aainstitutetext: Fermi National Accelerator Laboratory,
Batavia, IL, USAbbinstitutetext: University of Chicago,
Chicago, IL, USAccinstitutetext: University of Tennessee,
Knoxville, TN, USAddinstitutetext: Illinois Institute of Technology,
Chicago, IL, USAeeinstitutetext: Brown University,
Providence, RI, USAffinstitutetext: TRIUMF,
Vancouver, BC, Canada

# Optimizing Trigger-Level Track Reconstruction for Sensitivity to Exotic
Signatures

K. F. Di Petrillo c J. N. Farr d C. Guo c T. R. Holmes e J. Nelson f K. Pachal
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Many compelling beyond the Standard Model scenarios predict signals that
result in unconventional charged particle trajectories. Signatures for which
unusual tracks are the most conspicuous feature of the event pose significant
challenges for experiments at the Large Hadron Collider (LHC), particularly
for the trigger. This article presents a study of track-based triggers for a
representative set of long-lived and unconventional signatures at the upcoming
High Luminosity LHC, as well as resulting recommendations for the target
parameters of a hardware-based tracking system. Scenarios studied include
large multiplicities of low-$p_{\mathrm{T}}$ tracks produced in a soft-
unclustered-energy-pattern model, displaced leptons and anomalous prompt
tracks predicted in a Supersymmetry model with long-lived staus, and displaced
hadrons predicted in a Higgs portal scenario with long-lived scalars.

##  1 Introduction

Decades of experimental tests have demonstrated the predictive power of the
Standard Model (SM). However, there are a myriad of phenomena it does not
explain, including dark matter, baryon asymmetry, naturalness, and more. Many
beyond the Standard Model (BSM) scenarios that address these inefficiencies
predict unconventional track signatures that could be observed at high energy
colliders such as the Large Hadron Collider (LHC). For example, long-lived BSM
particles can travel a measurable distance before decaying, resulting in
displaced track or anomalous prompt track signatures Lee_2019 (1).
Alternatively, models with a strongly coupled hidden valley can result in
large multiplicities of soft charged particles Knapen_2017 (2). These
unconventional track signatures pose tremendous challenges for general purpose
experiments at the LHC, particularly for the trigger.

Due to bandwidth and storage constraints, the ATLAS and CMS experiments both
rely on real-time reconstruction to select interesting collisions before
detector data can be read out and stored. The system that performs this task
is called the trigger. A particular challenge for trigger systems at the LHC
is differentiating signal processes from a large background of low energy
proton-proton interactions, including the multiple interactions that occur per
event, referred to as pile-up. The trigger is primarily designed to select
events with high-$p_{\mathrm{T}}$ jets, photons, or leptons that originate
directly from a $pp$ collision of interest, or events characterized by large
missing transverse momentum. These prompt signatures are easily identified in
the first, hardware-based, stage of the trigger, which makes use of
calorimeter and muon system information. Limited tracking information is only
available at later stages of the trigger decision.

Unfortunately, standard triggers often result in low trigger efficiency for
unconventional BSM signatures. If a BSM particle decays at a distance and
produces jets or leptons, the decay products will not point back to the $pp$
collision. Decay products can be misidentified as pile-up or fail standard
quality criteria. Charged BSM particles which travel a long distance before
decaying can result in anomalous signals in the detector that can be confused
with noise. Models with many low energy charged particles can be difficult to
differentiate from pile-up.

When searching for these unconventional BSM signatures, the most distinctive
features of signal events are located in the tracker. Dedicated track-based
triggers are necessary to access these challenging scenarios, and would
require track information is available at an early stage of the trigger
decision. A long history of hardware-level or intermediate stage tracking in
various experiments provides a guideline for what rates and performance can be
realistically anticipated at ATLAS and CMS. While possible implementations and
track reconstruction algorithms are discussed elsewhere Louise (3), including
unconventional tracking, there has been no explicit study of how best to
allocate the limited resources available to optimize sensitivity across a
range of signatures thus far. This paper addresses that need.

###  1.1 History and current state of track-triggers

High energy physics experiments typically use multi-step trigger systems to
select events which are saved for analysis. The first stage, Level 1 (L1),
performs an initial reduction using simple decisions often implemented in
custom hardware. The following stages (collectively, the High Level Trigger or
HLT) have a longer latency budget and use standard software routines to
further reduce the event rate.

The usage of track information in the trigger at previous experiments has
prioritized the identification of Standard Model processes. Searches for long-
lived or displaced signatures have typically used triggers based on
calorimeter or muon detector information lhcb_aprime (4, 5). The CDF and D0
experiments, which can be considered the predecessors to ATLAS and CMS,
operated in conditions with a 2.5 MHz event rate and matched tracking detector
hits to predefined patterns or equations for fast track reconstruction at L1
cdf-svt (6, 7). Experiments specializing in $b$ physics have long prioritized
hardware-level track-triggers. BABAR babar (8), HERA-B hera_l1 (9), and Belle-
II belle_ii_paper (10) all included track reconstruction at L1, though in some
cases restricted to regions of interest identified by other event activity.
The L1 track reconstruction in these three cases constrains tracks to
originate from the $z$ axis, diminishing sensitivity to displaced decays.
Track reconstruction at HLT can be more sophisticated. LHCb contends with an
L1 event rate of 40 MHz, like ATLAS and CMS but with much lower instantaneous
luminosity. In Run 2 LHCb did not have any tracking at L1, but has moved to an
HLT-only design for Run 3 that will permit full event tracking with high
quality on each event lhcb_i (11, 12).

ATLAS and CMS currently use similar two-step trigger systems. In both
experiments, the L1 stage reduces the event rate from $40$ MHz to $100$ kHz
within a latency of $\sim$3\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ while the
second (HLT) stage reduces the event rate from $100$ kHz to $\sim 1$ kHz, with
$\sim 200$ ms on average to make a decision Sirunyan:2721198 (13, 14, 15). Due
to bandwidth and latency constraints, tracking information is only produced at
the HLT, where it is the most CPU-intensive reconstruction process. There,
limited track reconstruction routines are run in software, balancing the
amount of information necessary to make a decision with the amount of time
needed to reconstruct the event. For example in lepton triggers, track
reconstruction is often only performed in small regions of interest that were
identified by the calorimeter or muon system at L1. Full detector tracking is
possible with fast algorithms and a reduced event rate. These fast algorithms
result in reduced efficiency across wide ranges of track transverse momentum,
$p_{\mathrm{T}}$, and transverse impact parameter, $d_{0}$ with respect to
offline reconstruction.

The upcoming High Luminosity LHC (HL-LHC) will see significant upgrades to the
track-based triggering capacity of both experiments. ATLAS and CMS both plan
to install new tracking detectors and completely overhaul their trigger
designs. In CMS, tracker $p_{\mathrm{T}}$-modules will enable sufficient
bandwidth reduction such that tracks with $p_{\mathrm{T}}>2$ GeV can be
reconstructed at L1 CERN-LHCC-2017-009 (16, 17). While the ATLAS tracker will
not participate in the hardware-level trigger decision at the HL-LHC, it will
participate at the HLT, where tracking will be run on a selected subset of
events within the full HLT input rate of 1 MHz. Under the baseline plan, track
reconstruction will be performed at the HLT with a CPU based processing farm
potentially complemented by commodity accelerator hardware
Collaboration:2285584 (18).

With additional effort, it may be possible to extend these hardware track-
triggers to include a range of long-lived or unconventional signatures
Gershtein_2017 (19, 20) by allocating some fraction of the available tracking
resources to unconventional track reconstruction and a fraction of the rate to
triggers based on these tracks. Tracking performance is limited by detector
readout rates, the hardware-based tracking technology used, output rate, and
latency constraints. In general, increasing the maximum curvature,
displacement, or target efficiency of reconstructed tracks increases
complexity, rate, latency, or a combination thereof. In order to maintain a
constant latency but extend its $d_{0}$ range, a track-trigger could increase
its $p_{\mathrm{T}}$ threshold or decrease its target efficiency. Depending on
the track-trigger design, it may be possible to alter these requirements
independently for prompt and displaced tracks, resulting in a highly efficient
prompt track-trigger with a low $p_{\mathrm{T}}$ threshold, while also
exploring a large range of displacements.

###  1.2 Expanding track-triggers for future experiments

There has been a recent proliferation of studies on the efficacy of track-
based triggers for long-lived particles and other exotic signatures, including
rare Higgs decays Gershtein_2017 (19), soft-unclustered-energy-patterns
(SUEPs) Knapen_2017 (2), and generic displaced signatures M_rtensson_2019 (21,
22). ATLAS demonstrated that a hardware-based track finder algorithm could
extend its impact parameter range from $|d_{0}|<2$ mm to $|d_{0}|<10$ mm
without significantly degrading efficiency for prompt tracks ftk_results (20).
Displaced hardware-based tracking has been extensively studied for the HL-LHC
on CMS Gershtein_2017 (19).

However, because the signatures of these models are so varied, it is
challenging to translate individual studies into an optimal design for a
future trigger. In particular, for a realistic rate- and latency-limited
trigger, it is unclear which efficiency trade-offs are most beneficial across
a range of signatures: is it better to expand $d_{0}$ coverage, reduce
$p_{\mathrm{T}}$ thresholds, or maximize overall tracking efficiency? The
answer to this question depends on the design of the trigger being considered,
but it also depends on what kind of exotic signatures are being targeted. For
hadronic decays, low $p_{\mathrm{T}}$ thresholds are important, but high
efficiency may not be needed. For leptonic decays the reverse is true.

This study aims to provide a recommendation on how to prioritize a track-
trigger’s $p_{\mathrm{T}}$ and $d_{0}$ coverage in order to maximize
sensitivity to a range of unconventional signatures. This aim is based on the
idea that most tracking algorithms are configured to reconstruct particles
within a restricted range of tracking parameter space to meet bandwidth and
latency constraints, and that increasing coverage in one tracking parameter
will require decreasing coverage elsewhere. This study will not discuss
specific tracking algorithms, as these vary widely between experiments and are
discussed elsewhere. The overall efficiency to reconstruct the unconventional
tracks expected in a variety of BSM models will be presented as a function of
track parameters and simple trigger concepts, which can act as a guide for
future trigger design. Because the practicality of such triggers depends on
the rates at which they can operate, this study will also show expected
Standard Model backgrounds for possible triggers based on these tracking
parameters, demonstrating their feasibility.

