluded without the need for a dedicated search. This
would liberate person power to study the significantly displaced regions in
more detail, making better use of ATLAS and CMS analyst resources. We will
describe a method for estimating these constraints using existing open source
software, and produce new limits on a variety typical benchmark LLP models
which are studied at the LHC, sometimes constraining the same models in the
low-lifetime region for the very first time.

The rest of this paper is organised as follows. Section 2 reviews the
established Contur method to extract constraints on new-physics models from
the existing bank of LHC measurements. Section 3 describes how this method
normally breaks down for LLP models, and describes a proof-of-principle
workflow to remedy this drawback. Sections 4, 5, 6 and 7 then illustrate the
application of the modified method to several benchmark LLP models commonly
probed by ATLAS and CMS LLP searches, extracting new constraints which
complement the direct search programme. The models are chosen to illustrate a
breadth of experimental signatures: they include examples of single production
and pair production; leptonic and hadronic decays; and scenarios where the LLP
is electrically charged or neutral. Our choice of template models and
parameter ranges is not guided by theoretical considerations. Instead we
choose to work with models which are already employed by LHC experimental
collaborations to be able to establish comparisons with existing searches.
Finally, we present our conclusions in Section 8.

##  2 Re-interpretation of measurements with Contur

There exist broadly two ways to extract constraints on new models using
existing LHC results. The first is to embed the original benchmark model (on
which constraints have been set by an experimental paper) within a wider
theoretical framework (examples include SModelS Altakach:2023tsd  or
standalone analyses like Ref. Haisch:2023rqs ). The second option is to
determine the kinematics of new final states predicted by a model, and
approximate the signal efficiency for this new model in a target analysis.
This can be achieved using either efficiency maps provided by the experimental
collaborations, or runnable code snippets which reproduce the signal region,
if necessary with relevant smearing applied to approximate the effect of the
detector. In the latter category, many open source tools provide libraries of
re-interpretable analyses, usually exploiting searches. One tool however,
makes use instead of the bank of detector-corrected measurements at the LHC,
thus avoiding the need to approximate the detector response (often the most
costly and approximate step). This method also benefits from the fact that
measurement papers almost invariably provide experimentalist-produced
definitions fo the signal regions in the Rivet Bierlich:2024vqo  library,
which is by construction synchronised with the digitized measurement
information on the HEPData Maguire:2017ypu ; HEPData  analysis preservation
portal. This tool is called Contur Butterworth:2016sqg ; Buckley:2021neu , and
is able to provide rapid insights into constraints on new models (see Refs.
Butterworth:2024eyr ; Altakach:2022nuo ; Butterworth:2022dkt  for recent
examples using models predicting new prompt particles).

The Contur workflow is detailed in Ref. Buckley:2021neu  but the key points
are summarised here. The starting point is a new physics model encapsulated in
the standard Universal FeynRules Output (UFO) format Degrande:2011ua . The UFO
can be passed to a Monte Carlo Event Generator (MCEG) (with suitable model
parameter choices) to produce HEPMC Verbytskyi:2020sus  files, which contain
the particle-level history of the interaction for a given number of generated
collision events, after hadronisation. In the next step, the HEPMC events are
analysed in Rivet, passing them through dozens of “routines” that encapsulate
the definitions of the fiducial regions of as many measurements of SM
processes. The result is a set of histograms in the YODA Buckley:2023xqh
format which show where the signal would have shown up in the LHC
measurements. Since Rivet is automatically synchronised with HEPData, it is
trivial to then compare the size of the new signal (stacked on SM predictions)
to the measured cross-sections in each bin, within the experimental
uncertainties. The final step is to run a statistical analysis, grouping
measurements into orthogonal pools (defined by final state, experiment and
centre-of-mass energy) to avoid double counting, and produce a CLs Read:2002hq
exclusion value. Contur acts as a book-keeping and steering layer for this
workflow, and provides the final statistical analysis. The power of this
method is that while a dedicated search for a new signal with ATLAS or CMS can
take years, existing constraints can already be determined in a few days using
Contur. Indeed, running Contur on a single set parameter point takes usually a
few hours, and a grid of a few hundred parameter points can easily be done
overnight on a standard batch computing farm.

The same method can also be used to extract estimates of the constraints on a
given model after the end of the high-luminosity LHC (HL-LHC) lifetime in an
automated way, as first demonstrated in Ref. Butterworth:2024eyr . Indeed, if
we assume that 3000/fb are collected by the end of the HL-LHC programme, we
can scale the uncertainties by the square root of the ratio of the increase in
integrated luminosity for a given analysis and recalculate the CLs values.
Indeed, statistical uncertainties scale with the square root of integrated
luminosity. The extrapolation relies on the assumption that experimental
systematic uncertainties would also scale this way, which can be deemed a
little optimistic. This optimistic assumption is counterbalanced by
pessimistic assumptions that no improvements in theoretical uncertainties
would take place in the intervening decades, that no new final states would be
measured, and that no additional phase space would be explored. Assuming that
the pessimistic and optimistic assumptions cancel out, we can take the HL-LHC
extrapolations from Contur with a certain degree of confidence so long as
these caveats are kept in mind.

##  3 Adapting Contur to LLP topologies

Until now, the Contur method has not been usable for LLP signatures. Indeed,
the definitions of the analysis signal regions encapsulated in Rivet always
assume prompt behaviour (as they should, since the measured SM processes are
prompt). Hence, the detector corrections for those analyses would not account
for jets or tracks that would be lost due to a displaced starting point. The
consequence is an over-estimate of the selection efficiency for LLP signals,
and hence unreliable CLs exclusions. For this reason, models (or regions of
the model space) predicting LLPs could not be probed using the technique
described in Section 2. This prohibition is easier to state than to enforce
however: it may not be immediately obvious to a user what combinations of
parameters would lead to particles with small widths and therefore long
lifetimes. Indeed, the width of a particle can depend on the interplay between
the masses of daughter and mother particles, and the values of various
couplings and masses of new particles. When performing a scan over a range of
parameters in a model with Contur, it is not trivial to determine which
regions should be avoided, unless special care is taken, as was done for
example in Ref. Amrith:2018yfb . A workflow to automatically detect LLPs in
new models and apply a specific treatment is clearly beneficial to the Contur
method. Conversely, if such a special treatment can be determined, then it
opens the door to probing LLP models with Contur.

We propose the following approach to treat LLPs in the Rivet and Contur
machinery. The basic idea is to evaluate, for a given set of model parameters,
what the lab-frame decay length distribution is for all particles with small
widths (and therefore long lifetimes) as determined by the event generator. We
can then determine what fraction of particles decayed early enough to be
treated as prompt. These fractions can be used to calculate a scale factor,
with which to down-weight the signal yields predicted by the Rivet routines.
The rest of the Contur workflow can then run normally. We have implemented a
“naive” version of this workflow as a proof of concept, with which we have
extracted new constraints on LLP models as discussed in Sections 4, 5, 6 and
7. We discuss at the end of this section how similar results could be achieved
more efficiently, but the purpose of this implementation is to show the power
of the method and highlight the kind of new constraints that can be obtained.
We will now review the step-by-step implementation of our naive method,
including technical details.

