#  Triggering on Emerging Jets

Dylan Linthorne and Daniel Stolarski Ottawa-Carleton Institute for Physics,
Carleton University,
1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>

###### Abstract

Confining dark sectors at the GeV scale can lead to novel collider signatures
including those termed emerging jets with large numbers of displaced vertices.
The triggers at the LHC experiments were not designed with this type of new
physics in mind, and triggering can be challenging, especially if the mediator
is relatively light and/or has quantum numbers such that additional jets are
not automatically produced in each event. We show that the efficiency and the
total event rate at current triggers can be significantly improved by
considering initial state radiation of the events, with the largest increase
in rate coming from simulation of two additional jets. We also explore
possible new triggers that employ hit counts in different tracker layers as
input into a machine learning algorithm. We show that these new triggers can
have reasonably low background rates, and that they are sensitive to a wide
range of new physics parameters even when trained on a single model.

##  I Introduction

Confining hidden sectors with a confinement scale around the GeV scale have
received significant attention for potential discovery at colliders Schwaller
et al. (2015); Cohen et al. (2015); Csaki et al. (2015); Cohen et al. (2017);
Knapen et al. (2017); Pierce et al. (2018); Burdman and Lichtenstein (2018);
Cheng et al. (2019) (for a recent review, see chapter 7 of Alimena et al.
(2020a)), building on the seminal hidden valley work Strassler and Zurek
(2007, 2008); Han et al. (2008). Besides providing interesting signatures at
colliders, they can also be motivated by asymmetric dark matter Bai and
Schwaller (2014); Lonsdale et al. (2017); Lonsdale and Volkas (2018) and by
neutral naturalness Craig et al. (2015); Curtin and Verhaaren (2015); Cheng et
al. (2017); Kilic et al. (2019). This has led to several phenomenological
studies at the LHC Park and Zhang (2019); Cohen et al. (2020); Mies et al.
(2020); Knapen et al. (2021), flavour experiments Renner and Schwaller (2018),
and future proposed experiments Alekhin et al. (2016); Curtin et al. (2019).
There is also an experimental search Sirunyan et al. (2019) that puts direct
limits on certain regions of parameter space.

If the dark confining sector has a mediator to the SM whose mass is much
larger than the confining scale, then the lifetime of the lightest dark
hadrons that are not stable will be parametrically larger than their inverse
mass. One particularly interesting region of parameter space is where the
lifetime of the decaying dark hadrons is in the mm–m range, leading to
particularly spectacular signatures at the LHC’s detectors Schwaller et al.
(2015). For example, if the dark gauge group is QCD-like, then when dark
quarks are produced, they will shower and hadronize, producing dark jets. This
is analogous to the production of ordinary quarks at a high energy lepton
collider which then produce SM jets. Each of the dark hadrons will decay at a
different position in the detector, and the energy of the dark jet will
“emerge” into the detector, and this signature was thus termed an emerging jet
Schwaller et al. (2015). At distances long compared with the typical $c\tau$
of the dark hadrons, the dark jet will look like a QCD-jet.

Motivated by the asymmetric dark matter scenario of Bai and Schwaller (2014),
the work (Schwaller et al., 2015) considered a scalar mediator $X$ that is
charged under QCD and dark-QCD. That means that the dominant collider signal
will be pair production of the mediator, and each collider event will contain
two emerging jets and two SM jets. This is also the scenario experimentally
constrained in Sirunyan et al. (2019). Another well motivated possibility is
the one considered in the original hidden valley literature Strassler and
Zurek (2007, 2008); Han et al. (2008): a vector mediator $Z^{\prime}$ that
couples to both quark and dark quark currents. A third possibility, the one
expected in the neutral naturalness scenarios Craig et al. (2015); Curtin and
Verhaaren (2015); Cheng et al. (2017); Kilic et al. (2019), is the SM Higgs or
another scalar as the mediator ($H$) to the dark sector. In both of the latter
cases, we see two important differences between the original case of the $X$:

  * •

The production cross section of dark hadrons is a free parameter and not set
by pair production via SM QCD.

  * •

The production does not automatically include the associated production of SM
jets.

These cases are therefore more challenging experimentally.

A particularly important challenge with exotic signatures at the LHC is
triggering. The event rate at the LHC is extremely high, and a trigger is
employed to reduce the event rate by several orders of magnitude and attempt
to record all the events of interest. The triggers were designed to be
extremely efficient on many types of events and new physics models, but they
are not designed for more exotic scenarios such as emerging jets. In the case
of the $X$ mediator, if its mass is $\mathcal{O}$(TeV), then the associated
jets in combination with the very high total energy in the event makes
triggering relatively straightforward.

In this work we study on the significantly more difficult case of triggering
on the $Z^{\prime}$ mediator, focusing on the relatively lower mass regime,
$M_{Z^{\prime}}\sim$ 100 GeV – 1 TeV. We will take a two-pronged approach.
First, we will consider how well current triggers can capture these events,
and explore how the addition of initial state radiation can increase the
efficiency. The so-called mono-$X$ strategy Bai et al. (2010); Goodman et al.
(2010) has been used extensively to search for invisible states, with a broad
range of experimental searches including mono-jet Aad et al. (2021), mono-
photon Sirunyan et al. (2017a), mono-$W$ and mono-$Z$ Sirunyan et al. (2017b),
placing constraints on many different types of models. In this work we show
that both QCD and electroweak radiation can increase the trigger efficiency
and increase the total number of events recorded.

The second strategy we employ is to consider new triggers using modern machine
learning (ML) techniques. The landscape of machine learning applications
within particle physics is becoming ever broader and more complex. Its
utilities ranges from substructure classifications Larkoski et al. (2020),
such as jet discrimination Metodiev et al. (2017); Metodiev and Thaler (2018),
to multivariate analysis techniques explored at both CMS and ATLAS as the LHC
moves towards the energy and intensity frontiers. Guest et al. (2018); Radovic
et al. (2018). We look to investigate novel triggers based on simple machine
learning methods that can be applied to the triggering stream. Complementary
studies include Alimena et al. (2020b) where a deep neural-net (DNN) is
implemented at L1 resulting in high trigger efficiencies for a HL-LHC 15 KHz
target output, as well as the study of a more traditional trigger for
displaced vertices Gershtein et al. (2020). CMS (ATLAS) has been slowly
conducting analyses of this type on low level information reconstructed from
the triggering stream under the guise of Data scouting (Online trigger
analysis) (Mukherjee, 2020; Anderson, 2016; ATL, 2017).

One significant downside to the use of ML techniques in particle physics is
that the physical intuition of traditional analyses can be lost when using
complicated non-linear cuts dictated by ML algorithms. It is often difficult
to determine which aspects of events the algorithms are using to, for example,
distinguish signal from background, which in turn makes it difficult to
account for effects such as Monte Carlo mismodeling and detector
uncertainties, although there is some recent progress Faucett et al. (2020).
This is a significantly less important problem when considering triggers where
the most important task is to get interesting events recorded quickly. One can
then use a more careful offline analysis to gain physical insight and more
completely characterize things like systematic errors.