##  2 Benchmark model performance

This work studies a representative set of unconventional signatures motivated
by three beyond the Standard Mode scenarios. Figure 1 shows Feynman diagrams
corresponding to the benchmark models considered. Events with a large
multiplicity of prompt low $p_{\mathrm{T}}$ tracks are motivated by a soft-
unclustered-energy-pattern (SUEP) model. Displaced decays to hadrons are
motivated by a model with long-lived scalars coupling to the Higgs boson. A
gauge mediated Supersymmetry breaking scenario with direct production of long-
lived staus results in displaced leptons as well as heavy (meta)stable charged
particles (HSCPs) which can be detected directly.

Figure 1:  Feynman diagrams for benchmark processes. From left to right:
SUEPs, Higgs portal, and long-lived staus.

Together, these four signatures span a wide range of possible track momenta
and displacement, as well as event-level track multiplicity. In order to
directly compare these vastly different signatures, a common strategy is used
to evaluate the acceptance and efficiency of possible track-trigger
configurations. The following sections outline this overall analysis strategy
as well as the specific studies performed for each model.

###  2.1 Analysis strategy

All efficiency studies are performed assuming HL-LHC conditions, using truth-
level simulation of $pp$ collisions at a center of mass energy of
$\sqrt{s}=14$ TeV. Charged particles considered for analysis are stable (or
metastable) decay products of short-lived BSM particles, except for the HSCP
case in which direct detection of the metastable particle is considered. For
simplicity, there is no simulation of detector or material interactions, and
pile-up collisions are not included in signal simulation. Discussion of how
signal efficiencies and background rates are influenced by pile-up is included
in Section 3.

The acceptance and efficiency of possible trigger configurations are evaluated
for each signature. First, baseline criteria are defined based on potential
detector specifications (e.g. $\eta$ coverage, tracker radius). Only stable
charged particles satisfying these criteria are considered to be in
acceptance. This geometric acceptance simply asks whether particles could be
reconstructed by an offline tracking algorithm with the specified detector
layout, and is independent of what could then be defined by a track-trigger’s
configuration. Certain effects of adjusting the detector layout are explored
for the HSCP signature. For other models the detector layout is kept fixed.
The acceptance for a signal point is then the fraction of events with at least
$n$ particles in the final state that meet the tracking criteria, where $n$ is
also model dependent.

For displaced signatures, only tracks with $|d_{0}|>1$ mm are counted in the
acceptance criteria. This displacement requirement separates tracks which
would likely be picked up by a prompt track reconstruction algorithm from
those which require a displaced tracking algorithm, and also defines a region
in which SM backgrounds are small.

Efficiency is then defined to capture the effects of track reconstruction
criteria which could be set at the track-trigger level rather than enforced by
the physical detector. The two most important criteria studied are the minimum
$p_{\mathrm{T}}$ and maximum |$d_{0}$| of tracks which would be reconstructed
by the track-trigger. These factors have different effects in different BSM
models, depending on whether the final state tracks are prompt or displaced,
what the average decay angles of long-lived particles may be, and how much
momentum the final state particles carry. To study the interplay of these
effects, for a fixed acceptance, the $p_{\mathrm{T}}$ and $d_{0}$ thresholds
are varied, and the fraction of surviving tracks is measured. An event-level
efficiency is then defined, as for the acceptance, based on whether at least
$n$ particles in the final state pass these thresholds.

In realistic tracking detectors, although the reconstruction efficiency for
prompt isolated tracks is very close to $100\%$ independent of track
$p_{\mathrm{T}}$, the efficiency tends to decrease roughly linearly with
increasing |$d_{0}$| ATL-PHYS-PUB-2017-014 (23). To emulate this effect in the
following studies, the likelihood of reconstructing a given track is set not
as a binary by the $d_{0}$ threshold (e.g. all tracks up to $|d_{0}|=5$ mm are
reconstructed; all above 5 mm are not reconstructed), but instead as a
linearly decreasing function. The likelihood of reconstructing the track
begins at unity for $d_{0}=0$ and decreases linearly with increasing |$d_{0}$|
until it reaches zero at the specified |$d_{0}$| threshold. This linearly
falling assumption is a good match for the behavior observed in offline track
reconstruction in ATLAS, though the maximum |$d_{0}$| values explored here for
trigger-level reconstruction are more conservative.

The efficiency definition considered in this section is designed to test if a
hypothetical track-trigger can reconstruct a sufficient number of
unconventional tracks such that a dedicated trigger could be defined. For all
BSM scenarios considered, background rates in HL-LHC conditions for dedicated
triggers are studied in Section 3. For the heavy stable charged particle
scenario and displaced lepton scenarios, simple track counting requirements
will already result in negligible backgrounds. For the SUEP case, a large QCD
background has the potential to mimic signal if the track-trigger is unable to
differentiate between pile-up vertices. For the lower $p_{\mathrm{T}}$ Higgs
portal scenario, counting displaced tracks at large pile-up is not sufficient
to reduce background from Standard Model long-lived particles and randomly
crossing displaced tracks produced in high pile-up conditions, and more
complex selections are studied.

###  2.2 SUEPs

Motivated in certain hidden valley models, SUEPs represent a worst case
scenario for triggers at hadron colliders Knapen_2017 (2, 24). In SUEP events,
low-$p_{\mathrm{T}}$ final state particles are not collimated into Standard
Model-like jets. The most conspicuous feature is a large multiplicity of
diffuse low $p_{\mathrm{T}}$ tracks. This scenario is very difficult for the
trigger to distinguish from pile-up, unless the many soft tracks can be
reconstructed and associated to a $pp$ collision of interest. As a result,
there are currently no constraints on SUEP scenarios from the LHC.

In SUEP models, a dark SU(N) sector is connected to the Standard Model by a
mediator with a mass much greater than the mass splittings amongst the dark
sector particles. Energy from the production of the mediator is distributed
amongst a large multiplicity of light dark hadrons, which decay back to SM
particles. The ’t Hooft coupling of the dark sector,
$\lambda_{D}\equiv\alpha_{D}N_{C_{D}}$, where $\alpha_{D}$ is the dark
coupling constant and $N_{C_{D}}$ is the number of colors, determines the
angular emission of partons during showering. In SUEP scenarios this coupling
is assumed to be large, and dark mesons are isotropically distributed in the
mediator’s rest frame.

SUEP samples are generated in Pythia8 Sjostrand:2014zea (25) using the custom
plugin described in Ref. Knapen_2017 (2). Events are generated assuming a
scalar mediator, $\Phi$, with a range of masses: 125 GeV (matching the SM
Higgs boson) and then 200 GeV to 1000 GeV in 200 GeV intervals. The SUEP
shower consists of a large multiplicity of a single flavor of dark mesons with
mass $m_{\pi_{d}}=1$ GeV and temperature $T=1$ GeV.

In principle, dark mesons may decay to a variety of final state SM particles.
Possible decay modes include pairs of leptons, hadrons, photons, and light or
heavy flavor quarks. Decay products may be also be displaced, or the final
state may also include undetected dark matter particles. For simplicity, this
study focuses on a scenario in which each dark meson decays promptly to a pair
of light quarks, $d\bar{d}$.

Figure 2(a) shows charged particle transverse momenta for the SUEP benchmark
models under consideration. Unlike most BSM scenarios, the $p_{\mathrm{T}}$
spectra of final state particles do not depend on the mediator mass, but on
the dark meson mass and temperature. As a result, all generated momenta
distributions are similar in shape, and peak sharply in the lowest
$p_{\mathrm{T}}$ bins.

(a) Charged particle $p_{\mathrm{T}}$

(b) Charged particle multiplicity

Figure 2: Distribution of all charged SM particle transverse momenta (2(a))
and multiplicity of stable charged particles with $p_{\mathrm{T}}>0.5$ GeV.
(2(b)) in SUEP events for a range of mediator masses.

The number of final state particles depends on mass of the scalar mediator and
dark meson mass, $n\sim m_{\Phi}/m_{\pi_{d}}$, as shown in Figure 2(b). For a
fixed mediator and dark meson mass, increasing the temperature reduces the
number of final state particles and increases their mean momenta, as described
in Ref. Cesarotti:2020hwb (26).

Because all tracks from the model considered here are prompt, the selection
strategy is relatively straightforward. The number of stable charged particles
within tracker acceptance and above a given $p_{\mathrm{T}}$ threshold are
counted per event. It is assumed that the track-trigger would have sufficient
longitudinal impact parameter resolution to associate tracks to the correct
primary vertex, and neutral final state particles are neglected.

Table 1 summarizes SUEP object-level acceptance and efficiency definitions.
Tracks within $|\eta|<2.5$ are considered to be within the detector
acceptance, which is consistent with the $\eta$ range of the CMS L1 track
finder Tomalin:2017hts (27). The event-level acceptance is $100\%$ in all
cases because of the large multiplicity of charged particles. The efficiency
is defined as the fraction of events where the number of tracks,
$n_{\mathrm{Track}}$, that pass the specified $p_{\mathrm{T}}$ and $\eta$
requirements is above a certain threshold.

Variable | Requirement
---|---
Acceptance
$|\eta|$ | < $2.5$
Efficiency
$p_{\mathrm{T}}$ |  $>0.5,1,2$ GeV
$n_{\mathrm{Track}}$ | $>100,150,200$
Table 1:  Acceptance and efficiency requirements for a track-trigger targeting
events with a high multiplicity of soft tracks

The most important parameter to consider is the $p_{\mathrm{T}}$ threshold at
which track reconstruction begins. For SUEP signatures, signal to background
discrimination decreases with increasing track $p_{\mathrm{T}}$ requirements,
because track multiplicity is the most distinctive feature of these events.
This study investigates a range of $p_{\mathrm{T}}$ thresholds, from
$0.5<p_{\mathrm{T}}<2$ GeV. The lowest threshold represents the minimum
charged particle $p_{\mathrm{T}}$ for most standard offline track
reconstruction algorithms, aimed at charged particles that traverse a
sufficient number of tracking detector layers ATLAS_tracking_today (28, 29).
The highest $p_{\mathrm{T}}$ threshold examined, $p_{\mathrm{T}}>2$ GeV, is
based on the minimum L1 tracking threshold of 2 to 3 GeV estimated by CMS for
their Phase 2 trigger upgrade CERN-LHCC-2020-004 (17). For the temperature
considered, a threshold above $p_{\mathrm{T}}$ $=2$ GeV, counting the
$n_{\mathrm{Track}}$ is no longer expected to be a powerful discriminator.
Figure 3 shows the number of reconstructed tracks in SUEP events satisfying
different track $p_{\mathrm{T}}$ thresholds.