### Step 1

A regular Contur scan on a pair of parameters of a model is performed,
resulting in a YODA file for each point in the grid. We ensure that the events
(in HEPMCs format) are kept rather than deleted, as they normally would be to
save disk space.

### Step 2

At each point in the grid, the generator log files are parsed to determine the
list of particle species, in terms of their unique particle data group
identifier (PDGid), which have small widths and therefore macroscopic decay
lengths. If one already knows which particles are LLPs in the model, one need
only save the width of those particles at this stage. But alternatively, one
can select all particles with a width less than $\sim 10^{-6}$ eV (for
displacements above the millimeter scale).

### Step 3

For the list of LLP PDGids at each point, we parse the HEPMC files to extract
the LLP kinematics for each event, such that time dilation factors can be
calculated. Multiple particle species can be long-lived at a given point, and
there can be multiple instances of each species in a given event. For each
LLP, a random decay length is calculated, sampling from a falling exponential
distribution whose mean is the inverse of that particle species’ width (as
saved in Step 2). The time dilation factor for that particular particle is
applied to obtain a lab-frame decay position for each LLP, separately for each
event. In principle, this step can be performed by the MCEG programme, but we
choose to do it by hand such that we can treat the lifetime as an independent
parameter.

### Step 4

We determine a “prompt threshold” $p_{th}$ before which particles can be
considered indistinguishable from prompt. Indeed, if the decay of the LLP
takes place before $p_{th}$, then tracks and deposits from the decay products
should be reconstructible using the standard ATLAS and CMS algorithms. To
choose the value of the threshold, we are guided by Refs. ATLAS:2023nze  and
CMS:2014pgm  and the fact that vertices can only be reconstructed in ATLAS and
CMS up to about 10 cm in the barrel and around 60 cm in the end-caps. If a
vertex cannot be reconstructed, then tracks cannot be either, and the same
goes for jets that are typically vetoed in the standard algorithms if they do
not contain any tracks. We therefore choose $p_{th}$ to be 10 cm for particles
in the barrel ($|\eta|<1.5$, where $\eta$ is the pseudo-rapidity) and 60 cm
for particles in the endcaps ($|\eta|\geq 1.5$).

### Step 5

For each particle species, we build up the distribution of lab-frame decay
positions across all generated events. Multiple LLPs of the same species in
the same event can contribute to the distribution, assuming that their decay
positions are uncorrelated in the lab frame. We then calculate the fraction of
LLPs of species $i$ where the decay is below the prompt threshold, to
calculate a “prompt fraction” $f_{i}$ :

$f_{i}=\sum^{j=0}_{N_{i}}(L^{j}<p_{th})/N_{i}$

where $N_{i}$ is the number of LLPs of species $i$ in the sample (there can be
multiple per event), and $L^{j}$ is the lab-frame decay position of LLP $j$ in
either the transverse or longitudinal direction. If the pseudorapidity of the
LLP satisfies $\eta_{j}<1.5$, then the transverse decay position is taken and
it is compared with the transverse prompt threshold; otherwise if
$\eta_{j}\geq 1.5$, the longitudinal decay position and prompt threshold are
used.

### Step 6

We then calculate an overall scale factor $s$ taking the product of prompt
fractions across species. To account for the fact that LLPs of a same species
can occur multiple times per event, the prompt fractions are exponentiated by
the average number of LLPs of species $i$ per event. Finally:

$s=\prod^{i}_{\textrm{species}}f_{i}^{\frac{N_{i}}{N_{\textrm{events}}}}.$

The purpose of $s$ is to scale the final predicted yields to those that would
have been obtained if the unreconstructable events were filtered out before
processing in Rivet.

### Step 7

The resulting $s$ is calculated separately for each model parameter point in
the scan, and the corresponding YODA file is scaled by that factor. This has
the effect of reducing the predicted signal yields according to the chance
that the LLPs in that part of parameter space would have decayed promptly
enough to be reconstructed and selected in the standard measurements.

### Step 8

The rest of the Contur pipeline is executed as normal on the scaled YODA
files. The effect of this method is always to decrease the exclusion
determined by Contur: $s$ is always less than 1 by construction. Effectively,
we are down-weighting the claimed Contur sensitivity in regions where LLPs can
occur. Further, this method is likely somewhat conservative: since we apply
the same scale factor uniformly to the output histograms of all measurements,
we are assuming that all of the selections would rely specifically on the
reconstruction of all the LLPs to select an event. In reality there may be
other prompt objects in an event that might cause it to be selected even if
the LLPs were not reconstructed. Nevertheless, we prefer to be over-
conservative for this proof of principle.

###

More efficient workflows to achieve similar results can be envisaged. For
example, as the HEPMC event format is capable of recording vertex positions
calculated by the MCEG, one could bypass Step 3. Further, instead of
calculating a scale factor as in Step 6, it would be possible to remove events
containing unreconstructible particles completely, for example using the
PyHEPMC BUCKLEY2021107310  python module. Another possible option is to build
a pipeline that discards individual particles that are displaced (rather than
removing the whole event), ensuring that only prompt particles get used in the
Rivet routines. This would be a less conservative solution, but requires a
much more complex implementation. These alternative approaches can be
considered fully in Rivet internally if it allows an earlier processing stage
that scans particles’ vertices before running the routines. We leave these
workflows to be further investigated in future studies.

Finally, since this method takes the same final states that would be predicted
by a model, and down-weights the predicted yields, it does not incur any
additional model dependence. Further constraints on LLP models could likely be
obtained from the other interactions that are possible in the same model, but
do not involve LLPs. A good example is if a pair of LLPs is produced from the
decay of a scalar mediator, which can be produced via gluon–gluon fusion, then
that same mediator must necessarily be able to decay back into gluons (and
with no LLPs involved at all in that reaction). One could then check the
standard dijet measurements and searches for the corresponding bump at the
scalar mediator mass. This second method, involving only prompt particles,
could then use the standard Contur workflow without any special treatment, but
it would involve additional assumptions on what is or is not allowed by a
particular model. We do not explore this avenue further, and reserve it for
future work.

##  4 Constraints on feebly interacting DM

Figure 1: Diagram for the main production process for FIMPs at the LHC,
assuming the leptonic version of the model. The blue lines and text indicate
the long-lived particle(s).