In this work we use hits in the tracker as input into a potential new trigger
discriminator. Tracking is traditionally not used in triggering because track
reconstruction is too computationally time consuming (Owen, 2018; Khachatryan
et al., 2017). We skirt this problem by not reconstructing tracks, but rather
by simply counting hits in a region of the detector. This can be effective in
distinguishing emerging jets because the dark sector particles will not leave
hits but their decay products will. Therefore, if the dark sector particles
lifetime $c\tau\sim$ mm – m, an emerging jet will have increasing numbers of
hits in detector layers further from the interaction point. QCD jets, on the
other hand, will typically have the same number of hits in most layers because
unstable SM hadrons will typically decay before hitting the first or second
layer of the detector, with the exception of a few strange mesons. We will
show that this type of observable fed into a machine learning algorithm can be
an effective trigger for a wide class of model parameters. We explore the
universality of such strategies and show that a trigger trained on one
parameter point can be sensitive to a broad swath of parameter space.

The remainder of this paper is structured as follows: in Section II we
describe the concrete model we use for our analysis, and in Section III we
describe our simulation pipeline. In Section IV we describe how the mono-X
strategy can be used to improve event collection with current triggers, and in
Section V we outline how new triggers can also be used to explore new
parameter space. Conclusions are given in Section VI.

##  II Models for emerging jets

The study of generic hidden sectors at the LHC is an interesting and important
question. For concreteness we will specify a class of models and leave the
more general case to future work. We consider an $SU(N_{d})$ gauge group with
confinement scale $\Lambda_{d}\simeq$ GeV and $n_{f}$ flavours of vectorlike
quarks with masses below confinement scale. The dark quarks are singlets under
all SM gauge groups. If there is an accidental baryon number symmetry
analogous to QCD, then the baryons of this sector could be dark matter Bai and
Schwaller (2014).

Unlike previous work which studied a scalar mediator Schwaller et al. (2015),
we consider a vector mediator, $Z^{\prime}_{\mu}$ as in some of the original
hidden valley literature (Strassler and Zurek, 2007), whose mass $M$ is larger
than the dark confinement scale, $M\gg\Lambda_{d}$. The UV theory and
mechanism to give mass to the $Z^{\prime}$ is left unspecified, but we assume
the additional states needed to not affect the phenomenology. This mediator
couples to the quark and dark quark currents:

$\mathcal{L}\supset\frac{1}{2}M^{2}Z^{\prime\mu}Z^{\prime}_{\mu}+Z^{\prime\mu}\big{(}g_{q}\,\overline{q}\gamma_{\mu}q+g_{d}\,\overline{Q}\gamma_{\mu}Q\big{)}\,,$
(1)

where $q$ are SM quarks, $Q$ are dark quarks, and $g_{q/{d}}$ are coupling
constants. The $Z^{\prime}$ is a singlet under SM and unbroken dark gauge
groups, so gauge indices among the quarks and dark quarks are contracted in
the standard way. For simplicity we have assumed flavour universality for both
quarks and dark quarks with a universal coupling for each, and flavour indices
are summed and not written.

This model contains a rich spectrum of dark hadrons, with a multiplet of dark
pions, $\pi_{d}$ expected to be the lightest. All heavier mesons have a
lifetime of order $\Lambda_{d}^{-1}$ and decay to dark pions if kinematically
allowed (i.e. $\rho\rightarrow\pi\pi$ in the SM). Dark baryons in these models
are often very long lived. In the parameter regions we consider, dark pions
are significantly lighter than dark baryons (as in QCD) and thus are typically
produced in much larger abundances than dark baryons. This can be confirmed
with SM data Zyla et al. (2020) as well as a large $N_{c}$ expansion Witten
(1979) of QCD. Therefore we ignore the effects of dark baryons in this study,
but of course these assumptions can be violated if the hadron spectrum is
significantly different from that of the SM.

For simplicity, we take a common mass of the dark pions, $m_{\pi_{d}}$, but
for a study of non-trivial dark flavour dynamics, see Renner and Schwaller
(2018). The $Z^{\prime}$ mediates a decay of the dark pions that can be
computed using dark chiral perturbation theory with a width given by

$\Gamma(\pi_{d}\rightarrow\bar{q}q)\approx\sum_{q}\frac{g_{q}^{2}\,g_{d}^{2}\,N_{c}\,m_{q}^{2}\,f^{2}_{\pi_{d}}}{32\,\pi\,M^{4}}\,m_{\pi_{d}}$
(2)

where $N_{c}=3$ is the number of SM colours, $f_{\pi_{d}}$ is the dark pion
decay constant which we take to be $\sim\Lambda_{d}$, and $m_{q}$ is the mass
of the SM quark in the final state. The sum over all SM quarks that are
kinematically accessible, and we have ignored phase space effects, but they
can be trivially added. The factor of $m_{q}^{2}$ is a spin-parity affect
analogous to the decay of the charged pion in the SM, implying that the decay
will be dominated by the heaviest quark kinematically accessible. We can
estimate the proper decay length for a benchmark pion mass of 2 GeV:

$\begin{split}c\tau_{0}\approx 80\,{\rm
mm}\times\frac{1}{g_{d}^{2}\,g_{q}^{2}}\times\left(\frac{2~{}{\rm
GeV}}{f_{\pi_{d}}}\right)^{2}\\\ \times\left(\frac{100~{}{\rm
MeV}}{m_{q}}\right)^{2}\left(\frac{2~{}{\rm
GeV}}{m_{\pi_{d}}}\right)\left(\frac{M_{Z^{\prime}}}{1~{}{\rm
TeV}}\right)^{4}\,.\end{split}$ (3)

which we see can be macroscopic but smaller than the size of an LHC detector
for a wide range of parameter space.

Dark quarks (and thus dark jets) are produced at colliders like the LHC via an
$s$-channel $Z^{\prime}$. If kinematically accessible, resonant production
where the $Z^{\prime}$ is on shell will dominate the production. The lowest
order cross section for this production process at a proton-proton collider is

$\sigma(pp\rightarrow Z^{\prime}\rightarrow Q\bar{Q})=\sum_{f=u,d}\int
dx_{1}f_{f}(x_{1})\int dx_{2}f_{\bar{f}}(x_{2})\\\
\times\frac{g_{d}^{2}g_{q}^{2}}{72\pi}\bigg{(}\frac{x_{1}x_{2}s}{(x_{1}x_{2}s-M_{z^{\prime}}^{2})^{2}+\Gamma^{2}M_{Z^{\prime}}^{2}}\bigg{)}.$
(4)

Where $f_{i}(x_{i})$ is the parton distribution function for fermion $i$ and
momentum fraction $x_{i}$. The Mandelstam variable $s$ is set to the centre of
mass energy $\sqrt{s}=13$ TeV. The total decay width of the $Z^{\prime}$ is
given by $\Gamma$, which has two contributions $Z^{\prime}\rightarrow
q_{i}\bar{q}_{i}$ and $Z^{\prime}\rightarrow Q_{i}\bar{Q}_{i}$,

$\Gamma(Z^{\prime}\rightarrow
X\bar{X})\simeq\frac{Nng^{2}M_{Z^{\prime}}}{24\pi}$ (5)

where for X being a SM (dark) quark, $N$ is the number of (dark) colours which
we take to be 3, $n$ is the number of accessible flavours, and $g$ is the
coupling to (dark) quarks. We have ignored kinematic factors which are only
important if the $Z^{\prime}$ is approximately degenerate with a pair of
(dark) quarks.