(a)

(b)

Figure 3: Number of charged particles per event which pass the specified track
$p_{\mathrm{T}}$ threshold, for mediators of mass 200 GeV (3(a)) and 800 GeV
(3(b)).

Efficiencies for all simulated mediator masses are shown as a function of the
minimum $n_{\mathrm{Track}}$ requirement for three different $p_{\mathrm{T}}$
thresholds in Figure 4. The efficiency increases with mediator mass, due to
the larger number of tracks produced. The track $p_{\mathrm{T}}$ spectra do
not change with mediator mass, and this effect is simply because a sufficient
number of particles are reconstructed more frequently for heavy mediators.
With a track reconstruction threshold of $p_{\mathrm{T}}>0.5$ GeV,
efficiencies of $50\%$ are obtained for mediators with masses above $200$ GeV
with a $n_{\mathrm{Track}}>100$ trigger, or for mediators with masses above
$400$ GeV with a $n_{\mathrm{Track}}>150$ trigger. If the track reconstruction
threshold increases to $p_{\mathrm{T}}>1$ GeV, $50\%$ efficiency is only
possible for mediator masses above $\sim 600$ GeV. With a reconstruction
threshold of $p_{\mathrm{T}}>2$ GeV, there is negligible efficiency for all
signals with any $n_{\mathrm{Track}}$ threshold considered.

(a)

(b)

(c)

Figure 4: Efficiency, defined as number of events for which at least
$n_{\mathrm{Track}}$ within $\eta<2.5$ pass the track reconstruction
$p_{\mathrm{T}}$ threshold, as a function of mediator mass for minimum track
$p_{\mathrm{T}}$ of 0.5 GeV (4(a)), 1 GeV (4(b)), and 2 GeV (4(c)).

For a track-trigger similar to the CMS HL-LHC upgrade, where
$p_{\mathrm{T}}>2$ GeV is the baseline threshold, a trigger requiring a large
track multiplicity is inefficient for most SUEP signal models. Other possible
metrics for a SUEP trigger should be investigated, such as the scalar sum of
jet energy, the scalar sum of track $p_{\mathrm{T}}$ ($H_{\mathrm{T}}$, shown
in Figure 5), or event shapes such as isotropy or sphericity Cesarotti_2021
(30, 31). At the LHC, SUEP final state particles are expected to be
isotropically distributed in $\phi$, but localized in $\eta$, resulting in a
“belt of fire”. This characteristic shape has been shown to be an efficient
discriminator at the HLT Knapen_2017 (2). The most optimal trigger available
to the LHC experiments will likely use $H_{\mathrm{T}}$ or an event shape in
combination with the number of reconstructed tracks.

(a)

(b)

Figure 5: $H_{\mathrm{T}}$ of charged particles passing the specified track
$p_{\mathrm{T}}$ threshold, for mediators of mass $200$ GeV (5(a)) and $800$
GeV (5(b)).

###  2.3 Higgs portal to long-lived scalars

In this model, referred to here as the “Higgs portal”, a SM Higgs boson decays
to a pair of long-lived scalar particles. Each scalar subsequently decays to a
pair of SM particles. Since the long-lived particle is not electromagnetically
charged, no direct detection trigger strategies are available. The
identifiable characteristic of this scenario is several relatively
low-$p_{\mathrm{T}}$ displaced jets. The low-$p_{\mathrm{T}}$ nature of these
jets presents a challenge for current L1 triggers because it is difficult for
the calorimeter to distinguish these displaced jets from a large multi-jet
background. Instead, this scenario can be targeted with a trigger that selects
events with a number of displaced tracks.

The benchmark model considered in this paper explores the well-motivated
possibility that the SM Higgs boson acts as a mediator to a hidden sector via
mixing with a new neutral scalar ATLAS_displacedjets (32, 33, 34). A dark U(1)
gauge group with a dark Higgs potential is introduced, leading to a total of
two new particles, of which only the scalar has a mass low enough to be
accessible at the LHC. This model serves as a good benchmark for a wide
variety of Higgs portal scenarios, which will be a high priority target for
the HL-LHC. If the only available decay channel for the dark scalar is via its
(small) mixing with the Higgs, this process will be suppressed, and the scalar
will travel a macroscopic distance before decaying in to SM particles. The
resulting signature consists of low $p_{\mathrm{T}}$ displaced tracks arising
from the point at which the scalar decayed. The light mass of the scalar also
means these displaced tracks tend to have modest impact parameters, even when
the scalar decays at a high radius. The branching ratio for the decay of the
dark scalar follows that of the Higgs, restricted by its mass.

Higgs portal events are generated using MadGraph5_aMC@NLO 2.9.3 Alwall:2014hca
(35) and the SM + Dark Vector + Dark Higgs model described in Refs.
Curtin:2013fra (36, 37). The Higgs mixing parameter is taken to be $10^{-4}$
and the kinetic mixing parameter to be $10^{-10}$. The dark Z mass is set to 1
TeV, and effectively decoupled. In the generated scenario the dark scalar,
$s$, is the only relevant new particle and it decays only into Standard Model
particles. The properties of the dark scalar are varied across the mass range
$m_{s}=5$, 8, 15, 25, 40, and 55 GeV and proper lifetime range 0.01, 0.1, and
1 ns. The scalar’s branching ratio follows those of the Higgs, such that the
scalars decay predominantly to $c\bar{c}$ and $\tau\bar{\tau}$ for $m_{s}=5$
and 8 GeV and predominantly to $b\bar{b}$ for heavier $m_{s}$.

The number of displaced charged particles per event and their $p_{\mathrm{T}}$
is shown in Figure 6. The low parent masses and fully hadronic final state
result in a moderate number of low-$p_{\mathrm{T}}$ particles. However, there
are a sufficient number of displaced tracks to make a track-based trigger
feasible.

Figure 6:  (Left) The number of displaced charged particles per event.
Particles are required to be decay products of the long-lived scalar and have
$p_{\mathrm{T}}>0.5$ GeV. (Right) Displaced charged particle $p_{\mathrm{T}}$
for each scalar mass.

A baseline track-level acceptance for this scenario is defined using
requirements described in Table 2. In addition to minimum $p_{\mathrm{T}}$ and
$\eta$ requirements, displaced charged particles are required to be produced
within a minimum radius and pass a minimum length requirement. These
requirements ensure that displaced particles will traverse a sufficient number
of tracker layers to be considered for reconstruction.

Variable | Requirement
---|---
Acceptance
$L_{\mathrm{xy}}$ | < 300 mm
$L_{\mathrm{track}}$ | > 200 mm
$|d_{0}|$ |  $\geq 1$ mm
$p_{\mathrm{T}}$ | > 0.5 GeV
$|\eta|$ | < 2.5
Efficiency
$p_{\mathrm{T}}$ | > 0.5, 1, 2, 5, 10 GeV
$|d_{0}|$ | < 10, 20, 50, 100 mm
Table 2:  Details of the displaced track acceptance and efficiency definitions
for the Higgs portal model. The $L_{\mathrm{xy}}$ parameter refers to the
maximum allowed production radius of the track, while the $L_{\mathrm{track}}$
variable refers the minimum track length.

For this model, two event-level acceptance definitions are considered due to
the low momenta of the displaced tracks. The first requires a minimum of five
tracks per event to pass the acceptance criteria defined in Table 2, and the
second requires least two tracks per event. Event-level acceptances for these
two definitions are shown in Figure 7.

Figure 7:  Acceptances for the Higgs portal scenario requiring at least two
tracks (left) and least five tracks (right) passing the requirements listed in
Table 2.

Discontinuities in the event-level acceptances and efficiencies around scalar
masses of $m_{s}=10$ GeV are a result of the sudden change in dominant decay
modes, from pairs of charm quarks and tau leptons to $b\bar{b}$. The
acceptance requirement has the largest impact at low masses, where there are
fewer tracks likely to pass the minimum $p_{\mathrm{T}}$ threshold.

Track-level efficiency requirements corresponding to potential online tracking
configurations are defined in Table 2. The track-trigger parameters considered
include a minimum $p_{\mathrm{T}}$ and maximum |$d_{0}$| requirement, where
the exact values are varied to understand the impact each selection has on the
efficiency. Event-level efficiencies are defined as the fraction of events in
acceptance which also have at least $n_{\mathrm{Track}}$ passing efficiency
requirements.

As with the acceptance definitions, two different values of
$n_{\mathrm{Track}}$ are studied for the Higgs portal efficiencies.
Efficiencies are defined on top of the corresponding $n_{\mathrm{Track}}$
acceptance. Figures in this section largely correspond to
$n_{\mathrm{Track}}\geq 5$, a likely realistic baseline for a trigger
targeting hadronic long-lived particle decays. The additional study performed
with $n_{\mathrm{Track}}\geq 2$ is to allow for direct comparison to the stau
model. Results for this two-track trigger are used in Section 4 and directly
compared in Figure 8.

Figure 8:  Event-level efficiency as a function of scalar mass for a range of
minimum track $p_{\mathrm{T}}$ values. Efficiencies are shown for the two
(left) or five (right) scenarios. In all cases, the track reconstruction
efficiency is assumed to decrease linearly with $|d_{0}|$ up until
$|d_{0}|=100$ mm.

As expected, the Higgs portal efficiencies are very sensitive to the minimum
$p_{\mathrm{T}}$ requirement, as shown in Figure 8. The overall trend in
efficiency is the same for each $n_{\mathrm{Track}}$ considered. For a minimum
$p_{\mathrm{T}}$ of 2 GeV, efficiencies for $n_{\mathrm{Track}}\geq 5$ drop to
around $50\%$ of their maximum value, and $p_{\mathrm{T}}$ thresholds any
higher make this strategy ineffectual.