To illustrate how the method described in Section 3 can be used to obtain new
constraints on LLP models, we first consider a simple freeze-in model
explaining the observed DM abundance through the decay of an electrically-
and/or colour-charged long-lived parent particle into a Feebly Interacting
Massive Particle (FIMP), as proposed in Contribution 6 of Ref.
Brooijmans:2020yij  and described in more detail in Ref. Belanger:2018sti .
Freeze-in is an alternative to the frozen-out Weakly Interacting Massive
Particle (WIMP) picture which, despite its attractive features, is under some
tension with null experimental evidence. This FIMP model introduces a real
scalar DM candidate, denoted $S$, which is neutral under the SM gauge group.
Additionally, a vector-like fermion $F$ plays the role of mediator. In this
model, $F$ is electrically charged and can acquire a long lifetime if the
coupling governing its decay is small. We consider here the leptonic version
of this minimal freeze-in model, where the DM candidate is coupled to the SM
through Yukawa-type terms involving the vector-like fermion and the SM charged
leptons. The LHC signature of the leptonic model is then the Drell–Yan pair-
production of the vector-like fermion (with electric charge $\pm 1$), followed
by its decay into the scalar DM candidate along with a charged lepton, as
shown in Figure 1. This last decay can be displaced or even take place outside
the detector. Another version of the model, where $F$ decays hadronically,
exists but is not studied further here.

This FIMP model was confronted with LHC search results in Ref.
Brooijmans:2020yij ; Belanger:2018sti  for different assumptions of LLP
lifetime: an ATLAS search for prompt leptons and missing transverse momentum
ATLAS:2019lff  was used for short lifetimes, CMS searches for displaced
leptons CMS:2016isf  or displaced tracks CMS:2018rea  were used for
intermediate lifetimes, and an ATLAS search for heavy stable charged particles
ATLAS:2019gqq  was used for detector-stable LLPs. Together, these searches
provided coverage for $c\tau$ values ranging from tens of micrometers to
kilometers, excluding LLP masses between about 150 GeV and 350-700 GeV
depending on the lifetime.

To determine any additional constraints that could be extracted from existing
measurements using Contur, we simulated the process encoded in the publicly
available UFO file for the model FIMP_UFO  using MadGraph5_aMC@NLO v3.4.2
Alwall:2014hca  and Pythia v8 Sjostrand:2007gs  for hadronisation. For each
point, 10 000 events were generated. The grid was composed of LLP masses in
the range of 100 to 1000 GeV in steps of 45 GeV. The lifetime of the $F$ was
assumed to be independent of its mass, and 20 values of $c\tau$ were scanned
on a logarithmic scale between 0.01 mm and 10 m. The mass of the DM candidate
was set to 12 KeV, although the analysis presented here is not strongly
sensitive to this choice. Since the LLP $F$ is electrically charged, it would
in principle form a reconstructible track before its decay. After the decay, a
SM charged lepton may continue the track, but since the unseen $s$ is emitted,
one would expect the track from the $F$/$\ell$ to be “kinked”: in other words
for the hits not to be aligned in the expected pattern for a charged particle
trajectory, but instead be the discontinuous union of two curved tracks. Such
signatures would be thrown away by ATLAS and CMS, as standard tracking
algorithms look for a single particle trajectory. If the decay takes place
before the end of the inner tracking system however, the lepton track would
still be reconstructed as normal. The values for the prompt threshold
described in Section 3 are therefore still applicable. We then follow the
procedure set out previously to determine the exclusions from Contur.

$\ell^{+}\ell^{-}$+jet ATLAS:2017nei  |  $\ell_{1}\ell_{2}$+$E_{T}^{\rm miss}$+jet ATLAS:2021jgw  |  $\ell_{1}\ell_{2}$+$E_{T}^{\rm miss}$ ATLAS:2019rob  |  $\ell^{+}\ell^{-}\gamma$
---|---|---|---

Figure 2: (Left) Exclusion contours for the FIMP model as a function of the
$F$ mass and logarithm of $c\tau$, assuming $m_{S}=12$ keV. The solid (dotted)
lines represent the observed (expected) exclusions at 95% CL. The dashed line
represents the observed 68% CL exclusion. The red dotted lines represent the
95% CL expected exclusion extrapolated to the HL-LHC. The area below the
curves is excluded. The colours of the cells represent the dominant analysis
pool for each point in the grid. (Right) The dominant exclusion histogram for
a particular excluded point, where $m_{F}=147$ GeV and $c\tau=0.06$ m. The
exclusion comes from the dilepton invariant mass spectrum measured in the
$e\mu\nu\nu$ channel of an ATLAS $W^{+}W^{-}$ measurement ATLAS:2019rob ,
specifically, Figure 7c of that paper. The black markers represent the
observed data, and the red line shows the SM prediction. The blue line shows
the SM prediction summed with the new physics prediction, which in this case
results in a significant disagreement with the observation.
Figure 3:  Comparison of Contur results with current measurements,
extrapolated Contur results for HL-LHC, and recasted search results for the
FIMP model from Ref. Brooijmans:2020yij .

The results are shown in Figure 2, indicating that $F$ masses from 100 GeV up
to $\sim$250 GeV can be excluded with 95% confidence level (CL) so long as the
mean proper lifetime times the speed of light is below $\sim$20 cm. The
colours in the figure represent the type of measurement that brings the
exclusion power: for masses below about 300 GeV, the exclusion is dominated by
analyses that produce two (different-flavour) leptons and missing transverse
momentum. For higher masses, other final states dominate the exclusion,
notably opposite-flavour leptons with missing transverse energy and the
presence of one of more jets, or alternatively measurements of dilepton
production with an additional photon. Focusing on the excluded region, taking
a point where the mass of $F$ is approximately 150 GeV, Figure 2 also shows
the exclusion histogram from the ATLAS measurement of $W^{+}W^{-}$ production
ATLAS:2019rob  that generated the highest exclusion. Indeed, the presence of
such a signal would have significantly altered the predicted the shape of the
spectrum.

In Figure 3, we compare the results obtained from Contur to limits derived
from direct searches in Ref. Brooijmans:2020yij . In this case, the direct
searches provide more stringent exclusions at high LLP masses, which is
perhaps not surprising given that they were originally searches for high-mass
supersymmetric particles, and would be more optimal in such a regime. However,
all these searches have a cut-off around 150 GeV below which they do not set
exclusions. This region is filled in by the Contur exclusion lines. Further,
the projection of these exclusions to the HL-LHC indicate that even without
new searches, the limits on the FIMP model could be extended up to around 600
GeV and to $c\tau$ up to around 4 m, if the same measurements are repeated
during the HL-LHC era.