From these equations, assuming resonant production is dominant, we can show
that to leading order the cross section for the process $q\bar{q}\rightarrow
Z^{\prime}\rightarrow Q\bar{Q}$ depends only on the mass of the $Z^{\prime}$
and the variable

$\eta^{2}\equiv\frac{g_{d}^{2}\cdot
g_{q}^{2}}{g_{d}^{2}+\big{(}\frac{n_{f}}{n_{d}}\big{)}g_{q}^{2}},$ (6)

scaling with $\eta^{2}$. We have assumed the number of dark colours is also 3.
The production cross section at a centre of mass energy $\sqrt{s}=13$ TeV is
shown in Fig. 1 as a function of mass for a couple of benchmark values of
$\eta^{2}$. We see that number of such events at the LHC with an integrated
luminosity $\sim 100-3,000$ fb-1 can be very large.

Figure 1: Cross section of $pp\rightarrow Z^{\prime}\rightarrow Q\bar{Q}$ for
a leptophobic $Z^{\prime}$ as a function of its mass $M_{Z^{\prime}}$.
Generated by convolving the partonic cross sections of $u,d$-quarks with their
respective parton distribution functions at a centre of mass energy of
$\sqrt{s}=13$ TeV. $\eta$ is a function of the $Z^{\prime}$ Lagrangian
parameters defined in Eq. (6).

Alongside the LO contribution to the $Z^{\prime}$ production are higher order
terms from initial state radiation (ISR) coming off of the incoming quark
lines. In Fig. 2 we show this for QCD gluon radiation. These gluons will
hadronize into additional hard QCD jets affecting the overall event’s
topology. ISR is not exclusive to QCD, the quarks may radiate a hard $W$, $Z$
or photon. These ISR contributions have a smaller rate than the leading order
process, but they can be easier to detect experimentally.

Figure 2: Higher order QCD initial state radiation diagrams of $q\bar{q}$
induced $Z^{\prime}$ resonant production. Gluons are radiated from the quark
lines in the form of detectable hard QCD jets. EW splittings are also possible
with diagrams scaling with the EW couplings.

In the scalar mediator case explored in (Schwaller et al., 2015), collider and
direct detection experiments put on stringent constraints on hidden sector
spectrum (Mies et al., 2020). In our case, these $Z^{\prime}$ models are of
large interest for detector searches because of the freedom in the production
cross section of dark hadrons. This model, however, does have resonant dijet
production proportional to $g_{q}^{4}$ which can place constraints (Phy, 2017;
Aaboud et al., 2019a; Sirunyan et al., 2017c). Most of these studies at modern
collider experiments are only sensitive to the higher mass regime
$M_{Z^{\prime}}>1$ TeV because of the filtering of lower mass events by their
triggers. Some searches have employed data scouting techniques and trigger
level analysis to probe lower masses Sirunyan et al. (2017d); Phy (2018).
Typical upper bounds at the lower mass regime are $g_{d}<0.1$. We do demand
that the dark quarks produce emerging jets with lifetimes between [1mm -
1000m]. Doing so puts constraints on the product of couplings $g_{d}\cdot
g_{q}$ from Eq. (1). Assuming lower mediator masses, dark pion masses, and
decay constant at $\mathcal{O(\textrm{GeV})}$ emerging jet events are
consistent with $g_{d}\cdot g_{q}\lesssim 0.2$.

For the rest of this work, we take $c\tau$ to be a free parameter. The mass of
the $Z^{\prime}$ and $c\tau$ are related by Eq. (3), but there are enough
parameters in the models such that we can tune each variable independently.
Therefore we vary the mass of the $Z^{\prime}$ as it broadly controls the
total energy in the event. For the dark QCD parameters, we use six benchmark
parameter points shown in Table 1. The first two, Model A and Model B, have
different dark pion masses of 2 GeV and 5 GeV and 2 different lifetimes of 5
mm and 150 mm, respectively. These models are used in studying current
triggers in Section IV.111These were also the benchmarks used in Schwaller et
al. (2015). Models C, D and E, are identical to Model A but with lifetimes
ranging from 5mm to 500 mm. Model F has the same lifetime as Model D but a
heavier hadron spectrum. Models A, C, D, E and F are considered in the machine
learning trigger analysis of Section V.

Model | A | B | C | D | E | F
---|---|---|---|---|---|---
$\Lambda_{d}$ | 10 GeV | 4 GeV | 10 GeV | 10 GeV | 10 GeV | 20 GeV
$m_{V}$ | 20 GeV | 8 GeV | 20 GeV | 20 GeV | 20 GeV | 40 GeV
$m_{\pi_{d}}$ | 5 GeV | 2 GeV | 5 GeV | 5 GeV | 5 GeV | 10 GeV
$c\,\tau_{\pi_{d}}$ | 150 mm | 5 mm | 50 mm | 500 mm | 5 mm | 500 mm
Table 1: Dark sector parameters for our benchmark models. $\Lambda_{d}$ is the
dark confinement scale, $m_{V}$ is the mass of the dark vector mesons, and
$m_{\pi_{d}}$ is the pseudo-scalar mass. $c\,\tau_{\pi_{d}}$ is the rest frame
decay length of the pseudo-scalars. We take $N_{c}=3$ and $n_{f}=7$ in both
benchmarks. Models A and B are considered in Section IV, while Models A, C, D,
E and F are considered in Section V.

##  III Event generation

Here we describe our simulation pipeline to generate Monte Carlo events for
the models described in the previous section. The events were generated using
a modified spin-1 mediator model222https://github.com/smsharma/SemivisibleJets
Cohen et al. (2017) implemented using the `FeynRules` Alloul et al. (2014)
package. The model is outputted as a UFO Degrande et al. (2012) file which
allows generation of hard processes with `Madgraph5_aMC@NLO` (Alwall et al.,
2014), and we use LHC conditions with a centre of mass energy of 13 TeV. This
output is interfaced to the Hidden Valley Carloni and Sjostrand (2010a, b)
module of `Pythia8` (Sjöstrand et al., 2015), which simulates showering and
hadronization in the dark sector as well as decays of dark hadrons to either
other dark hadrons or to SM states. Initial state radiation (ISR) in QCD or
EW, i.e jets, leptons and EW gauge bosons, are included in the hard processes
in `Madgraph5_aMC@NLO`, and we use `Pythia8` to shower and hadronize SM
quarks.

Double counting of jets can occur when introducing ISR at the matrix element
(ME) level and then subsequently hadronizing the partons in `Pythia8`. To
avoid this, we use the MLM matching and merging procedure Mangano et al.
(2007). An `XQCut` of $M_{Z^{\prime}}$/10 is used at the matrix element level
which forces the production of only partons with a minimum $K_{T}$ separation.
Matching in `Pythia8` is done by enforcing $\verb!QCut!>\verb!XQCut!$. For
photon ISR, a minimum transverse momentum cut is placed on additional photon
radiation of $P_{T}>10\text{ GeV}$ to avoid soft and collinear divergences.
The resulting hadrons are clustered into jets using the Anti-$k_{t}$ algorithm
Cacciari et al. (2008) implemented in `FASTJET` (Cacciari et al., 2012), where
the ATLAS tracker’s pseduorapidty is $|\eta|<2.49$ and the jet angular
parameter $R=0.4$.