The Higgs portal efficiency’s dependence on the |$d_{0}$| range, shown in
Figure 9 is more modest than the dependence on $p_{\mathrm{T}}$. Endpoints of
$|d_{0}|<1$ cm are at worst about $50\%$ as efficient as endpoints of
$|d_{0}|<10$ cm for the lifetimes considered. Given the high track
multiplicity per event and long-lived particle’s boost, it is easier to find
tracks with small |$d_{0}$| for this model, even for very displaced decays.

Figure 9:  Event-level efficiency as a function of long-lived scalar lifetime
for different $|d_{0}|$ endpoints and $p_{\mathrm{T}}$ larger than 2 GeV.
Efficiencies are shown for the two (left) or five (right) $n_{\mathrm{Track}}$
scenarios. In all cases, the track reconstruction efficiency is assumed to
decrease linearly with $|d_{0}|$ up until the endpoint specified.

Figure 10 shows the event-level efficiency as a function of scalar mass and
lifetime for a variety of minimum $p_{\mathrm{T}}$ thresholds. The nominal
threshold of $p_{\mathrm{T}}>2$ GeV for the CMS Phase 2 tracker results in a
peak efficiency of around $50\%$ for higher mass scalars.

Figure 10:  Efficiencies for the Higgs to long-lived scalar model for a
variety of track $p_{\mathrm{T}}$ thresholds: 0.5 GeV (top left), 1 GeV (top
right), 2 GeV (bottom left), 5 GeV (bottom right). Across a wide range of
masses and lifetimes, efficiency falls off steeply as the $p_{\mathrm{T}}$
threshold increases above 2 GeV. In all cases, a linearly parameterized
|$d_{0}$| efficiency is used with an endpoint at $50$ mm, and the five-track
selection is used.

###  2.4 Long-lived staus

Long-lived staus serve as a useful benchmark for two long-lived particle
detection techniques. Staus can be either be directly detected as HSCPs or
indirectly identified via displaced decay products. The current most
comprehensive limits are from the OPAL Experiment, which exclude staus with
mass $m_{\tilde{\tau}}<90$ GeV for all possible lifetimes OPAL:2005pwy (38).
LHC searches for highly ionizing particles, disappearing tracks, and displaced
leptons extend exclusions to larger stau masses ATLAS:2022pib (39, 40, 41, 42,
43). However, these searches do not cover the full lifetime space. Searches
which aim to indirectly detect displaced decay products are limited by trigger
$p_{\mathrm{T}}$ thresholds and efficiency for displaced particles. Searches
which aim to directly detect charged particles such as the stau itself are
currently trigger-limited when a track is not reconstructed in the muon
system.

The benchmark model considered here assumes direct production of a pair of
staus in a gauge mediated Supersymmetry breaking (GMSB) scenario evans-staus
(44, 45, 46). While some GMSB slepton models assume the three sleptons are
mass degenerate, here the stau alone is taken to be the next-to-lightest
supersymmetric particle (NLSP). The gravitino is the lightest supersymmetric
particle in GMSB models. The stau decays to a tau and a gravitino via a small
gravitational coupling, thereby gaining a significant lifetime.

Stau samples are generated and decayed with MadGraph5_aMC@NLO 2.9.3. A
simplified model is used wherein the left-handed and right-handed staus are
assumed to be mass degenerate and have a mixing angle $\sin\theta=0.95$
ATLAS:2020wjh (42). The gravitino is given a negligible mass of 1 GeV. Stau
samples are generated for masses $m_{\tilde{\tau}}=100$, 200, 300, 400, 500,
and $600$ GeV and lifetimes 0.001, 0.01, 0.1, 1 ns. For these lifetimes,
displaced tracks from the decay of the tau (pions and light leptons) can be
reconstructed. Additional samples are generated with 10 ns and infinite
lifetimes. For these lifetimes, the stau is sufficiently long-lived that it
can be directly identified as an anomalous prompt track with slow velocity or
high ionization.

####  2.4.1 Displaced leptons

For lifetimes below $1$ ns, long-lived staus are most easily identified by
their displaced decay products. In the GMSB model considered, where the light
gravitino does not interact with the detector, the only visible signatures of
the stau decays are the displaced taus. In the case of pair produced staus,
two taus are present in the final state, each the only visible product of its
own displaced vertex. The taus then decay as usual, leading to a mix of final
state scenarios ranging from fully leptonic when both taus decay leptonically
to fully hadronic.

While relatively rare (12% of di-tau events), the fully leptonic scenario
produces the distinctive signature of a pair of displaced, high-quality
leptons, and is the easiest signature to target. The fully leptonic decay mode
is the only scenario explored at the LHC thus far, with Run 2 results from
both ATLAS ATLAS:2020wjh (42) and CMS CMS:2021kdm (43). Triggering on leptonic
tau decays is also the most straightforward. Standard photon triggers can be
used to recover displaced electrons. Triggers which only require a track
reconstructed in the muon system, and are agnostic to tracker activity or
impact parameter requirements, are highly efficient for displaced muons up to
$|d_{0}|\lesssim 10$ cm ATLAS:2020xyo (47). In Run 2, displaced electrons and
muons were both required to pass a relatively high $p_{\mathrm{T}}$ threshold,
upwards of $40$ GeV. Because each lepton carries roughly 1/6 of the parent
stau’s energy, these thresholds are prohibitively high for the identification
of low mass staus.

Hadronic decays of displaced taus are harder to identify. The complex
algorithms that are used to identify prompt taus must be adapted to recover
efficiency for taus with larger impact parameters. Approximately $15\%$ of
taus decay to three charged particles. In this unique case, a visible
displaced vertex can be reconstructed. The remaining $50\%$ of taus decay
hadronically to a single charged particle. In both cases, charged particles
may be accompanied by one or more neutral hadrons, and are always accompanied
by a single tau neutrino. The fraction of the tau’s momentum carried by the
charged particles depends on the decay mode.

Figure 11:  (Left) Displaced charged particle $p_{\mathrm{T}}$. Particles are
required to be decay products of the long-lived stau. (Right) Displaced
charged particle |$d_{0}$| for varied lifetimes.

In all scenarios, at least two displaced tracks are produced per signal event.
Figure 11 shows the $p_{\mathrm{T}}$ and |$d_{0}$| of displaced charged
particles produced in the stau decay. For staus with masses above $100$ GeV,
displaced tracks typically have $p_{\mathrm{T}}$ values well above the minimum
thresholds for potential track-triggers. Displaced tracks tend to have larger
impact parameters than in the Higgs portal model as a result of the larger
parent mass.

This section follows a similar strategy to the Higgs portal model, as
described in Section 2.3. The baseline track acceptance is described in Table
2. Events are considered to pass acceptance if they contain at least $2$
tracks passing all acceptance requirements. The resulting event-level
acceptance, for a range of masses and lifetimes, can be seen in Figure 12.

Figure 12:  Acceptance for displaced lepton signature requiring at least two
tracks and passing the requirements listed in Table 2.

Efficiency requirements for individual tracks are also described in Table 2.
The event-level efficiency requires events have at least two tracks passing
the per-track efficiency, with respect to events passing acceptance. A range
of minimum $p_{\mathrm{T}}$ thresholds are explored, as shown in Figure 13. In
this scenario $p_{\mathrm{T}}$ thresholds have a minimal impact on overall
efficiency, especially for higher stau masses. For the CMS track-trigger
baseline $p_{\mathrm{T}}$ threshold of 2 GeV, there is no significant
reduction of efficiency compared to lower thresholds.

Figure 13:  Efficiency as a function of stau mass for a range of minimum track
$p_{\mathrm{T}}$ values. Events are defined to have passed if they contain two
tracks with a $p_{\mathrm{T}}$ value larger than the minimum value, as well as
a |$d_{0}$| larger than 1 mm and smaller than 100 mm. The proper lifetime is
$\tau=0.1$ ns. Track reconstruction efficiency decreases linearly from $100\%$
for prompt tracks to $0\%$ at $|d_{0}|=100$ mm.

The event-level efficiency for this model is much more sensitive to changes in
the track-trigger efficiency as a function of |$d_{0}$|. In this study, two
parameterizations of efficiency are compared. The first parameterization is a
binary model in which all tracks with |$d_{0}$| less than a given value are
assumed to be reconstructed. The second scenario follows the more realistic
model described in Section 2.1. The efficiency for prompt tracks is assumed to
be $100\%$, decreasing linearly to $0\%$ at the $|d_{0}|$ endpoint. The two
scenarios are compared in Figure 14, which shows event-level efficiencies for
a range of stau lifetimes. Though the binary model results in a higher
efficiency, by definition, the difference is modest ($<30\%$) for most
lifetimes.

Figure 14:  Event-level efficiency as a function of stau lifetime for a
$m_{\tilde{\tau}}=100$ GeV stau. (Left) Assumes $100\%$ efficiency up until a
$|d_{0}|$ endpoint. (Right) Efficiency decreases linearly from $100\%$ for
prompt tracks to $0\%$ at the $|d_{0}|$ endpoint. In both cases, a minimum
track $p_{\mathrm{T}}$ of $2$ GeV and a minimum track |$d_{0}$| of $1$ mm are
required.

Figure 15 shows the event-level efficiency as a function of stau mass and
lifetime for a variety of |$d_{0}$| endpoints, assuming a linearly decreasing
efficiency as a function of |$d_{0}$|. For endpoints below $2$ cm,
efficiencies are small ($\leq 30\%$) across the plane, but for endpoints at
$10$ cm, closer to typical offline efficiencies, efficiencies are high
($30-80\%$) for the full range of lifetimes. Given the low multiplicity of
tracks expected per event with impact parameters roughly consistent with the
stau decay radius, this sensitivity to |$d_{0}$| endpoint is expected.

Figure 15:  Event-level efficiencies for the stau model for a variety of track
$|d_{0}|$ endpoints: $10$ mm (top left), $20$ mm (top right), $50$ mm (bottom
left), $100$ mm (bottom right). In all cases, a linearly parameterized
|$d_{0}$| efficiency is used. Tracks are required to have $p_{\mathrm{T}}>2$
GeV and events are required to have $n_{\mathrm{Track}}\geq 2$.

####  2.4.2 Direct detection

For long-lived staus with proper lifetimes of $\tau\gtrsim 1$ ns, the charged
stau will traverse a sufficient number of tracker layers such that its
trajectory can be directly reconstructed as a prompt, high-$p_{\mathrm{T}}$,
isolated track. These staus will be slowly moving and highly ionizing with
respect to Standard Model charged particles produced at the LHC. Most SM
charged particles have mass $m<1$ GeV, $\beta\sim 1$, and can be approximated
as minimum ionizing particles. In contrast, staus with masses $m\geq 100$ GeV
will most often be produced with $\beta\gamma\sim 1$ and be slowly moving,
$\beta<1$.