##  5 Constraints on hidden sector models with heavy mediators decaying into
long-lived scalars

Hidden sector models Strassler:2006im ; Strassler:2006ri ; Chan:2011aa ;
Curtin:2013fra  postulate that the SM is connected to a dark sector of new
particles via some scalar mediator $\Phi$, which can be the Higgs boson or
some new boson. If the mediator is a scalar boson, it can be produced (amongst
others) via gluon–gluon fusion (ggF) and associated vector boson production
($V\Phi$) just like the Higgs boson. It can decay into a pair of neutral
scalars from the dark sector, denoted $S$, which would eventually decay back
into the SM via a Yukawa interaction, as shown in Figure 4. The $S$ can become
long-lived because its decay back into the SM occurs through small couplings.
The Hidden Abelian Higgs Model (HAHM) Wells:2008xg ; Curtin:2013fra  can
describe this model. The HAHM introduces two new scalars $\Phi$ and $S$, and a
dark photon $Z_{d}$. In this section, we focus on the decays of the $\Phi$ to
$S$ only (as done by ATLAS).

Figure 4: Diagrams for (left) gluon–gluon fusion and (right) associated $Z$
boson production of a HS mediator $\Phi$ at the LHC, assuming decays into
neutral scalars $S$, which later decay into fermions $f$ via a Yukawa-type
interaction. In this paper, the $S$ decays chiefly into $b$-quark or $t$-quark
pairs if kinematically allowed.

Generic HS models have been studied by both ATLAS and CMS, for example in
recent searches for displaced activity in the tracking system ATLAS:2024qoo ,
calorimeters ATLAS:2024ocv  or muon system ATLAS:2022gbw , ensuring a coverage
from millimeters to tens of meters in c$\tau$ of the LLP $S$, and covering
many scenarios for the masses of the particles. Phenomenological re-
interpretations of other searches into this model have also been performed,
for example in Ref. Wang:2024ieo . Summaries of experimental constraints are
available for ATLAS ATL-PHYS-PUB-2022-007  and CMS CMSsummar . CMS has mostly
focused on the case where the mediator is the Higgs boson, while ATLAS has
also considered heavier mediators, up to 1 TeV in mass. For both experiments,
the searches at low lifetimes have studied the $V\Phi$ production mode,
exploiting the vector boson for triggering and background reduction. Those
searches focused on the Higgs-boson-mediated case. For higher mediator masses,
no constraints exist for $c\tau$ below the centimeter range. We address this
lack of coverage using the modified Contur workflow.

$\ell^{+}\ell^{-}$+jet ATLAS:2017nei ; ATLAS:2020juj  |  $\ell_{1}\ell_{2}$+$E_{T}^{\rm miss}$+jet ATLAS:2019ebv  |  $\ell$+$E_{T}^{\rm miss}$+jet ATLAS:2019hxz ; CMS:2018htd ; ATLAS:2017cez ; CMS:2016oae  |  4$\ell$ ATLAS:2021kog
---|---|---|---

Figure 5: (Left) Exclusion contours for the HS model as a function of the
logarithms of cross-section times branching fraction of $Z\Phi(SS)$ and
$c\tau$, assuming $m_{S}=475$ GeV. The solid (dotted) lines represent the
observed (expected) exclusions at 95% CL. The dashed line represents the
observed 68% CL exclusion. The red dotted lines represent the 95% CL expected
exclusion extrapolated to the HL-LHC. The area above the curves is excluded.
The colours of the cells represent the dominant analysis pool for each point
in the grid. (Right) The dominant exclusion histogram for a particular
excluded point, where the cross-section is around 2.15 pb and $c\tau=0.31$ m.
The exclusion comes from an ATLAS measurement of the cross-section for
electroweak production of dijets in association with a $Z$ boson ATLAS:2017nei
, specifically, Table 3 of that paper. The black markers represent the
observed data, and the red line shows the SM predictions. The blue line shows
the SM prediction summed with the new physics prediction, which in this case
results in a significant disagreement with the observation.

We focus on the case of a heavy mediator of mass $1000$ GeV produced via
associated $Z$ boson production only. The signature consists of pairs of
opposite-sign same-flavour leptons in association with up to four $b$-jets or
products of top quark decays if the LLP mass is above the relevant kinematic
threshold. The events are generated with using MadGraph5_aMC@NLO v3.4.2 and
Pythia v8 for hadronisation, using the existing UFO implementation of the HAHM
HAHM_UFO  to establish direct comparisons to direct searches. The coupling of
the dark photon to the SM and mediator is set to zero. Several values of the
neutral scalar $S$ mass are considered, between 50 and 475 GeV, to match the
parameter values chosen by ATLAS. The $Z$ boson is allowed to decay into
electrons, muons or taus. For each assumption, the grid scan consisted of 10
values of $c\tau$ on a logarithmic scale between 0.01 mm and 100 m, and 10
values of cross-section times branching fraction ($\sigma\times B$) on a
logarithmic scale between 0.001 pb and 1000 pb, with 10000 events generated at
each point. We note here the high end of the considered cross-sections could
be in a non-perturbative regime, but we ignore this effect in this study. The
kinematics of the events are assumed to be uncorrelated to the LLP mean proper
lifetime, as was done for the ATLAS searches.

The workflow described in Section 3 is exploited to determine what constraints
can be extracted from the bank of LHC measurements for HS models with heavy
mediators. Since the displaced hadronic activity from LLPs would lead to
anomalous jets, which would be trackless and therefore vetoed by the standard
experimental reconstruction algorithms, we maintain the prompt threshold
values described previously. An example of the results for the case where the
$S$ mass is set to 475 GeV is shown in Figure 5. In this case, one expects the
LLP to decay mostly into top quarks. Cross-sections below around 0.7 pb are
excluded up to 10 cm in $c\tau$. The exclusion is mostly driven by 13 TeV
ATLAS measurements involving one lepton, missing transverse momentum and jets
or measurements of pairs of leptons (from a $Z$ boson decay) and extra jets.
Figure 5 focuses on one of the excluded points: the existence of this signal
would have caused divergences in the Baseline and High-$p_{\textrm{T}}$ bins
of the measurement presented in Ref ATLAS:2017nei . Figure 6 shows equivalent
information for a case where the LLP mass is below the top-quark threshold,
such that most of the decays are into $b$-quarks. In this case, the excluded
cross-sections peak at around 1 pb for nearly-prompt particles before dropping
off after around 2 cm. The dominant analysis is a measurement of $t\bar{t}$
production ATLAS:2018orx , which would indeed yield the same final state of a
pair of leptons accompanied by $b$-jets.

hadronic $t\bar{t}$ ATLAS:2018orx  |  $\ell^{+}\ell^{-}$+jet |
---|---|---

Figure 6: (Left) Exclusion contours for the HS model as a function of the
logarithms of cross-section times branching fraction of $Z\Phi(SS)$ and
$c\tau$, assuming $m_{S}=150$ GeV. The solid (dotted) lines represent the
observed (expected) exclusions at 95% CL. The dashed line represents the
observed 68% CL exclusion. The red dotted lines represent the 95% CL expected
exclusion extrapolated to the HL-LHC. The area above the curves is excluded.
The colours of the cells represent the dominant analysis pool for each point
in the grid. (Right) The dominant exclusion histogram for a particular
excluded point, where the $\sigma\times B$ is around 10 pb and $c\tau=0.31$ m.
The exclusion comes from an ATLAS measurement of boosted $t\bar{t}$ production
ATLAS:2017nei , specifically, Figure 4a of the auxiliary material of Ref.
ATLASttbarAuxMat . The black markers represent the observed data, and the red
line shows the SM predictions. The blue line shows the SM prediction summed
with the new physics prediction, resulting in a significant disagreement with
the data.