For the analysis of current triggers in Section IV, one million events are
generated for each $Z^{\prime}$ mass within the range [50 GeV, 1500 GeV], with
a step size of 50 GeV, for Models A and B in Table 1. A $Z^{\prime}$ width of
$\Gamma_{Z^{\prime}}=M_{Z^{\prime}}/100$ is used throughout. A crude detector
volume cut is implemented at the `Pythia8` stage. All particles that have not
decayed outside of a cylinder of $(r=3000$ mm$,z=3000$ mm) are considered
stable. This cut was placed to mimic the detector volume’s reach up to the
muon spectrometer. For the analysis of current triggers, backgrounds rates are
already known and do not have to be estimated.

In Section V, the focus is on using a machine learning approach for novel
triggers. We are less interested in hard ISR events and therefore use
`Pythia8`’s hidden valley production process $f\bar{f}\rightarrow Z_{v}$
processes instead of `Madgraph5_aMC@NLO`. Hits on the ATLAS inner tracker are
used as discriminating variables. A proper detector simulation of the inner
tracker is outside of the scope of this work, but we use a crude detector
simulations with code used in (Knapen et al., 2017) which encompasses the
ATLAS tracker from the Inner Bilayer (IBL) to the Transition Radiation Tracker
(TRT). This detector simulation assumes simple models of energy loss through
each thin layer of the detector. An emerging jet registers various hits as
function of the radial distance from the interaction point. Section V
considers Models A, C, D, E and F, and we choose a $Z^{\prime}$ mass of
$M_{Z^{\prime}}=500$ GeV. This set of models span a wide range of lifetimes
while keeping all other model parameters equal.

When considering new triggers, we must also estimate the background. Various
background processes are considered, but the backgrounds are dominated by
$pp\rightarrow b\bar{b}$ events which have high multiplicity and hadrons with
longer lifetimes producing many displaced vertices that can mimic an emerging
jet signal. We simulate $10^{5}$ events of $gg\rightarrow b\bar{b}$ using
`Pythia8`’s heavy flavour hard $b\bar{b}$ processes. Pileup added to both
signal and background events with `Pythia8`’s minimum bias events. For each
signal or background event, a number of minimum bias events are added randomly
sampled from a poisson distribution with mean of $\mu=50$, approximately
corresponding to expected pileup contributions for recent and near future
ATLAS runs. The inclusive background cross section $\sigma_{\text{bkg}}$ is
taken from the `Pythia8` event generation stage, which is used to estimate the
instantaneous background rate.

##  IV Current triggers

A consequence of having high instantaneous luminosities, such as at the LHC,
is the necessity of implementing triggering streams on specified event
criteria. ATLAS/CMS produce event rates in the MHz range, which is far too
large to write every event onto record. Triggers were introduced to greatly
reduce the event rate that is written for offline use, to a manageable $\sim$
1 kHz. Although emerging jet experimental searches do exist (Sirunyan et al.,
2019), models with unique phenomenology such as emerging jets may not be
visible to the current dedicated trigger sets used at ATLAS and CMS (Owen,
2018; Khachatryan et al., 2017). In this section, we quantify the efficiency
of current triggers for emerging jet phenomenology.

In addition to leading order production, we also study the effects of ISR on
trigger efficiencies which come from Feynman diagrams of the type shown in
Fig. 2. Multiple QCD jets (and/or electroweak gauge bosons) can modify the
naive expectation of a two-pronged emerging jet scenario. Each additional hard
object will change the event’s topology from back to back scattering in the
transverse plane. Emerging jets on their own provide unique detector signals
that may have difficulty passing trigger selections in various parameter
spaces. We will show that the inclusion of hard SM objects, such as jets and
leptons, increases the likelihood of passing the triggers. Numerical trigger
thresholds used in ATLAS ATL (2018) and used for this analysis are shown in
Table 2.

Triggers | Lower (GeV) | Higher (GeV) | Offline (GeV)
---|---|---|---
Single Jet | 100 | 420 | 435
Single $\gamma$ | 20* | 140 | 145
Single e | 22* | 26* | 27
Single $\mu$ | 20 | 26* | 27
MET | 50 | 110 | 200
$H_{T}$ | // | 850 | //
Table 2: The 2017 ATLAS triggers ATL (2018) used in Section IV analysis.
Triggers are separated by lower level (L1) thresholds, Higher level (HLT)
thresholds and offline selection criteria. Reconstructed jets used in the
Single Jet, MET and $H_{T}$ trigger have R = 0.4. Thresholds with (*) must
additionally satisfy the isolation cone criteria in Eq. (8). A lower level
threshold isn’t given for the $H_{T}$ as we seed it from the lower level
Single Jet trigger instead.

###  IV.1 Description of triggers

Figure 3: The Mercedes topology of three jets recoiling off of each other
within the $p_{T}$ plane. In this case, two emerging jets carrying missing
transverse energy recoil off of a visible hard QCD jet. This configuration
produces more missing transverse energy as the jet momentums are balanced in
opposite directions.

Figure 4: ATLAS trigger efficiencies, at truth level, for various processes
outlined in Section III. The first (second) row corresponds to Model A (B)
from Table 1. In the first column, we have the MET triggers; in the second
column, the $H_{T}$ triggers; and, in the third column, the lepton and photon
triggers. Both the electron and muon trigger efficiencies considered $W/Z$ ISR
while the photon trigger considered photon ISR only. All processes were
generated under LHC conditions of a centre of mass energy of 13 TeV. The
trigger thresholds were taken from the ATLAS trigger menu summarized in Table
2.

The trigger systems at both experiments are separated into two disjoint online
subsystems: the low level hardware trigger system (L1), and the high level
software trigger system (HLT). L1 primarily deals with low level information
from energy depositions in the calorimeter towers and minimally reconstructed
jet variables. Track reconstruction and jet algorithms are available at HLT
triggers for more sophisticated triggering criteria. Here, we will describe
the triggers that are relevant to our analysis.

MET Triggers: Dark sector mesons with sufficiently long lifetimes $\tau_{d}$
will typically escape the detector before decaying and thus contribute to
Missing Transverse Energy (MET). Energy deposited within the calorimeter
towers is reconstructed for MET calculations at both ATLAS and CMS. In the
plane transverse to the beam, the transverse momentum $P_{T}$ is conserved
with a zero net $P_{T}$, and thus this observable is sensitive to production
of invisible particles. Energy deposition in each tower is summed and a
transverse momentum vector is constructed using

$P_{T}=\sqrt{\big{(}\sum P_{x}\big{)}^{2}+\big{(}\sum P_{y}\big{)}^{2}}.$ (7)

Any non-zero contribution is taken to be MET $=|P_{T}|$.