This study considers long-lived staus with masses between $100\leq
m_{\tilde{\tau}}\leq 1000$ GeV and proper lifetimes between
$0.01\leq\tau_{\tilde{\tau}}\leq 10$ ns. Two possible trigger strategies are
studied. The first scenario looks for at least one high-$p_{\mathrm{T}}$
isolated track, and investigates the impact of geometric acceptance on trigger
efficiency. This scenario is meant to be as inclusive as possible to direct
long-lived particle detection, and can serve as a baseline for triggers which
additionally incorporate delay or anomalous ionization measurements to reduce
backgrounds. The second scenario assumes the outermost layer of the tracker
also serves as a time of flight layer, and incorporates a timing measurement
to further distinguish staus from SM tracks.

The selections considered for the inclusive trigger scenario are summarized in
Table 3, and require at least one stau to decay within detector acceptance.
The first geometric effect considered is the pseudorapidity range of the
tracking detector or track-trigger. Typically the tracker is designed to
emphasize efficiency and performance in the barrel. In comparison the endcap
and further forward regions have degraded performance and higher background
rates. The maximum pseudorapidity is varied from $|\eta|<1.0,~{}2.5,~{}4.0$ to
represent a barrel only scenario (if endcap rates are prohibitively high), a
nominal scenario, and an extended scenario, respectively.

Another geometric constraint is that the stau must traverse a sufficient
number of tracker layers to be reconstructed. To investigate this effect, the
minimum $L_{\mathrm{xy}}$ beyond which the stau must decay is varied from
$600<L_{\mathrm{xy}}<1200$ mm. The minimum $L_{\mathrm{xy}}$ corresponds to
particles which would traverse the pixel detector and at least three
$p_{\mathrm{T}}$-layers of the upgraded CMS tracker, or at least seven silicon
layers in the upgraded ATLAS tracker. The maximum $L_{\mathrm{xy}}$ roughly
corresponds to the full tracker for both experiments. In all cases the minimum
$|z|$ of the endcap is kept fixed at 3000 mm.

Signatures of disappearing tracks with shorter $L_{\mathrm{xy}}$ could be
considered, and have been used for offline discrimination of signal processes
in both ATLAS and CMS CMS:2020atg (40, 41). However, this method is prone to
high fake rates due to the reduced number of track hits, and is therefore
outside of the scope of this study.

At the HL-LHC, the inclusive trigger studied here would likely include a
track-based isolation requirement in order to reduce backgrounds. This study
assumes sufficient tracker longitudinal impact parameter resolution such that
any track-based stau isolation requirement would be fully efficient.

Variable | Requirement
---|---
Acceptance
$|\eta|$ | $<1.0,2.5,4.0$
$L_{\mathrm{xy}}$ |  $>600,800,1000,1200$ mm
OR $|z|$ |  $>3000$ mm
Inclusive Efficiency
$p_{\mathrm{T}}$ |  $>10,20,50,100$ GeV
Time-of-flight Efficiency
delay |  $>0.25,0.33,0.50$ ns
OR $\beta_{\mathrm{TOF}}$ | $<0.96,0.95,0.9$
OR $m_{\mathrm{TOF}}$ |  $>15,30,60$ GeV
Table 3:  Track acceptance and efficiency requirements considered for the
inclusive high-$p_{\mathrm{T}}$ charged particle selection and for the time-
of-flight selection. Events are required to have at least one stau passing
these selections.

Figure 16 shows event-level acceptance as a function of varying the track-
trigger maximum $|\eta|$ for a fixed $L_{\mathrm{xy}}=1200$ mm requirement.
Varying the track-trigger pseudorapidity has a larger effect on the event-
level acceptance for lower mass staus. Extending the track-trigger to
$|\eta|<4.0$ would improve the acceptance for $m_{\tilde{\tau}}=100$ GeV by
$30\%$ but provides negligible benefit for $m_{\tilde{\tau}}=1$ TeV. Extending
the track-trigger acceptance to the far forward region would be extremely
challenging from a technical perspective, and offer little benefit in terms of
physics reach for metastable charged particles because lower mass BSM
particles also benefit from larger cross sections. In contrast, reducing the
pseudorapidity coverage from $|\eta|<2.5$ to the barrel only scenario,
$|\eta|<1.0$, would reduce the overall acceptance by approximately $50\%$ or
more, and provides strong motivation for including the endcaps in any track-
trigger.

Figure 16:  Stau model event-level acceptance for a variety of minimum tracker
$|\eta|$ values. Acceptance is shown as a function of stau mass for proper
lifetime $\tau=10$ ns. The stau is required decay beyond
$L_{\mathrm{xy}}=1200$ mm.

The impact of varying the minimum number of layers to reconstruct a track is
shown in Figure 17. This study assumes a fixed pseudorapidity requirement of
$|\eta|<2.5$. Reducing the minimum $L_{\mathrm{xy}}$ from $1200$ mm to $600$
mm, or requiring fewer layers per track, improves the overall acceptance for
stau lifetimes of $\tau_{\tilde{\tau}}=1$ ns by roughly a factor of four.
However, the event-level efficiency for this lifetime never exceeds that of a
displaced track trigger, as shown in Figure 14. Varying the number of layers
required per track does not significantly improve the acceptance for lifetimes
$\tau_{\tilde{\tau}}>1$ ns, because most staus decay well beyond the tracker,
$L_{\mathrm{xy}}\sim 1.2$ m.

(a)

(b)

Figure 17: Stau event-level acceptance for a variety of minimum tracker
$L_{\mathrm{xy}}$ values. Acceptance is shown as a function of stau lifetime
for staus with mass $m_{\tilde{\tau}}=100$ GeV (17(a)) and
$m_{\tilde{\tau}}=500$ GeV (17(b)). The track-trigger covers $|\eta|<2.5$.

Increasing the minimum $p_{\mathrm{T}}$ required per track is a useful handle
to reduce background rates while retaining nearly full signal efficiency. The
impact of varying the minimum track $p_{\mathrm{T}}$ threshold on the event-
level efficiency is shown in Figure 33 of Appendix A.3. All $p_{\mathrm{T}}$
thresholds considered are nearly fully efficient except for the lowest stau
mass considered.

The second track-trigger scenario investigates using a time-of-flight detector
as an additional handle to improve signal to background discrimination. A
timing layer similar to the planned CMS MIP timing detector is considered
Butler:2019rpu (48). The timing layer is assumed to be located at roughly
$L_{\mathrm{xy}}=1150$ mm, $z=3000$ mm, covering $|\eta|<2.5$, and provide a
$50$ picosecond resolution timestamp for all charged particles passing through
the detector. ATLAS also plans to incorporate precision timing at the HL-LHC,
but only at larger pseudorapidity CERN-LHCC-2020-007 (49). For ATLAS, it may
be more optimal to consider calorimeter or muon spectrometer timing
measurements to target heavy metastable charged particles.

The time-of-flight detector’s measurement for a given particle is computed
according to

$t_{\mathrm{hit}}=\frac{L(p_{\mathrm{T}},\eta)}{c\cdot p}\sqrt{p^{2}+m^{2}}$

An increase in path length, $L$, due to the magnetic field is negligible for
charged particles with $p_{\mathrm{T}}>10$ GeV in a $4$ Tesla solenoidal
magnetic field, and therefore neglected by this study. The detector resolution
is assumed to have a Gaussian uncertainty with a width of $\sigma=50$ ps. The
spread of collisions in $z$ and the spread in collision time are assumed to
follow a Gaussian distribution with widths $\sigma=50$ mm and $\sigma=200$ ps,
respectively ATLAS:2016nnj (50). This study assumes the hardware track-trigger
is able to measure the track’s longitudinal impact parameter, or origin in
$z$, but not the primary vertex time. It is likely that computing the primary
vertex time will only be possible in the HLT or offline, and that the
uncertainty in the L1’s time-of-flight measurement will be dominated by the
beamspot’s spread in time. For simplicity, this study focuses on stable
lifetimes, and the selections considered are summarized in Table 3.

Time-of-flight and delay distributions are shown in Figure 18 for different
stau masses. For comparison, a “background” sample is simulated by using SM
particles with $p_{\mathrm{T}}>10$ GeV in the signal samples as a proxy. The
time-of-flight measurements contain two peaks, at $t_{\mathrm{hit}}\sim 3$ and
$t_{\mathrm{hit}}\sim 10$ ns, corresponding to the difference in path length
for tracks which traverse the barrel or the endcaps, respectively. The delay
is defined to be the time-of-flight measurement subtracted by the time it
would take a particle traveling at the speed of light to reach the same region
of the detector. Staus arrive noticeably later than background particles, and
the measured delay in arrival increases with stau mass.

(a)

(b)

Figure 18: Timestamp as measured by the time-of-flight detector (18(a)) and
delay in time of arrival with respect to particles travelling at the speed of
light (18(b)). These distributions incorporate a realistic spread in collision
time and $z$ position, as well as the timing layer’s temporal resolution. SM
particles with $p_{\mathrm{T}}>10$ GeV from the signal samples are shown for
comparison.

With the time-of-flight measurement it is also possible to extract the stau
velocity, $\beta$, and mass as shown in Figure 19. Computing
$\beta_{\mathrm{TOF}}$ accounts for differences in path length, due to the
stau $\eta$ and the longitudinal position of the primary vertex $z$. As
expected, background tracks peak at $\beta_{\mathrm{TOF}}=1$, while the
measured velocity decreases with increasing mass.

Computing a charged particle’s mass requires a track $p_{\mathrm{T}}$
measurement, and the mass resolution can be described as

$(\Delta m_{\mathrm{TOF}})^{2}=m^{2}\Bigg{[}\Big{(}\frac{\Delta
p_{\mathrm{T}}}{p_{\mathrm{T}}}\Big{)}^{2}+\Big{(}\frac{1}{1-\beta^{2}}\Big{)}^{2}\Big{(}\frac{\sigma_{t_{\mathrm{hit}}}}{t_{\mathrm{hit}}}\Big{)}^{2}\Bigg{]}$

For simplicity, this study assumes a $p_{\mathrm{T}}$ resolution of $1\%$
which increases up to $10\%$ for particles with $p_{\mathrm{T}}$ of $1$ TeV.
This assumption is somewhat optimistic for the endcaps, but reasonable for the
barrel CERN-LHCC-2017-009 (16, 17). Measured masses peak at the actual stau
mass, while background is peaked near zero, with long tails due to resolution
effects.