Figure 7: Comparison of Contur results for the HS model for a mediator of mass
1 TeV with current measurements, extrapolated Contur results for HL-LHC, and
comparable results from an ATLAS search ATLAS:2024ocv  and a phenomenological
re-interpretation study Wang:2024ieo  for the same model and parameter
choices.

The comparison of the results using Contur with the best results from ATLAS is
shown in Figure 7. The regions excluded by Contur are exactly those where no
constraints from direct searches exist. Although for light LLP mass scenarios
the Contur results are comparatively weak, the results for heavier LLPs are of
the same order of magnitude as the direct search and provide complementary
constraints in a region which the direct searches were unable to touch.

##  6 Constraints on heavy, long-lived dark photon models

We now consider a model where the dark photon $Z_{d}$ is the long-lived
particle, as in Refs. Curtin:2014cca ; Davoudiasl:2012ag . In this model, the
$Z_{d}$ is produced in association with a regular $Z$ boson from the decay of
the mediator $\Phi$ described in Section 5. It acquires a long lifetime
through suppressed couplings to the SM, and decays mostly into quark–antiquark
pairs or pairs of leptons. A Feynman diagram showing the processes under study
is shown in Figure 8. This model was studied by ATLAS in Refs. ATLAS:2018niw ;
ATLAS:2024ocv . The most constraining results from those searches restrict
cross-sections to be smaller than around 0.1 pb in the 10 cm to 10 m range,
but no constraints exist so far for $c\tau$ values below 10 cm from ATLAS or
CMS.

Figure 8: Diagram for the production of a dark photon $Z_{d}$ via a scalar
mediator $\Phi$ at the LHC. The blue lines and text indicate the long-lived
particle(s).

Events were generated with MadGraph5_aMC@NLO v3.4.2 and Pythia v8 for
hadronisation, starting from the HAHM UFO HAHM_UFO  file used in Section 5 but
turning the couplings of the mediator $\Phi$ to $S$ particles to zero and
setting the $S$ mass to a very high scale. The grid is composed of 10 points
spaced logarithmically in $\sigma\times B$ between 0.01 mm and 10 m and 10
points spaced logarithmically in $c\tau$ between 1 mm and 1 km. The cross-
section and kinematics of the model are assumed to be independent of the
$c\tau$, with 10000 events generated for each combination. Several options of
the mediator mass between 400 and 600 GeV, and the dark photon mass between 50
and 400 GeV, are considered, to match those probed by the ATLAS experiment.

The results of the Contur study are presented in Figure 9 as a function of
$c\tau$ and cross-section times branching fraction, for the case where the
mediator mass is 600 GeV and the dark photon mass is 400 GeV. For low
lifetimes, the model can be excluded down to the 0.01 pb level, exploiting
measurements of the four-lepton invariant mass and dilepton+jets measurements.
The sensitivity starts to decrease for $c\tau$ values around 10 cm, in a way
that is complementary to the ATLAS search programme. The red dotted line
indicates the sensitivity that could be obtained if the measurements were
repeated at the high-luminosity LHC, with constraints down to 0.001 pb
possible. Figure 9 also shows an example of a histogram from the four-lepton
invariant mass measurement ATLAS:2021kog , where the invariant mass of the
second lepton pair further away from the $Z$ boson mass shows a large excess
compared with the measured data. So, this model can be excluded by the fact
that the dark photon can delay leptonically sufficiently often that it would
lead to differences in this spectrum beyond what was measured by ATLAS.

$\ell_{1}\ell_{2}$+$E_{T}^{\rm miss}$+jet ATLAS:2019hau  |  $\ell^{+}\ell^{-}$+jet ATLAS:2017nei  |  4$\ell$ ATLAS:2021kog  |
---|---|---|---

Figure 9: (Left) Exclusion contours for the dark photon model as a function of
the $F$ mass and logarithm of the $c\tau$, assuming $\Phi$ has a mass of 600
GeV and $Z_{d}$ has a mass of 400 GeV. The solid (dotted) lines represent the
observed (expected) exclusions at 95% CL. The dashed line represents the
observed 68% CL exclusion. The red dotted lines represent the 95% CL expected
exclusion extrapolated to the HL-LHC. The area above the curves is excluded.
The colours of the cells represent the dominant analysis pool for each point
in the grid. (Right) The dominant exclusion histogram for a particular
excluded point, where $\sigma\times B=0.001$ pb and $c\tau=1$ mm. The
exclusion comes a measurement of the four-lepton invariant mass spectrum
ATLAS:2021kog , specifically, from the measurement of the mass of the second
lepton pair in Figure 7d of that paper. The black markers represent the
observed data, corrected for detector effects, and the red line shows the SM
predictions. The blue line shows the SM prediction summed with the new physics
prediction, which in this case results in a significant disagreement with the
observed data in the last bin.

A comparison of the Contur results for various mass hypotheses against the
results obtained by ATLAS in direct searches can be seen in Figure 10. In this
model, the Contur results are complementary to the direct searches,
effectively ruling out a region not covered by the search programme at a
comparable cross-section. Further, measurements at the HL-LHC could even
eventually exclude the region covered by the direct searches.

Figure 10:  Comparison of Contur results for the $Z_{d}$ model, extrapolated
Contur results for HL-LHC, and the results of a direct search from ATLAS
ATLAS:2024ocv ,

##  7 Constraints on long-lived photophobic axion-like particles

The final model we consider is a “photo-phobic” axion-like particle (ALP)
model where the ALPs couple exclusively to gluons (and so decay into jets), as
proposed for example in Ref. Brivio:2017ije . In some parts of the parameter
space, the ALP can become long-lived due to small couplings mediating the
decay. We consider a case where the ALP is produced via associated $Z$ boson
production, as shown in Figure 11. This model was studied by ATLAS in Refs
ATLAS:2024qoo ; ATLAS:2024ocv  exploiting decays of the ALP in the tracker and
calorimeters respectively. The tracker-based search considered ALPs with
masses of 40 GeV and 55 GeV, and set constraints at the level of 0.1 pb in the
millimeter to 10 cm range. The calorimeter-based search considered ALPs in
masses ranging from 100 MeV to 40 GeV, giving constraints for the 40 GeV mass
point in the 10 cm to 10 m range. Lighter ALP mass choices give similar
constraints but shifted towards lower $c\tau$ values due to increased time
dilation factors. In both cases, the sub-millimeter range however is not well
covered.