Triggers cut on an event’s MET according to the trigger menu thresholds in
Table 2, at both L1 and HLT. L1 thresholds are lower than the HLT, although
the calculated MET might differ between reconstructed calorimeter tower
energies at both levels. It is usually precise enough to assume that HLT is
seeded from L1 with $100\%$ efficiency, therefore we only consider the HLT
efficiencies.

Typical $Z^{\prime}$ events, even if the dark hadrons have long lifetimes,
tend to have relatively low MET because the two emerging jets are produced
back to back, so there can be significant cancellation between them. Hard
initial state radiation can qualitatively change this picture as shown in Fig.
3. With additional radiation the two emerging jets are no longer collinear,
and their MET will to some extent add. Furthermore, the additional radiation
means that the energy of each jet will be larger (for fixed $Z^{\prime}$
mass), which also tends to increase the MET. Fig. 3 shows QCD radiation, but
the same logic applies to EW radiation.

$H_{T}$Triggers: The $\textit{H}_{T}$ trigger is a threshold on the scalar
$P_{T}$ sum of the event’s reconstructed objects. These triggers help reduce
rates by focusing on events with large final state transverse energy
$\textit{E}_{T}$. The $\textit{H}_{T}$ is built from objects with $|\eta|<2.5$
and jets are only included if they have $\textit{P}_{T}\geq 50\textrm{ GeV}$.

For the $Z^{\prime}$ model here, as the mediator mass increases, the trigger
is more likely to be satisfied due to the larger final state momentum imparted
onto the constituents. At lower mediator masses, the efficiency can be
increased with additional hard QCD radiation. The QCD jets are more visible
than the dark sector jets and lead to a higher $\textit{H}_{T}$. This is also
realized in the EW ISR case, as the hard photon/$W$/$Z$ caries all of the ISR
energy,333In leptonic decays of the $W$ or decays of the $Z$ to neutrinos, the
energy of the neutrino does not contribute to $H_{T}$.  and if properly
reconstructed, contributes significantly to final state $\textit{E}_{T}$. This
effect can be seen in Fig. 3.

Lepton and Photon Triggers: ISR of a $W^{\pm},Z$ can add additional hard
leptons and/or missing energy with neutrinos, and radiation of photons can
also be used. Triggers that cut on identified leptons ($e$ and $\mu$) and
photons, are considered. The trigger menu hosts a range of triggers depending
on the level of reconstruction necessary for the identification of the event’s
particles.

The rate of leptons produced within jets is extremely high, so lepton triggers
also have isolation requirements. ATLAS identifies leptons under various
classifications. These criteria are known as tight (loose) Isolation Aad et
al. (2020); Aaboud et al. (2019b), which are defined by

$\sum_{i\;\in\;\textrm{cone}}\frac{P_{T}^{i}(\Delta
R<R^{l})}{P_{T}^{l}}<\mathcal{I},$ (8)

where in Eq. (8), $\mathcal{I}=0.6$ $(1.5)$ for tight (loose) isolation,
$P_{T}^{l}$ is the transverse momentum of the candidate lepton, $P_{T}^{l}$
are the transverse momentum of the visible non-candidate objects within the
isolation cone, $\Delta R$ is the distance between the $i^{th}$ particle and
the candidate lepton $l$, and $R^{l}=0.2$ $(0.3)$ is the cone radius for
electrons (muons). This is accomplished with $P_{T}$ calculated from lepton
track measurements.

In the case of photons, the condition is given by

$\sum_{i\;\in\;\textrm{cone}}E_{T}^{i}(\Delta R<R^{\gamma})<0.022\cdot
E_{T}^{\gamma}+2.45\text{ GeV}.$ (9)

The photon isolation uses calorimeter measurements of the transverse energy
$E_{T}$ since photons do not leave tracks, $E_{T}^{\gamma}$ is the transverse
energy of the candidate photon, and $E_{T}^{i}$ of the $i^{th}$ cone
constituent. Since we are simulating events without full detector effects, we
assume that the truth level transverse energy of the photon is equal to that
of the reconstructed calorimeter energies.

Trigger menus may demand different levels of isolation strictness between L1
and HLT. In Table 2, for the single electron trigger considered, both L1 and
HLT must adhere to tight isolation criteria where as the muon trigger has
isolation only at L1. It is important to consider these drastic differences of
kinematic acceptance between L1 and HLT when calculating the total efficiency.
Because of this, we do not assume that the lepton triggers are seeded from an
L1 trigger with $\epsilon_{L1}=1$ (100$\%$ efficiency). Instead, we calculate
the L1 efficiency and project the product
$\epsilon=\epsilon_{\text{L1}}\cdot\epsilon_{\text{HLT}}$ in our results.

###  IV.2 Results with current triggers

We first calculate the trigger efficiency for different triggers in Model A
and Model B. In this section we do not use a detector simulation as the output
of `Pythia` should be a reasonable estimate of these simple variables. The
efficiency is the number of events that pass the threshold, and therefore get
written for offline use, over the total number of events, and these are shown
in Fig. 4, with the top (bottom) row being for Model A (B). In the first
column we see that for Model A, any kind of radiation increases the MET
trigger efficiency because in Model A the dark hadrons have long lifetimes,
and initial state radiation increases their momentum and makes them not back
to back. Radiation of $W/Z$ does the best because of the presence of
neutrinos. For Model B, notice that we only get significant improvement with
$W/Z$ radiation.

In the second column of Fig. 4 we see that QCD radiation can significantly
increase the efficiency of the $H_{T}$ trigger for both lifetime benchmarks,
and two hard jets does better than a single extra jet. Both $W/Z$ and photon
radiation do better than QCD radiation because of the the clean visible
momentum carried by the EW radiation. At low masses $M_{Z^{\prime}}\lesssim
500\text{ GeV}$, the efficiencies are similar for both models while at higher
masses $M_{Z^{\prime}}\gtrsim 500\text{ GeV}$, Model B becomes far more
efficient as it carries more visible particles in the final state. The
improvement due to extra radiation is very important at low mass, a section of
the $Z^{\prime}$ parameter space not easily probed, but less so at high mass
because the trigger can already be quite efficient at leading order in that
case.

We also show the trigger efficiency as a function of dark pion lifetime for a
fixed $Z^{\prime}$ mass of 800 GeV in Fig. 5. As expected, as the lifetime
increases, more of the energy escapes the detector and the MET trigger gets
better while the $H_{T}$ trigger gets worse. This explains the differences of
the first and second columns of Fig. 4 between both Models A and B.

Figure 5: ATLAS trigger efficiencies as a function of the dark pion’s lifetime
$c\tau$. The LO process is shown in the solid lines, the LO + j process is
shown in the dashed lines. A inverse relationship is exhibited between MET and
$H_{T}$ efficiencies.

Figure 6: Cross section times efficiency of various processes (leading order,
1-jet ISR, 2-jet ISR, electro weak ISR) scaled by their respective leading
order process. The left plot uses the MET trigger and Model A, while the right
uses $H_{T}$ and Model B. The dotted line is the leading order process, which
when normalized to itself, is just unit as a reference for the rate boost
obtained from additional processes. On the right plot, the blue region on the
left has zero events that we simulated pass the trigger,
$\epsilon\lesssim\epsilon_{\rm{min}}=1/400$ $000$.