(a)

(b)

Figure 19: Measured $\beta$ (19(a)) and mass (19(b)) for different stau
masses. These distributions incorporate a realistic spread in collision time
and $z$ position, as well as the timing layer’s temporal resolution. SM
particles with $p_{\mathrm{T}}>10$ GeV from the signal samples are shown for
comparison.

The event-level efficiency for staus to pass a variety of
$\beta_{\mathrm{TOF}}$ and $m_{\mathrm{TOF}}$ requirements are shown in Figure
34 of Appendix A.3. Efficiencies are shown for stable staus with respect to
having at least one track in geometric acceptance. Selections on
$\beta_{\mathrm{TOF}}<0.96,0.95,0.90$ or $m_{\mathrm{TOF}}>15,30,60$ GeV were
chosen to provide comparable background efficiencies of $10\%$, $5\%$, and
$1\%$ per track, respectively. In both cases, the event-level efficiency
increases with stau mass. Requirements on $\beta_{\mathrm{TOF}}$ are less
efficient than the measured mass.

##  3 Background rates and trigger feasibility

Future track-triggers must operate within strict latency constraints and
reject HL-LHC backgrounds to reduce data flow to a manageable rate. The
required latency depends on the overall design of the trigger system, and the
ability to meet latency requirements depends on the exact implementation of
the hardware track-trigger. Any latency related studies are therefore beyond
the scope of this publication, which intends to remain independent of
implementation details. However, it is possible to investigate whether HL-LHC
backgrounds would overwhelm the BSM signals given the the track-trigger
reconstruction parameters considered. While the previous section focused on
the efficiency of a potential track-trigger to reconstruct anomalous tracks in
BSM signal events, this section investigates signal efficiency and background
rates for specific trigger selections. The goal of this section is not to
identify an optimal selection, but to demonstrate trigger feasibility.

The vast majority of background events input to any track-trigger at the HL-
LHC will consist of low-$p_{\mathrm{T}}$ QCD processes with $\sim 200$
overlapping pile-up collisions. Cross sections for electroweak processes are
several orders of magnitude lower than QCD and can be safely neglected.

Displaced tracks additionally have backgrounds from long-lived SM particles,
material interactions inside the detector, cosmic ray muons, and tracks which
are reconstructed from an incorrect combination of hits. Material interactions
and cosmic ray backgrounds can be rejected with fairly simple selections
ATLAS:2020xyo (47). Fake tracks due to the incorrect combination of hits are
highly dependent on tracker geometry and the exact implementation of the
track-trigger. In CMS’s L1 track finder, fake rates are expected to be around
5% for prompt tracks after applying quality cuts Bartz:2019dkp (51). Prompt
signatures are therefore dominated by SM backgrounds and fake contributions
can be safely ignored.

Due to reduced SM backgrounds, fake tracks are a more significant background
for displaced track reconstruction. To minimize this effect, this study
considers only displaced tracks with relatively small production radii and
long path-lengths, such that a sufficient number of hits and quality of fit
requirements could be applied per track to keep combinatoric backgrounds low.
While these quality requirements are not explicitly addressed in this study,
the shape of the efficiency vs. |$d_{0}$| parameterization is taken from
existing algorithms that employ these cuts to keep fake rates low. Additional
requirements on vertexing or more complex topological requirements could be
added to further reduce the contribution of fake tracks for a given trigger
selection.

In order to estimate SM background rates, a minimum bias sample of 100,000
events is generated using Pythia8. Half of the sample consists of events with
$p_{\mathrm{T}}<20$ GeV, which are produced via Pythia’s non-diffractive soft
QCD class. The other half of events are produced with Pythia’s hard QCD class,
and required to have $p_{\mathrm{T}}>20$ GeV. Because higher $p_{\mathrm{T}}$
events are more likely to pass a given trigger, the rate of hard QCD events
produced is biased by a ${p_{\mathrm{T}}}^{4}$ selection factor. Event weights
are applied to correct for this bias.

Tracks studied in this section are required to pass the criteria outlined in
Table 4. A threshold of $p_{\mathrm{T}}>0.5$ GeV was found to result in an
overwhelming rate of QCD background tracks. Therefore thresholds of
$p_{\mathrm{T}}>1$ GeV or higher are considered. Requirements are designed to
be consistent with previous studies in this paper. Particles are required to
be produced within $300$ mm in $L_{\mathrm{xy}}$ and $|z|$, and decay outside
the tracker barrel $L_{\mathrm{xy}}>1$ m or endcap $|z|>3$ m. In this section,
the linearly falling efficiency in $|d_{0}|$ with a 100 mm endpoint presented
in previous sections is used, which is pessimistic with respect to offline
displaced tracking efficiency shown in Ref ATL-PHYS-PUB-2017-014 (23). The
resulting track $p_{\mathrm{T}}$ and |$d_{0}$| for the minimum bias sample are
compared to representative signal samples in Figure 20. As expected, QCD
tracks tend to be much lower in $p_{\mathrm{T}}$ than signal tracks,
especially stau samples, and QCD |$d_{0}$| drops off rapidly compared to long-
lived signals.

Variable | Requirement
---|---
$p_{\mathrm{T}}$ | > 1.0, 2.0, 10
$|\eta|$ | $<2.5$
production vertex |  $L_{\mathrm{xy}}<300$ mm AND $|z|<300$ mm
decay vertex |  $L_{\mathrm{xy}}>1$ m OR $|z|>3$ m
Table 4:  Track criteria for minimum bias background studies.

(a)

(b)

Figure 20: Track $p_{\mathrm{T}}$ (20(a)) and |$d_{0}$| (20(b)) for QCD and
representative signal samples.

HL-LHC pile-up conditions are mimicked by overlaying minimum bias collisions
on each generated event. For each event of interest, 199 pile-up collisions
are randomly chosen from the minimum bias sample with a probability consistent
with the event weight. The $z$ position of each primary and pile-up collision
is sampled from a Gaussian distribution with a width $\sigma_{z}=50$ mm in
order to account for the longitudinal spread of the beamspot ATLAS:2016nnj
(50). The production and decay vertex positions of each truth particle in the
collision are adjusted accordingly.

The total number of tracks passing the criteria described in Table 4 with
$p_{\mathrm{T}}>1$ GeV are shown in Figure 21. Tracks which additionally pass
a displacement requirement of $|d_{0}|>1$ mm are shown for comparison. On
average $\sim 1400$ tracks are expected per event, and $50$ of those tracks
are displaced.

(a)

(b)

Figure 21:  The total number of tracks per event(21(a)) and total number of
displaced tracks per event (21(b)) for minimum bias QCD and representative
signal samples at pile-up=200. Slashed bands on the QCD distribution represent
the statistical uncertainty.

For the purpose of this study, an acceptable background rate for an individual
trigger is assumed to be at most a few percent of the total L1 output. When
correcting for empty bunches, the LHC collides protons bunches at a rate of
$\sim 30~{}\mathrm{MHz}$. As currently planned, ATLAS and CMS L1 triggers will
have output rates of $1~{}\mathrm{MHz}$ and $750~{}\mathrm{kHz}$, respectively
Collaboration:2285584 (18, 52). To be considered acceptable for this study,
background rates must be below $10~{}\mathrm{kHz}$, corresponding to QCD
selection factors smaller than $\sim 1/3000$.

Figure 21 demonstrates that a primary vertex selection is essential to
separate prompt signals from QCD in high pile-up conditions. The track-trigger
is assumed to have a longitudinal vertex resolution of 1 mm, and merge truth-
level pile-up vertices with $|\Delta z|<1$ mm. Once the primary vertex
associated with the signal process is identified, any tracks with $|\Delta
z|<1$ mm and $|d_{0}|<1$ mm are defined to pass the prompt selection.

The primary vertex is typically identified as the vertex with the largest sum
of track ${p_{\mathrm{T}}}^{2}$, in order to prioritize higher
$p_{\mathrm{T}}$ tracks. This definition is highly efficient for the HSCP
scenario, and correctly selects the primary vertex $95-100\%$ of the time, for
lower and higher mass staus respectively. However, for signals with lower
$p_{\mathrm{T}}$ tracks, such as the SUEP scenario, this criteria selects the
correct primary vertex $30-70\%$ of the time, for the lowest and highest
mediator masses considered. For SUEPs, the vertex with largest number of
tracks is taken to be the primary vertex, which is the correct choice for
$70-100\%$ of events.

Figure 22 shows the number of prompt tracks for the minimum bias and SUEP
samples. It is possible to reject QCD to acceptable levels by requiring
$n_{\mathrm{Track}}>150$ or $n_{\mathrm{Track}}>50$ for $p_{\mathrm{T}}>1$ and
2 GeV tracks, respectively. Resulting efficiency would be between $1-10\%$
depending on the mediator mass.

(a)

(b)

Figure 22: The number of prompt tracks with $p_{\mathrm{T}}>1$ GeV (22(a)) and
$p_{\mathrm{T}}>2$ GeV (22(b)) for minimum bias and representative SUEP signal
samples at pile-up 200. Slashed bands on the QCD distribution represent the
statistical uncertainty.

Alternative handles to retain SUEP signal efficiency include the number of
track jets per event and event $H_{\mathrm{T}}$, which are shown in Figure 23.
To compute these observables, Anti-$k_{\mathrm{T}}$ $R=0.4$ jets are
reconstructed from prompt tracks with $p_{\mathrm{T}}>1$ GeV, and jets are
required to have $p_{\mathrm{T}}>25$ GeV. No requirements on
$n_{\mathrm{Track}}$ are applied. A requirement of $n_{\mathrm{jet}}>4$ or
$H_{\mathrm{T}}>300$ would enable SUEP efficiencies of a few percent. More
complex event-shape variables such as isotropy or sphericity could be
considered for additional discrimination Cesarotti_2021 (30, 31).