Figure 11: Diagram for the production of an axion-like particle $a$ in
association with a $Z$ boson. We consider the case where $a$ decays
exclusively into gluons. The blue lines and text indicate the long-lived
particle(s).

For the Contur study, events were generated using MadGraph5_aMC@NLO v3.4.2 and
Pythia v8 for hadronisation using a UFO file corresponding to the model
presented in Ref. Brivio:2017ije . The event generation is limited to cases
where the $Z$ decays into electrons or muons, to be consistent with the ATLAS
analyses. The grid is composed of 10 points spaced logarithmically in
$\sigma\times B$ between 0.1 mm and 10 m and 10 points spaced logarithmically
in $c\tau$ between 0.01 mm and 1000 m, with 10000 events generated at each
point. The exclusions obtained for this model using Contur are shown in Figure
12 for the case where the ALP mass is 10 GeV. The exclusion is dominated by
dilepton plus jet analyses everywhere in the plane, as would be expected since
the final state under study consists of two light leptons and two gluons. The
most sensitive analyses are the $Z$+jets measurements from ATLAS ATLAS:2017nei
(the same one already giving exclusion for the HS case in Section 5, but this
time in the electroweak-enriched bin) and an equivalent measurement by CMS
CMS:2019raw .

$\ell^{+}\ell^{-}$+jet CMS:2019raw ; ATLAS:2017nei
---

Figure 12: (Left) Exclusion contours as a function of the cross-section times
branching fraction for the production of an photo-phobic ALP with a $Z$ boson
as a function of the logarithm of the $c\tau$, assuming an ALP mass of 10 GeV.
The solid (dotted) lines represent the observed (expected) exclusions at 95%
CL. The dashed line represents the observed 68% CL exclusion. The red dotted
lines represent the 95% CL expected exclusion extrapolated to the HL-LHC. The
area above the curves is excluded. The colours of the cells represent the
dominant analysis pool for each point in the grid. (Right) The dominant
exclusion histogram for a particular excluded point, where the cross-section
is around 0.02 pb and $c\tau=0.01$ mm. The exclusion comes from an ATLAS
measurement of the cross-section for electroweak production of dijets in
association with a $Z$ boson ATLAS:2017nei , specifically, Table 3 of that
paper. The black markers represent the observed data, with the red line
showing the SM predictions. The blue line shows the SM prediction summed with
the new physics prediction, which in this case results in a significant
disagreement with the observation.

A comparison of the Contur results with the best-available constraints from
ATLAS are shown in Figure 13. They show that the constraints extracted from
the measurements are of the same order of magnitude as the direct searches,
and completely ruling out the sub-millimeter part of the spectrum in a way
that is complementary and competitive with the direct search programme. The
plots also show the expected reach of measurements would increase the
sentivity by more than an order of magnitude over the course of the HL-LHC,
practically overtaking the existing direct search results.

Figure 13:  Comparison of Contur results for a photo-phobic ALP produced in
association with a $Z$ boson, extrapolated Contur results for HL-LHC, and
dedicated search results from ATLAS exploiting the calorimeter ATLAS:2024ocv
and tracker ATLAS:2024qoo .

##  8 Conclusions

In this work, a method to constrain long-lived particle models using the
fraction of particles that decay early was shown. It exploits the Contur
method to derive constraints from the bank of LHC measurements, but scaling
down the predicted signal yield by the relative fraction of reconstructible
long-live particle decays. As a proof of principle, the method was used to
determine new constraints on four popular long-lived particle models of
various types. The considered models are varied in the nature of the LLPs:
charged or neutral, singly-produced or pair-produced, leptonically or
hadronically decaying. In each case, new parts of the parameter space can be
excluded. Projections of how the constraints could evolve into the HL-LHC era
are also presented, which sometimes can match or even over-take direct
searches in some parts of the parameter space. This suggests than certain
regions need not have dedicated searches, freeing up human resources in the
large experimental collaborations to focus on the more displaced regions that
cannot be probed in any other way. Moreover, the HL-LHC projections show that
large parts of the so far unexplored parameter space could become accessible:
in other words, that new long-lived particle discoveries could be awaiting us
with the onset of Run 4 of the LHC.

## Acknowledgements

We wish to thank Jon Butterworth for fruitful discussions. SJ received funding
from the Université Clermont Auvergne Graduate Track for Mathematics and
Physics to pursue this research.