Finally in the third column of Fig. 4 considers the effect on lepton (photon)
triggers on the process with additional $W/Z$ (photon) radiation. The two
models are almost identical, showing that these triggers are picking up the
extra electroweak radiation. This assures us that the dark pion lifetimes and
parameters of the confined hidden sector have little bearing on the EW focus
of the lepton triggers. The slightly higher efficiency for lepton triggers
($e,\mu$) in Model A because of the dark pion long lifetimes leaving less
contaminants within the lepton isolation cone, i.e Eq. (8). In terms of the
dependence on $M_{Z^{\prime}}$, the electroweak radiation is roughly constant,
while the photons tend to be harder at larger $M_{Z^{\prime}}$ so the
efficiency increases.

Although a useful metric, the efficiency does not consider the overall
probability of the event occurring because it does not take into account that
extra radiation reduces the cross section. Therefore when looking at the total
rate, we consider the cross section of the process times its respective
efficiency. To get a proper sense of the rate independent of some of the
unknown particle physics such as the couplings of the $Z^{\prime}$, we take
the ratio of the cross section times efficiency with respect to the leading
order (LO) result with no additional radiation, and show the results in Fig.
6. In the left column we show the improvement achieved for the MET trigger in
Model A. We see that QCD radiation can lead to significant improvements at low
mass, and even at high mass simulating extra radiation increases the overall
rate by $\mathcal{O}(100)$%. We also see that simulating two additional jets
gives significant improvement relative to only a single jet at low mass.
Electroweak radiation only gives a modest improvement in the event rate,
roughly 10% at low mass and even more modest at high mass. This is because the
rate suppression due to $\alpha$ is significantly stronger than that from QCD
that goes like $\alpha_{s}$.

In the right column of Fig. 6 we show the improvement for the $H_{T}$ trigger
in Model B. At low mass, none of the $400$ $000$ events we simulate at leading
order pass the trigger, so considering radiation opens a new parameter regime
for discovery. Even at intermediate masses, $M_{Z^{\prime}}\sim 500$ GeV,
additional radiation gives orders of magnitude improvement in rate. As with
Model A, two additional jets gives the greatest improvement, but all processes
considered can be significant.

##  V Machine learning triggers

The current triggers of Section IV, although well understood, are still
limited in their capacity to find new physics. They have forced us to consider
subsets of model parameter space that best produce specific signals that
current triggers discriminate best on and create blind spots in other regions
of parameter space. The CMS and ATLAS experiments have been investigating more
novel methods of ML analyses at the trigger level Alimena et al. (2020b).
These non-linear algorithms are not limited to primary detector features, such
as hard jets and charged particle tracks, as they can be trained
(unsupervised) to converge on non-intuitive abstract features within an event.
These algorithms can be trained on a array of new physics simulations, opening
up the door for unique correlations between new physics models that could
prove generic for a dedicated trigger. Additionally, the computational
resources necessary in training and testing the ML method does not impact the
resources in-situ as the training/testing stage is done prior to
implementation on the trigger stream.

Emerging jets produce interesting signatures and are complimentary to other
models consisting of hidden sector portals with dark showers. Examples include
semi-visible jets Cohen et al. (2015, 2017) and SUEPS Knapen et al. (2017),
which exhibit extreme cases of a similar baseline theory to that of emerging
jets. In this section, we employ the use of the ATLAS inner trackers in
training ML methods for emerging jet signals for the purpose of triggering.
Like the complimentary models mentioned above, emerging jets produce a wealth
of uncharacteristic lifetimes as compared to the SM. Here we show that
recruiting the inner tracker allows an ML algorithm to converge on
discriminating features that exploit this gap in lifetimes between the new
physics and the SM. Whether ML methods converge on generic features of new
physics or those more specific to the training model, understanding what is
the physics behind these ML features has been a long standing question.
Answering this question requires solving an inverse problem of the ML output,
and proves more and more difficult as ML methods such as neural networks and
boosted decision trees become more complex, although there is some recent
progress Faucett et al. (2020). Fortunately we can largely disregard this
problem since the purpose of the trigger is to get as many interesting events
onto record as possible. Anything written onto record can then be properly
analyzed offline. So we take the pragmatic approach to ML and focus our
attention on producing the highest trigger efficiency independent of what it
is “seeing.”

As in Section IV, the trigger systems at both L1 and HLT are limited by their
allocated computational resources. These triggering operations must be fast
enough to reduce the input data stream to $\sim 1$ kHz. This can be
challenging when data from all areas of the detector package are
simultaneously used within the triggering systems in some form or another.
Fitting non-linear algorithms at the trigger level, such as ML methods can be
taxing on the available resources. In this case, we look at low level
variables such as hits on the tracking detectors and simple jet
reconstructions from L1. The lack of fully constructed particle tracks and
momentum measurements allows for fully trained algorithms to operate quickly
on the incoming data streams.

For concreteness, we will analyze the ATLAS tracker geometry in this section,
but we expect similar qualitative conclusions for CMS. In Fig. 7 we show the
fraction of the dark pions that decay in a given detector subsystem as a
function of lifetime, and we see that for all lifetimes, the largest fraction
is in the inner tracker. Therefore, we focus only on that system in this work,
though we note that there could be interesting improvements by including the
calorimeters and muon system.

Figure 7: The fraction of dark pions that decay into visible sector particles
within the primary ATLAS detectors. The green line represents the decay
fraction within the electromagnetic calorimeter, the orange for the hadronic
calorimeter, and the blue for the inner trackers. The blue line is the
fiducial boundary of the muon spectrometer which we take to be the boundary of
the ATLAS detector. We do not show that fraction that decay before the first
tracker layer, and this fraction is significant at short lifetime. These
fractions are computed from simulation with $M_{Z^{\prime}}=500$ GeV.

The ATLAS inner tracker comprises of, in increasing order of distance from the
beam pipe: the Inner Bi-layer (IBL), the Pixel Detectors, the silicon
detectors, and the Transition Radiation Tracker (TRT). Table 3 shows the
number of layers and radial distances from the interaction point as well as
whether the tracker has additional endcaps. The primary purpose of the inner
trackers is track reconstruction and particle identification. Tracks are
reconstructed at the HLT level where calorimeter information has been seeded
from L1. Although useful, track reconstruction is a very computationally
expensive operation as all possible track trajectories are back propagated
towards the interaction point. We propose using machine learning methods
trained on the hit patterns of the inner tracker’s layers while bypassing full
track reconstruction.

As noted in Section III, in our simulations we take the dark hadronization to
be dominated by dark pions. The proper lifetime of the dark pions depend
strongly on the $Z^{\prime}$ mass as shown in Eq. (3), with lower mediator
masses corresponding to shorter lifetimes. We assume a common lifetime for the
dark pions, but for a study of hierarchical lifetimes see Renner and Schwaller
(2018), and we take the lifetime to be a free parameter that we vary.