(a)

(b)

Figure 23: The number of prompt jets (23(a)) and $H_{\mathrm{T}}$ (23(b)) for
minimum bias QCD and representative SUEP signal samples at 200 pile-up. Jets
are reconstructed from prompt tracks with $p_{\mathrm{T}}>1$ GeV, and jets are
required to have $p_{\mathrm{T}}>25$ GeV. Slashed bands on the QCD
distribution represent the statistical uncertainty.

For the HSCP scenario counting the number of high-$p_{\mathrm{T}}$, isolated,
or slowly moving tracks per event is sufficient to enable a feasible trigger.
To compute isolation, prompt tracks within $\Delta R<0.4$ of the HSCP
candidate are considered. Tracks are considered isolated if the
$p_{\mathrm{T}}$-sum of other tracks in that cone divided by the HSCP
candidate $p_{\mathrm{T}}$ is less than $0.05$. The time of flight,
$\beta_{\mathrm{TOF}}$, and $m_{\mathrm{TOF}}$, are computed using the same
approach as described in Section 2.4.2. Figure 24 shows efficiencies of above
$80\%$ can easily be achieved.

(a)

(b)

Figure 24: The number of $p_{\mathrm{T}}>50$ GeV isolated tracks (24(a)) and
$m_{\mathrm{TOF}}>25$ GeV, isolated $p_{\mathrm{T}}>25$ GeV tracks (24(b)) per
event for minimum bias QCD and representative HSCP signal samples with pile-up
200. Slashed bands on the QCD distribution represent the statistical
uncertainty.

For signals with displaced tracks, a primary vertex requirement cannot be
applied, and high-|$d_{0}$| tracks from all pile-up vertices must be
considered. For higher mass scenarios, such as long-lived staus, a low rate,
efficient trigger can be defined by increasing the $p_{\mathrm{T}}$ threshold
to $10$ GeV and requiring $n_{\mathrm{Track}}\geq 2$, as shown in Figure 25.
For the lower mass Higgs portal scenario, this strategy would result in an
efficiency less than one percent, so a different approach is needed.

(a)

(b)

Figure 25: The number of displaced tracks with $p_{\mathrm{T}}>2$ GeV (25(a))
and $p_{\mathrm{T}}>10$ GeV (25(b)) per event for minimum bias QCD and
representative long-lived signals with pile-up 200. Slashed bands on the QCD
distribution represent the statistical uncertainty.

The Higgs portal signal is the most challenging scenario considered, but there
are several possible strategies to reduce background rates. In Run 2, CMS
operated with a dedicated trigger that selected events with
$H_{\mathrm{T}}>400$ GeV and trackless or displaced jets CMS:2020iwv (53).
ATLAS uses a similar concept for their displaced decay triggers based on
calorimeter and muon spectrometer information ATLAS:2013bsk (54). For the HL-
LHC, there are several strategies to identify displaced jets or displaced
vertices PhysRevD.96.035027 (55, 56).

For this study, a crude displaced vertex reconstruction algorithm is
considered. First, all possible pairs of two-track vertices are formed from
tracks with $p_{\mathrm{T}}>1$ GeV and $|d_{0}|>1$ mm. The distance of closest
approach between the two tracks is required to be $<1.5$ mm. Next, any two-
track vertices $<1$ mm apart are merged. Remaining tracks with
$p_{\mathrm{T}}>1$ GeV and $|d_{0}|>0.5$ mm are added to the highest
$n_{\mathrm{Track}}$ vertex within $<1.5$ mm, where one exists. Finally, any
ambiguity (tracks associated to multiple vertices) is resolved by uniquely
assigning them to the possible vertex with the highest $n_{\mathrm{Track}}$.

Displaced vertices are required to be located within $L_{\mathrm{xy}}<300$ mm,
and $|z|<300$ mm. To reject real SM long-lived particles, vertices must have a
sum of track ${p_{\mathrm{T}}}^{2}>10$ GeV2, $m_{\mathrm{DV}}>3$ GeV, and
$L_{\mathrm{xy}}>4$ mm. Figure 26 shows requiring displaced vertices with at
least four tracks results in a feasible trigger with $\sim 10\%$ efficiency.
For comparison, triggering on a prompt lepton from associated Higgs production
would be less than one tenth as efficient.

(a)

(b)

Figure 26: The number of tracks per displaced vertex (26(a)) and the number of
vertices with at least four tracks (26(b)) per event for minimum bias QCD and
representative Higgs portal signals with pile-up 200. Slashed bands on the QCD
distribution represent the statistical uncertainty.

An efficient trigger strategy that reduces SM backgrounds to a reasonable rate
is possible for all BSM scenarios and tracking parameters considered.
Incorporating realistic detector effects and a proper optimization is left for
future work.

##  4 Comparison of tracking parameters

The effect of varying primary tracking parameters is compared across BSM
models and signatures in the following sections. The minimum $p_{\mathrm{T}}$
threshold is relevant to some extent for all models, primarily affecting SUEPs
and the Higgs portal model. Maximum reconstructed |$d_{0}$| affects only
models with displaced tracks. The final factor considered is the detector
layout, as studied through the HSCP model.

###  4.1 Minimum transverse momentum

Event-level efficiencies as a function of the minimum track $p_{\mathrm{T}}$
threshold and BSM parent mass are shown for several different models in Figure
27. The $p_{\mathrm{T}}$ threshold has a negligible effect on the direct
detection of most stable charged particles, and is not considered here. For
this comparison, the Higgs portal trigger selection is loosened to consider an
event accepted if at least two, rather than five, tracks are successfully
reconstructed. This adjustment makes the two selections identical and the
results directly comparable.

(a)

(b)

(c)

Figure 27: Trigger efficiency as a function of BSM particle mass for the stau
(27(a)), Higgs portal (27(b)), and SUEPs (27(c)) models for a range of minimum
track $p_{\mathrm{T}}$ thresholds. The stau and Higgs portal samples both
correspond to a lifetime of $\tau=0.01$ ns in order to target displaced decay
products. A linearly decreasing efficiency is assumed up to a $|d_{0}|$
endpoint of $100$ mm, and both long-lived scenarios require
$n_{\mathrm{Track}}\geq 2$. In the SUEP model all tracks are prompt.

Displaced leptons, like heavy stable charged particles, are relatively robust
to the $p_{\mathrm{T}}$ threshold, though efficiencies would drop below $50\%$
for the lightest staus with a $p_{\mathrm{T}}$ $>10$ GeV requirement. The
$p_{\mathrm{T}}$ threshold has a larger effect on the Higgs portal scenario,
where the hadronic decay products are usually softer. Thresholds above 2 GeV
would reduce the efficiency of even this two-track trigger to below $20\%$ for
all Higgs portal scalar masses. The largest impact of the $p_{\mathrm{T}}$
threshold is on the SUEP signature, where all masses have negligible
efficiency for $p_{\mathrm{T}}\geq 2$ GeV. The lowest reconstruction threshold
considered, $p_{\mathrm{T}}\geq 0.5$ GeV, would result in a high rate of QCD
background. A practical minimum $p_{\mathrm{T}}$ for both models would be 1
GeV.

In most systems the reconstruction for prompt and displaced tracks is likely
to be controlled and parameterized separately. In these cases, it would be
advisable to set a lower $p_{\mathrm{T}}$ threshold for prompt track
reconstruction than for displaced tracks ATLAS:2021tfo (57). This would enable
triggering on prompt SUEP scenarios, where a low $p_{\mathrm{T}}$ requirement
is the most essential, without imposing the same low $p_{\mathrm{T}}$ on
displaced tracks. However, the $p_{\mathrm{T}}$ threshold for displaced tracks
would still need to be on the order of $2$ to $3$ GeV to achieve significant
efficiency for the Higgs portal model.

While the minimum track reconstruction $p_{\mathrm{T}}$ threshold should be
kept as low as possible within the balance of rate and other factors,
individual triggers tuned for particular BSM scenarios may benefit from using
a much higher minimum $p_{\mathrm{T}}$ requirement to suppress background. For
example, the efficiency for detection of HSCPs was found to be nearly
independent of minimum track $p_{\mathrm{T}}$ for cut values up to $100$ GeV.

###  4.2 Impact parameter dependence

Trigger efficiency dependence on the maximum reconstructed |$d_{0}$| is shown
as a function of long-lived particle lifetime for the stau and Higgs portal
models in Figure 28. As is the case in the $p_{\mathrm{T}}$ comparison, the
Higgs portal scenario requires $n_{\mathrm{Track}}\geq 2$, rather than five.
The requirements for an event to pass are therefore identical between these
two models. Two mass points for each scenario are chosen to illustrate the
full range of displaced track kinematics. For the stau model, one low- and one
high-mass point are chosen. For the Higgs portal, mass points are chosen such
that scalar decays are either primarily to $b$ quarks or to $c$ quarks and
taus.

(a)

(b)

(c)

(d)

Figure 28: Trigger efficiency as a function of lifetime at a range of maximum
track |$d_{0}$| values for staus with masses 100 GeV (28(c)) and 600 GeV
(28(b)) and a Higgs portal with scalar masses 8 GeV (27(b)) and 40 GeV
(28(d)). Only displaced tracks produced in the long-lived particle decay are
considered. Effects of the $p_{\mathrm{T}}$ threshold are minimized by
requiring $p_{\mathrm{T}}>0.5$ GeV, and both long-lived scenarios require
$n_{\mathrm{Track}}\geq 2$.

The trends in Figure 28 demonstrate that the largest change in efficiency as a
result of reduced $|d_{0}|$ endpoints occurs for intermediate lifetimes
($10^{-2}\lesssim\tau\lesssim 1$ ns) for both benchmark models. For proper
lifetimes of $\tau=0.1$ ns, Figure 29 shows the impact of varying the
|$d_{0}$| endpoint for a variety of stau and scalar masses. The difference in
efficiency for different endpoints is consistent for the range of long-lived
particle masses considered.

(a)

(b)

Figure 29: Event-level efficiency for different $|d_{0}|$ endpoints as a
function of long-lived particle mass. Both the stau (29(a)) and Higgs portal
model (29(b)) have lifetimes of $\tau=0.1$ ns. Only displaced tracks produced
in the long-lived particle decay are considered. Effects of the
$p_{\mathrm{T}}$ threshold are minimized by requiring $p_{\mathrm{T}}>0.5$
GeV, and both long-lived scenarios require $n_{\mathrm{Track}}\geq 2$.