## References

  * (1) N. Arkani-Hamed, A. Gupta, D. E. Kaplan, N. Weiner, and T. Zorawski, Simply Unnatural Supersymmetry, arXiv:1212.6971.
  * (2) G. F. Giudice and R. Rattazzi, Theories with gauge-mediated supersymmetry breaking, Phys. Rept. 322 (1999) 419–499, [hep-ph/9801271].
  * (3) R. Barbier et al., R-Parity-violating supersymmetry, Phys. Rept. 420 (2005) 1–202, [hep-ph/0406039].
  * (4) C. Csáki, E. Kuflik, O. Slone, and T. Volansky, Models of dynamical R-parity violation, JHEP 06 (2015) 045, [arXiv:1502.03096].
  * (5) J. Fan, M. Reece, and J. T. Ruderman, Stealth supersymmetry, JHEP 11 (2011) 012, [arXiv:1105.5135].
  * (6) J. Fan, M. Reece, and J. T. Ruderman, A stealth supersymmetry sampler, JHEP 07 (2012) 196, [arXiv:1201.4875].
  * (7) M. J. Strassler and K. M. Zurek, Echoes of a hidden valley at hadron colliders, Phys. Lett. B 651 (2007) 374–379, [hep-ph/0604261].
  * (8) M. J. Strassler and K. M. Zurek, Discovering the Higgs through highly-displaced vertices, Phys. Lett. B 661 (2008) 263–267, [hep-ph/0605193].
  * (9) Y. F. Chan, M. Low, D. E. Morrissey, and A. P. Spray, LHC Signatures of a Minimal Supersymmetric Hidden Valley, JHEP 05 (2012) 155, [arXiv:1112.2705].
  * (10) I. Brivio, M. B. Gavela, L. Merlo, K. Mimasu, J. M. No, R. del Rey, and V. Sanz, ALPs Effective Field Theory and Collider Signatures, Eur. Phys. J. C 77 (2017), no. 8 572, [arXiv:1701.05379].
  * (11) J. Alimena et al., Searching for long-lived particles beyond the Standard Model at the Large Hadron Collider, J. Phys. G 47 (2020), no. 9 090501, [arXiv:1903.04497].
  * (12) ATLAS Collaboration, G. Aad et al., Search for light long-lived neutral particles from Higgs boson decays via vector-boson-fusion production from $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector, arXiv:2311.18298.
  * (13) CMS Collaboration, Search for long-lived particles using displaced vertices and missing transverse momentum in proton–proton collisions at $\sqrt{s}=13\,\text{TeV}$, Phys. Rev. D 109 (2024), no. 11 112005, [arXiv:2402.15804].
  * (14) ATLAS Collaboration, G. Aad et al., Search for neutral long-lived particles that decay into displaced jets in the ATLAS calorimeter in association with leptons or jets using $pp$ collisions at $\sqrt{s}=13$ TeV, arXiv:2407.09183.
  * (15) CMS Collaboration, A. Hayrapetyan et al., Search for long-lived particles using displaced vertices and missing transverse momentum in proton-proton collisions at s=13 TeV, Phys. Rev. D 109 (2024), no. 11 112005, [arXiv:2402.15804].
  * (16) ATLAS Collaboration, G. Aad et al., Search for heavy long-lived multi-charged particles in the full LHC Run 2 pp collision data at s=13 TeV using the ATLAS detector, Phys. Lett. B 847 (2023) 138316, [arXiv:2303.13613].
  * (17) ATLAS Collaboration, G. Aad et al., Search for light long-lived particles in $pp$ collisions at $\sqrt{s}=13$ TeV using displaced vertices in the ATLAS inner detector, arXiv:2403.15332.
  * (18) ATLAS Collaboration, Long-lived particle summary plots for Hidden Sector and Dark Photon models, tech. rep., CERN, Geneva, 2022.  All figures including auxiliary figures are available at https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PUBNOTES/ATL-PHYS-PUB-2022-007.
  * (19) “CMS Higgs decays to LLPs summary.” https://twiki.cern.ch/twiki/bin/view/CMSPublic/SummaryPlotsEXO13TeV#Higgs_decays_to_long_lived_parti.
  * (20) M. M. Altakach, S. Kraml, A. Lessa, S. Narasimha, T. Pascal, T. Reymermier, and W. Waltenberger, Global LHC constraints on electroweak-inos with SModelS v2.3, SciPost Phys. 16 (2024), no. 4 101, [arXiv:2312.16635].
  * (21) U. Haisch and L. Schnell, Long-lived particle phenomenology in the 2HDM+a model, JHEP 04 (2023) 134, [arXiv:2302.02735].
  * (22) C. Bierlich, A. Buckley, J. Butterworth, C. Gutschow, L. Lonnblad, T. Procter, P. Richardson, and Y. Yeh, Robust Independent Validation of Experiment and Theory: Rivet version 4 release note, arXiv:2404.15984.
  * (23) E. Maguire, L. Heinrich, and G. Watt, HEPData: a repository for high energy physics data, J. Phys. Conf. Ser. 898 (2017), no. 10 102006, [arXiv:1704.05473].
  * (24) “The HEPData porta.” https://www.hepdata.net/.
  * (25) J. M. Butterworth, D. Grellscheid, M. Krämer, B. Sarrazin, and D. Yallup, Constraining new physics with collider measurements of Standard Model signatures, JHEP 03 (2017) 078, [arXiv:1606.05296].
  * (26) A. Buckley et al., Testing new physics models with global comparisons to collider measurements: the Contur toolkit, SciPost Phys. Core 4 (2021) 013, [arXiv:2102.04377].
  * (27) J. Butterworth, H. Debnath, P. Fileviez Perez, and Y. Yeh, Dark Matter from Anomaly Cancellation at the LHC, arXiv:2405.03749.
  * (28) M. M. Altakach, J. M. Butterworth, T. Ježo, M. Klasen, and I. Schienbein, Exploring Contur beyond its default mode: a case study, in 56th Rencontres de Moriond on QCD and High Energy Interactions , 4, 2022.  arXiv:2204.10577.
  * (29) J. Butterworth, J. Heeck, S. H. Jeon, O. Mattelaer, and R. Ruiz, Testing the scalar triplet solution to CDF’s heavy W problem at the LHC, Phys. Rev. D 107 (2023), no. 7 075020, [arXiv:2210.13496].
  * (30) C. Degrande, C. Duhr, B. Fuks, D. Grellscheid, O. Mattelaer, and T. Reiter, UFO - The Universal FeynRules Output, Comput. Phys. Commun. 183 (2012) 1201–1214, [arXiv:1108.2040].
  * (31) A. Verbytskyi, A. Buckley, D. Grellscheid, D. Konstantinov, J. William Monk, L. Lönnblad, T. Przedzinski, and W. Pokorski, HepMC3 Event Record Library for Monte Carlo Event Generators, J. Phys. Conf. Ser. 1525 (2020), no. 1 012017.
  * (32) A. Buckley, L. Corpe, M. Filipovich, C. Gutschow, N. Rozinsky, S. Thor, Y. Yeh, and J. Yellen, Consistent, multidimensional differential histogramming and summary statistics with YODA 2, arXiv:2312.15070.
  * (33) A. L. Read, Presentation of search results: The $CL_{s}$ technique, J. Phys. G 28 (2002) 2693–2704.
  * (34) S. Amrith, J. M. Butterworth, F. F. Deppisch, W. Liu, A. Varma, and D. Yallup, LHC Constraints on a $B-L$ Gauge Model using Contur, JHEP 05 (2019) 154, [arXiv:1811.11452].
  * (35) ATLAS Collaboration, G. Aad et al., Performance of the reconstruction of large impact parameter tracks in the inner detector of ATLAS, Eur. Phys. J. C 83 (2023), no. 11 1081, [arXiv:2304.12867].
  * (36) CMS Collaboration, S. Chatrchyan et al., Description and performance of track and primary-vertex reconstruction with the CMS tracker, JINST 9 (2014), no. 10 P10009, [arXiv:1405.6569].