As the emerging jets traverse the inner tracker, the invisible dark pions
decay into visible quarks which hadronize into SM jets with high particle
multiplicity creating a complex bundle of displaced tracks as they decay
throughout the detector volume. Ideally, without pileup and other secondary
detector effects, the number of hits registered on the tracker layers should
increase with radial distance.

Tracker | Layer | Radius (mm) | Geometry
---|---|---|---
IBL |  | 33.25 | Barrel
| First | 50.5 |
Pixel | Second | 88.5 | Barrel + Endcap
| Third | 122.5 |
| First | 299 |
SCT | Second | 371 | Barrel + Endcap
| Third | 443 |
| Fourth | 514 |
TRT | Start | 554 | Barrel + Endcap
Table 3: ATLAS inner tracker specifications taken from (Air, 1999; Capeans et
al., 2010). The barrel layers of each tracker section are shown with their
radial distance from the beam pipe/interaction point. Since the TRT is a more
complex tracking package, we only consider the hits on the initial layer of
the TRT. Endcaps also accompany most of the trackers but aren’t considered in
this analysis.

Signal and background events are simulated and explained in Section III.
Hadronized events are passed to a highly simplified detector simulation used
for Knapen et al. (2017) that attempts to model the ATLAS inner tracker. The
simulation accounts for updated particle trajectories from the bending of the
ATLAS toroidal magnetic field, $B\approx 2$T, as well as energy loss from
interactions with the material (assuming a thin layer approximation). The
simulation does not account for the production of secondary particles from
interactions with the layer materials. These secondaries are a possible source
of error as they could fake a displaced vertex around each tracker. We take
combinations of the concentric tracker volumes utilizing the IBLi, Pixeli,
SCTi and TRTi detectors where the subscript denotes the $i^{th}$ layer.
Results are represented as hit patterns in the form of heat maps in the
($\phi$, $\eta$) plane for each layer of the trackers.

A simple strategy that ultimately does not work is to train a classifier using
only the total number of hits recorded on each layer. This strategy reduces
the 2D map in ($\phi,\eta$) space to a single variable $N_{h,i}$ representing
the number of hits on the $i^{th}$ layer. ROOT’s TMVA machine learning toolkit
(Hoecker et al., 2007) is used to train and test the ML algorithm, in this
case a support vector machine, with both signal and $b\bar{b}$ background
events. The primary reason this strategy does not work is because of pileup.
Each LHC event contains pileup from multiple proton collisions in each event.
While there is only one hard collision responsible for the emerging jet
production, the remaining collisions produce a large number of soft particles
distributed approximately isotropically through the detector. This will
typically wash out large differences in hits on consecutive layers
$N_{h,i}-N_{h,i+1}\approx 0$. We simulate pileup by adding an average of 50
minimum bias collisions to both signal and $b\bar{b}$ background events. Since
pileup is relatively soft, the toroid magnet will deflect charged particles
with a radius of curvature inversely proportional to the particles momentum
$R_{\textrm{c}}\propto p^{-1}$. Pileup, in the form of minimum bias events,
has characteristically low momentum spectrum, so layers at larger radial
distances are less sensitive to pileup effects.

To minimize the influence of pileup, we refine our strategy by adding a
simplified jet reconstruction algorithm such that instead of counting hits on
the entire layer, we instead count tracker hits in the geometric vicinity of
hard jets. In detail, energy deposition in the calorimeter is used to
reconstruct jets in terms of topo clusters at L1 and at HLT. With jet
information at the trigger levels we use the truth level jet vectors $\hat{v}$
as well as the jet cone acceptance $\mathcal{R}=0.5$ to define $N_{cor}^{i}$,

$N_{cor}^{i}\equiv\sum_{j}N_{h,j}^{i}\left(\sqrt{(\Delta\eta)_{j}^{2}+(\Delta\phi)_{j}^{2}}\leq\mathcal{R}\right).$
(10)

Where the sum runs through all grid points $j$ of the layer $i$.
$(\Delta\eta)_{j}=\eta_{j}-\eta_{\hat{v}}$, where $\eta_{j}$ are
pseudorapidities at grid point j and $\eta_{\hat{v}}$ is the pseudorapidity of
leading jet direction $\hat{v}$. Similarly,
$(\Delta\phi)_{j}=\phi_{j}-\phi_{\hat{v}}$. The radius of each layer $R_{i}$
values can be found in Table 3. This approach allows the classifier to become
sensitive to a conical subset of hit patterns in the direction of an L1
topocluster jets. This substantially reduces the $4\pi$ reach of pileup.

For this analysis, we fix $M_{Z^{\prime}}=500$ GeV. Models A, C, D and E
(described in Table 1) are used as signal benchmarks. They vary in only the
lifetimes while keeping all other dark sector parameters equal. We also
included Model F, which has the lifetime of Model D but varies in the dark
sector parameters. The TMVA support vector machine was trained and tested
using Eq. (10) as the input variables. Testing and training sets were created
and randomly separated from the simulation results. The left panel of Fig. 8
shows the resulting SVM signal and background discrimination. We see that
there is excellent separation between signal background, and that the trained
and tested distributions look very similar. Only four layers of the trackers
were used, IBL, Pixel2, SCT2, TRT1, sampled from each of the four inner
tracker packages. This arrangement of layers allows the ML to train on
snapshots of the emerging jets evolution at substantially separated intervals,
but using only four layers reduces the required size of the training sample. A
proper study could simulate a much larger and realistic sample size while
incorporating additional layers.

Figure 8: On the left, Model E (blue) discrimination from $b\bar{b}$
background (green) using a support vector machine. The SVM was trained using
four layers IBL, Pixel2, SCT2, TRT1. An average pileup of $\mu=50$ was added
to both signal and background. The flat bars (points) correspond to the
training (test) set. On the right, Receiver Operation Characteristic ROC for
Models A, C,D, and E using the TMVA support vector machine. From Eq. (10) the
SVM was trained and tested using four layers: IBL, Pixel2, SCT2, TRT1. At a
given background efficiency the expected signal efficiencies increase as the
dark pion lifetimes lower. The required background rejection is estimated to
lie between the horizontal dotted lines.

Choosing a value for the SVM response dictates the Type I (false positive) and
Type II (false negative) errors which are related to the power of the
signal/background classification. Assuming that all events all background
saturated, meaning very little new physics events occur, implemented triggers
must reach background rates that do not exceed the allocated bandwidth. Unlike
in Section IV where we investigated pre-existing triggers, the background
rates of our proposed novel triggers are unknown and must be estimated. The
high level trigger rate usually allocated for new triggers is of $R\sim 1$ Hz
(Aad et al., 2013). The required background rejection is given by,

$\epsilon_{\textrm{bkg}}=\frac{R}{\sigma_{\textrm{bkg}}\mathcal{L}}\sim
10^{-3}-10^{-2}$ (11)

for a peak instantaneous luminosity of $\mathcal{L}=21\times 10^{33}\textrm{
cm}^{-2}\textrm{s}^{-1}$. The background rates were estimated assuming high
$p_{T}$ $b\bar{b}$ production, which primarily mimic the signal events.
Additional backgrounds were considered, such as inclusive hard QCD
backgrounds, generated through `Pythia8`. Since these additional background
sources had substantially smaller efficiencies than pure heavy flavour
$b\bar{b}$, we take them to be negligible to the total background rates.
Inclusive background cross sections were taken from the `Pythia8` hard event
generation as explained in Section III, but these background cross sections
are leading order and thus only order of magnitude estimates.