Figure 29 also shows that the impact of |$d_{0}$| endpoints is significantly
more important for the stau model than for the Higgs portal model. This effect
is due to a difference in boost. The long-lived scalar’s decay products are
more collimated and often point back to the $pp$ collision. As a result,
displaced tracks in the Higgs portal model tend to have smaller |$d_{0}$| for
a given long-lived particle decay position, with respect to the stau model. It
is therefore possible to achieve comparable efficiencies between tight and
extended |$d_{0}$| endpoints in the Higgs portal model. In contrast, the less
boosted stau model suffers major efficiency losses as the |$d_{0}$| endpoint
is reduced from $10$ cm to $10$ mm.

###  4.3 Number of tracking detector layers

The minimum number of tracking detector layers required per track was studied
for the heavy metastable charged particle scenario by requiring the stau to
decay beyond a varying minimum $L_{\mathrm{xy}}$. Reducing the minimum
$L_{\mathrm{xy}}$ from $1.2$ to $0.6$ m was not found to have a significant
effect on the event-level acceptance. For staus with a lifetime of
$\tau_{\tilde{\tau}}=1$ ns, the difference between the most stringent and
least stringent requirements on the effective number of tracker layers led to
a factor four improvement in event-level acceptance. However, the acceptance
remained low in all cases. For even longer lifetimes, most staus decay well
beyond $L_{\mathrm{xy}}\sim 1$ m. Reducing the minimum number of layers
required per track resulted in minimal acceptance improvements.

The effects of the number of tracking layers will still be significant for
displaced signatures. The radius at which a long-lived particle decays will
directly affect how many tracking layers are available to reconstruct the
displaced track. For example, if the particle decays after the first tracking
layer, only $n-1$ hits will be associated to the reconstructed track. The
long-lived particle’s decay distance also correlates to $|d_{0}|$, with the
mean decay distance longer than the mean $|d_{0}|$. If a long-lived particle
decays beyond multiple tracker layers, there may not be enough hits to
reconstruct displaced tracks. The efficiency for reconstructing tracks beyond
the equivalent |$d_{0}$| would drop to zero.

As seen in the Section 4.2, access to high-|$d_{0}$| tracks significantly
increases the efficiency for both stau and Higgs portal models. For staus,
reconstructing high-|$d_{0}$| tracks enables the displaced track trigger to
outperform the stable charged particle trigger at a lifetime of $1$ ns. For
the stable charged particle approach, staus with masses between $100$ GeV and
$500$ GeV at $1$ ns lifetimes all show an acceptance times efficiency below
$15\%$ regardless of the number of layers required per track. For the
displaced track approach, the acceptance times efficiency for a stau with a
$1$ ns lifetime increases from $\sim 0\%$ to $\sim 20\%$ if the |$d_{0}$|
endpoint is extended from $10$ mm to $100$ mm, assuming a linearly decreasing
efficiency (see Appendix A.2). Extending the efficiency to a larger |$d_{0}$|
endpoint relies on having a sufficient number of tracking layers at larger
radii.

##  5 Conclusions and Recommendations

Current ATLAS and CMS triggers are primarily designed to target prompt decays
of heavy particles to high-$p_{\mathrm{T}}$ Standard Model particles. However,
many compelling BSM scenarios lead to decays which are significantly displaced
from the interaction point or include low $p_{\mathrm{T}}$ decay products. For
these unconventional scenarios, the anomalous charged particle trajectories
are the most distinctive feature of the event. This study explored the impact
of varying tracking parameter requirements on trigger efficiency for a
collection of representative models.

In general, for these unconventional signatures, perfect track reconstruction
efficiency is rarely achievable. However, even introducing a partial track
reconstruction efficiency in a $p_{\mathrm{T}}$ or $d_{0}$ range can make a
substantial difference in sensitivity, and high efficiencies across these
parameter spaces are not required. Triggering on $20\%$ of events is
infinitely better than triggering on none.

HSCP identification is relatively insensitive to choice of tracker parameters,
due to their distinctive, prompt, and high-$p_{\mathrm{T}}$ tracks. If
available, a time of flight estimate is the most discriminating strategy. For
HSCPs, the most impactful variation studied was $\eta$ coverage, with typical
efficiencies from a barrel-only scenario at $\sim 30\%$, but those from an
extended endcap scenario as high as $\sim 90\%$.

To tackle highly diffuse models such as SUEPs, a track-trigger with a low
$p_{\mathrm{T}}$ threshold is required. For the SUEP models studied here, the
optimal threshold for prompt track reconstruction is $p_{\mathrm{T}}>1$ GeV.
Higher thresholds result in low signal efficiency while lower thresholds lead
to high QCD rates. For experiments where a low $p_{\mathrm{T}}$ threshold is
not feasible at L1, alternatives to track counting should be considered, and
could be complemented with lower track $p_{\mathrm{T}}$ thresholds at HLT. For
the SUEP scenario considered here, the overall efficiency is still likely to
be low, because high energy initial state radiation is required for any
alternate trigger strategy. A $p_{\mathrm{T}}$ threshold above 1 GeV will be
acceptable for SUEP models with higher temperatures or dark meson masses. In
all cases, a primary vertex requirement is essential to separate any SUEP
signal from QCD backgrounds at the HL-LHC. The vertex reconstruction method,
selection, and longitudinal resolution are intimately connected to a SUEP
trigger’s performance.

For the other models considered here, a tracking threshold of
$p_{\mathrm{T}}>2$ GeV is sufficient to provide useful sensitivity. To gain
sensitivity to the decay products of more massive particles, such as long-
lived staus, a large |$d_{0}$| range is more essential for overall trigger
efficiency than a low $p_{\mathrm{T}}$ threshold. Simply counting the number
of high-$p_{\mathrm{T}}$ displaced tracks is sufficient to reject QCD
backgrounds. To address models with light mediators, lower mass long-lived
particles, and hadronic decays, the most useful design strategy is a track-
trigger that pushes the $p_{\mathrm{T}}$ threshold as low as possible with
modest |$d_{0}$| coverage. In the case of the Higgs portal model, substantial
improvement could be gained from reduced $p_{\mathrm{T}}$ thresholds. QCD
backgrounds pose an additional challenge for these lower $p_{\mathrm{T}}$
scenarios, and simple track counting is not sufficient for separating signal
from background. Displaced vertex reconstruction or displaced jet tagging will
be essential to improve sensitivity to Higgs portal scenarios.

To address all of these cases simultaneously while keeping resource usage to a
minimum, the best design scenario is one that allows for increasingly large
|$d_{0}$| coverage as particle $p_{\mathrm{T}}$ increases. In track-trigger
designs using pattern banks, this strategy is relatively easy to employ
ATLAS:2021tfo (57). An approximately constant number of patterns is required
to cover a given area in the 2 dimensional space defined by maximum |$d_{0}$|
and $1/p_{\mathrm{T}}$ values. An optimal pattern bank could be defined by
sampling from a triangle rather than a rectangle in that 2D space. A more
generic strategy is to create a displaced tracking algorithm that is applied
only to tracks passing a minimum $p_{\mathrm{T}}$ threshold that is higher
than the nominal prompt one. For example, a track-trigger that reconstructs
prompt tracks with $p_{\mathrm{T}}>$ 1 GeV and displaced tracks with
$p_{\mathrm{T}}>2$ GeV would provide new sensitivity to all models considered
here. If the algorithm allows, the $p_{\mathrm{T}}$ threshold could be
increased along with the maximum |$d_{0}$|. This strategy is depicted in
Figure 30.

Figure 30: Recommended strategy for optimizing sensitivity to a large range of
unconventional signals while minimizing resource usage. Teal represents prompt
tracking, while orange represents displaced tracking. HSCP sensitivity is not
depicted, but relies on very accessible high-$p_{\mathrm{T}}$ prompt tracks.

Track-triggers at the High Luminosity LHC will provide a unique opportunity to
directly trigger on these challenging exotic scenarios for the first time.
Acting on the recommendations provided here will enable track-trigger designs
for the HL-LHC to strike a balance that can best serve a full range of
unconventional signatures. The same principles will apply to tracker and
trigger designs for future detectors.

###### Acknowledgements.

Tova Holmes and Jesse Farr’s research is supported by U.S. Department of
Energy, Office of Science, Office of Basic Energy Sciences Energy Frontier
Research Centers program under Award Number DE-SC0020267. Karri Folan
DiPetrillo and Chris Guo’s work utilizes resources of the Fermi National
Accelerator Laboratory (Fermilab), a U.S. Department of Energy, Office of
Science, HEP User Facility. Fermilab is managed by Fermi Research Alliance,
LLC (FRA), acting under Contract No. DE-AC02-07CH11359. DiPetrillo’s research
is also supported in part by the Neubauer Family Foundation Program for
Assistant Professors. Jess Nelson’s research is supported by the Duke Research
Experiences for Undergraduates program in particle physics, under National
Science Foundation Award Number 1757783. Katherine Pachal’s research is
supported by TRIUMF, which receives federal funding via a contribution
agreement with the National Research Council (NRC) of Canada.

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##  Appendix A Additional Distributions

###  A.1 Higgs portal

Figure 31:  Event-level acceptance times efficiency for Higgs portal model
with a constant |$d_{0}$| endpoint and $p_{\mathrm{T}}$ thresholds ranging
(clockwise from top left to bottom left) from 0.5 Gev, 1.0 GeV, 2.0 Gev, and
5.0 GeV. In all cases, the two-track selection and a linearly decreasing
efficiency is used.

###  A.2 Displaced Leptons

Figure 32:  Event-level acceptance times efficiency for a displaced track
trigger targeting the stau scenario. The constant $p_{\mathrm{T}}$ threshold
is kept constant and |$d_{0}$| endpoint is varied from (clockwise from top
left to bottom left) from 10 mm, 20 mm, 50 mm, and 100 mm. In all cases, the
two-track selection and a linearly decreasing efficiency is used.

###  A.3 HSCP

Figure 33:  Stau efficiency for a variety of minimum tracker $p_{\mathrm{T}}$
values. Efficiency is shown as a function of stau mass for a stable lifetime
and minimum $L_{\mathrm{xy}}=1200$ mm.

(a)

(b)

Figure 34: Efficiency with respect to various requirements on measured track
$\beta$ (34(a)) and measured mass (34(b)).