  * (37) A. Buckley, P. Ilten, D. Konstantinov, L. LÃ¶nnblad, J. Monk, W. Pokorski, T. Przedzinski, and A. Verbytskyi, The hepmc3 event record library for monte carlo event generators, Computer Physics Communications 260 (2021) 107310.
  * (38) G. Brooijmans et al., Les Houches 2019 Physics at TeV Colliders: New Physics Working Group Report, in 11th Les Houches Workshop on Physics at TeV Colliders: PhysTeV Les Houches, 2, 2020.  arXiv:2002.12220.
  * (39) G. Bélanger et al., LHC-friendly minimal freeze-in models, JHEP 02 (2019) 186, [arXiv:1811.05478].
  * (40) ATLAS Collaboration, G. Aad et al., Search for electroweak production of charginos and sleptons decaying into final states with two leptons and missing transverse momentum in $\sqrt{s}=13$ TeV $pp$ collisions using the ATLAS detector, Eur. Phys. J. C 80 (2020), no. 2 123, [arXiv:1908.08215].
  * (41) CMS Collaboration, Search for displaced leptons in the e-mu channel, .
  * (42) CMS Collaboration, A. M. Sirunyan et al., Search for disappearing tracks as a signature of new long-lived particles in proton-proton collisions at $\sqrt{s}=$ 13 TeV, JHEP 08 (2018) 016, [arXiv:1804.07321].
  * (43) ATLAS Collaboration, M. Aaboud et al., Search for heavy charged long-lived particles in the ATLAS detector in 36.1 fb-1 of proton-proton collision data at $\sqrt{s}=13$ TeV, Phys. Rev. D 99 (2019), no. 9 092007, [arXiv:1902.01636].
  * (44) “FIMP model UFO file.” https://feynrules.irmp.ucl.ac.be/wiki/FICPLHC.
  * (45) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro, The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations, JHEP 07 (2014) 079, [arXiv:1405.0301].
  * (46) T. Sjostrand, S. Mrenna, and P. Z. Skands, A Brief Introduction to PYTHIA 8.1, Comput. Phys. Commun. 178 (2008) 852–867, [arXiv:0710.3820].
  * (47) ATLAS Collaboration, M. Aaboud et al., Measurement of the cross-section for electroweak production of dijets in association with a Z boson in pp collisions at $\sqrt{s}$ = 13 TeV with the ATLAS detector, Phys. Lett. B 775 (2017) 206–228, [arXiv:1709.10264].
  * (48) ATLAS Collaboration, G. Aad et al., Measurements of $W^{+}W^{-}+\geq 1\leavevmode\nobreak\ $jet production cross-sections in $pp$ collisions at $\sqrt{s}=13\leavevmode\nobreak\ $TeV with the ATLAS detector, JHEP 06 (2021) 003, [arXiv:2103.10319].
  * (49) ATLAS Collaboration, M. Aaboud et al., Measurement of fiducial and differential $W^{+}W^{-}$ production cross-sections at $\sqrt{s}=13$ TeV with the ATLAS detector, Eur. Phys. J. C 79 (2019), no. 10 884, [arXiv:1905.04242].
  * (50) D. Curtin et al., Exotic decays of the 125 GeV Higgs boson, Phys. Rev. D 90 (2014), no. 7 075004, [arXiv:1312.4992].
  * (51) J. D. Wells, How to Find a Hidden World at the Large Hadron Collider, arXiv:0803.1243.
  * (52) ATLAS Collaboration, G. Aad et al., Search for events with a pair of displaced vertices from long-lived neutral particles decaying into hadronic jets in the ATLAS muon spectrometer in pp collisions at $\sqrt{s}$=13 TeV, Phys. Rev. D 106 (2022), no. 3 032005, [arXiv:2203.00587].
  * (53) Z. S. Wang, Constraining long-lived particles from Higgs decays at the LHC with displaced vertices and jets, arXiv:2406.16281.
  * (54) ATLAS Collaboration, G. Aad et al., Measurements of the production cross-section for a $Z$ boson in association with $b$-jets in proton-proton collisions at $\sqrt{s}=13$ TeV with the ATLAS detector, JHEP 07 (2020) 044, [arXiv:2003.11960].
  * (55) ATLAS Collaboration, M. Aaboud et al., Searches for scalar leptoquarks and differential cross-section measurements in dilepton-dijet events in proton-proton collisions at a centre-of-mass energy of $\sqrt{s}$ = 13 TeV with the ATLAS experiment, Eur. Phys. J. C 79 (2019), no. 9 733, [arXiv:1902.00377].
  * (56) ATLAS Collaboration, G. Aad et al., Measurements of top-quark pair differential and double-differential cross-sections in the $\ell$+jets channel with $pp$ collisions at $\sqrt{s}=13$ TeV using the ATLAS detector, Eur. Phys. J. C 79 (2019), no. 12 1028, [arXiv:1908.07305]. [Erratum: Eur.Phys.J.C 80, 1092 (2020)].
  * (57) CMS Collaboration, A. M. Sirunyan et al., Measurement of differential cross sections for the production of top quark pairs and of additional jets in lepton+jets events from pp collisions at $\sqrt{s}=$ 13 TeV, Phys. Rev. D 97 (2018), no. 11 112003, [arXiv:1803.08856].
  * (58) ATLAS Collaboration, M. Aaboud et al., Measurements of top-quark pair differential cross-sections in the lepton+jets channel in $pp$ collisions at $\sqrt{s}=13$ TeV using the ATLAS detector, JHEP 11 (2017) 191, [arXiv:1708.00727].
  * (59) CMS Collaboration, V. Khachatryan et al., Measurement of differential cross sections for top quark pair production using the lepton+jets final state in proton-proton collisions at 13 TeV, Phys. Rev. D 95 (2017), no. 9 092001, [arXiv:1610.04191].
  * (60) ATLAS Collaboration, G. Aad et al., Measurements of differential cross-sections in four-lepton events in 13 TeV proton-proton collisions with the ATLAS detector, JHEP 07 (2021) 005, [arXiv:2103.01918].
  * (61) “HAHM model UFO file.” https://feynrules.irmp.ucl.ac.be/wiki/HAHM.
  * (62) ATLAS Collaboration, M. Aaboud et al., Measurements of $t\bar{t}$ differential cross-sections of highly boosted top quarks decaying to all-hadronic final states in $pp$ collisions at $\sqrt{s}=13\,$ TeV using the ATLAS detector, Phys. Rev. D 98 (2018), no. 1 012003, [arXiv:1801.02052].
  * (63) “Auxiliary Material for ATLAS-TOPQ-2016-09.” https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PAPERS/TOPQ-2016-09/.
  * (64) D. Curtin, R. Essig, S. Gori, and J. Shelton, Illuminating Dark Photons with High-Energy Colliders, JHEP 02 (2015) 157, [arXiv:1412.0018].
  * (65) H. Davoudiasl, H.-S. Lee, and W. J. Marciano, ’Dark’ Z implications for Parity Violation, Rare Meson Decays, and Higgs Physics, Phys. Rev. D 85 (2012) 115019, [arXiv:1203.2947].
  * (66) ATLAS Collaboration, M. Aaboud et al., Search for the Production of a Long-Lived Neutral Particle Decaying within the ATLAS Hadronic Calorimeter in Association with a $Z$ Boson from $pp$ Collisions at $\sqrt{s}=$ 13 TeV, Phys. Rev. Lett. 122 (2019), no. 15 151801, [arXiv:1811.02542].
  * (67) ATLAS Collaboration, G. Aad et al., Measurement of the $t\bar{t}$ production cross-section and lepton differential distributions in $e\mu$ dilepton events from $pp$ collisions at $\sqrt{s}=13\,\text{TeV}$ with the ATLAS detector, Eur. Phys. J. C 80 (2020), no. 6 528, [arXiv:1910.08819].
  * (68) CMS Collaboration, A. M. Sirunyan et al., Measurements of differential Z boson production cross sections in proton-proton collisions at $\sqrt{s}$ = 13 TeV, JHEP 12 (2019) 061, [arXiv:1909.04133].