The Type I and Type II errors of Fig. 8 are closely related to the signal and
background efficiencies. A more robust visualization of this relationship is
seen when plotting the Receiver Operator Characteristic curve (ROC). The ROC
curve is calculated by plotting the background vs. signal efficiency for each
SVM response value as shown in the right panel of Fig. 8. The value of the
signal efficiency $\epsilon$ can be read off for each model at the highest
allowable ATLAS trigger background efficiency. In Table 4 we see the range of
$\epsilon$ for various lifetimes ranging from $5$ mm to $500$ mm. Shorter
lifetimes seem to out perform longer lifetimes. In Model E ($c\tau_{d}=5$ mm)
there are distinct hit patterns all within the IBL and the beginning of the
TRT. These dark pions are boosted such that their respective emerging jets
must go through majority of its evolution between these inner tracker slices.
Whereas for the higher lifetime models, their evolution from invisible to
visible is either skewed towards the later layers or even clipped beyond the
tracker limits. Efficiencies of $\mathcal{O}(0.1-0.3)$ can be found for a
comparable resonance mass of $M_{Z^{\prime}}=500$ GeV for the simple
$s$-channel process with no additional hard ISR. Comparing these to the signal
efficiencies found in Section IV (see Fig. 4) we see that for similar
topologies, training on detector hits can be advantageous as current triggers
have less reach in the low mass regime.

$c\tau_{d}$ |  $\epsilon$ ( Bkg rej $10^{-2})$ |  $\epsilon$ (Bkg rej $10^{-3})$
---|---|---
5 mm | 0.370 | 0.250
50 mm | 0.230 | 0.125
150 mm | 0.122 | 0.060
500 mm | 0.100 | 0.050
Table 4: Signal efficiencies for expected allowable background rates for new
ATLAS triggers for $M_{Z^{\prime}}=500$ GeV. The lifetimes represent Models A,
C, D, and E. Each value in the last two columns are extracted from Fig. 8 for
a bkg efficiency calculated using Eq. (11) assuming the events are fully
saturated by background.

So far we have discussed training and testing on the same model parameters.
Implementing an ML trigger would require training on some expected signal
model prior to integration on the trigger stream. As mentioned earlier,
emerging jets and other models of dark showers have a large available lifetime
parameter space. Training an ML trigger on a single model parameter point
could bias the trigger towards classifying only a small portion of this
parameter space. The overall classification power is related to the area under
the ROC curve (AUC). As a test of the universality of this method, we take an
unknown sample set from each of the five models and apply each of the trained
SVMs on them as was done in Cesarotti and Thaler (2020). The results are seen
in where the diagonal corresponds to the AUC of Fig. 9. Each row is of an SVM
trained on a single lifetime and then applied to an unknown signal set of
differing or same lifetime (columns). The deviations from each unknown
lifetime set is of order a few percent. This reinforces the insensitivity of
the trigger to the parameters it was trained on. Each rows’ average AUC value
does not have a substantial change. Model F was included to see how sensitive
this analysis is on the parameters of the hidden sector, such as the
hadronization scale $\Lambda_{d}$, and the dark composite masses
($\pi_{d},\rho_{d}$). The first thing to notice is that when trained on Model
F there is almost no change in the AUC when applied to the range of lifetime
models, much like Model D which shares the same lifetime. Secondly, when the
various trained models are applied to both Model F and Model D the AUC values
are almost identical. This similarity gives us more confidence in the
universality of the trigger, such that it is mostly sensitive to pion
lifetimes while being largely insensitive to other deviations in the hidden
sector.

Figure 9: Area Under the Curve (AUC) of the ROC curves. The SVM was trained on
specific models (vertical axis) and tested/applied on a unknown data set
(horizontal axis) of the same/alternative model. The diagonals are the AUC
values corresponding for training and testing on the same models i.e. the ROC
curves of Fig. 8. Trained models show little separation in classification
power when applied to a range of lifetimes and hidden sector parameters.

##  VI Summary and conclusions

In this work we explore ways to trigger on new physics models with a high
multiplicity of displaced vertex of the type explored in Schwaller et al.
(2015). If the mediator is uncharged under the SM gauge groups, then it can be
relatively light and associated SM radiation is not guaranteed, so triggering
on these events at the LHC is not straightforward. If events are not triggered
on, they are forever lost, so maximizing trigger efficiency is a necessary
step for maximizing discovery potential.

We have first explored how both efficiency and event rate can be increased
using current triggers with the addition of initial state radiation. Our main
results are given in Figs. 4 and 6. Both QCD and electroweak radiation
increase the total energy of the event and thus increases the efficiency of
$H_{T}$ triggers. The effects are the largest for relatively light mediators,
but they can also be relevant for heavy mediators. Other triggers, such as
those searching for missing energy, leptons, and photons can also have higher
efficiency with certain radiation. Processes with radiation have lower cross
sections than the leading order process, but we have shown that even taking
this into account, there are significant increases in event rate, especially
at low mass. This ISR process is guaranteed by gauge invariance to exist, and
we encourage experimentalists to simulate these processes in future studies.

We have also explored possible new triggers using modern machine learning
methods that use simple counts of hits in the tracker as input variables. The
new physics models considered here leave an increasing number of hits on each
tracker layer as the dark mesons decay in flight to visible SM states.
Counting hits in the tracker is significantly faster than performing track
reconstruction, making it an ideal technique for a trigger. The use of ML
techniques allows for a more sophisticated separation of signal and background
based on these hit counts, and the effectiveness of our proposed method is
shown in Fig. 8. Using ML for triggering can significantly reduce one of the
primary problems of ML techniques in particle physics that it is difficult to
determine which features the algorithms are training on, and thus difficult to
estimate systematic errors. For a trigger, one wants to maximize the events
recorded, and then a full study of systematics can be done at the analysis
stage.

Finally, we have explored the sensitivity of our ML techniques across
different model parameters as summarized in Fig. 9. Of course, the underlying
model parameters of the new physics are unknown, and the ideal trigger would
be sensitive to as much of the parameter space as possible and also to new
physics models not captured by the simulation framework used in this work. We
see that varying the particle physics parameters of the dark sector does
significantly affect the efficiency, so a realistic trigger can be trained on
one model parameter and still be sensitive to a broad class of new physics
models.

Acknowledgments: We thank Paul Archer-Smith, Kevin Earl, Dag Gillberg, Simon
Knapen, Siddharth Mishra-Sharma, Michael Ramsey-Musolf, Pedro Schwaller, Jesse
Thaler, and Andreas Weiler for helpful discussions. This work is supported in
part by the Natural Sciences and Engineering Research Council of Canada
(NSERC). DL acknowledges support from the NSERC Postgraduate Scholarships
Doctoral Program (PGS D). DS is grateful for the hospitality from the Kavli
Institute for Theoretical Physics and the Galileo Galilei Institute for
Theoretical Physics where part of this work was done.

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