# A guide to hunting long-lived particles at the LHC

Simon Knapen1,2 Steven Lowette3 1Lawrence Berkeley National Laboratory,
Berkeley, United States, 94720; email<EMAIL_ADDRESS>2Berkeley Center for
Theoretical Physics, UC Berkeley, Berkeley, United States, 94720  3Inter-
university Institute for High Energies (IIHE), Vrije Universiteit Brussel,
Brussels, Belgium; email<EMAIL_ADDRESS>
###### Abstract

This article is a pedagogical review to searches for long-lived particles at
the LHC, primarily aimed at experimentalists and theorists seeking to initiate
and/or deepen their research in this field. We cover general theory priors and
example models, the main experimental techniques for long-lived particles and
some of the subtleties involved with both estimating signal efficiencies and
background rates.

###### keywords:

Long-lived particle, Large Hadron Collider

††journal:

###### Contents

  1. 1 Introduction
  2. 2 Theory perspectives
    1. 2.1 General theory priors
    2. 2.2 Example models
  3. 3 Experimental signatures
    1. 3.1 Displaced tracks and vertices
    2. 3.2 Calorimeter signals
    3. 3.3 Other experimental handles
  4. 4 Signal selection
    1. 4.1 Signal reweighting and geometric acceptance
    2. 4.2 Triggering on long-lived particles
    3. 4.3 Offline signal reconstruction and selection
  5. 5 Backgrounds
    1. 5.1 Standard Model long-lived particles
    2. 5.2 Material interactions
    3. 5.3 Fake tracks and vertices
    4. 5.4 External particle backgrounds
  6. 6 Dedicated detectors
  7. 7 Closing thoughts

##  1 Introduction

The discoveries of the muon (1936) [1] and the kaon (1947) [2] marked the
emergence of particle physics as a new field of physics, as they were the
first novel particles which did not fit in the contemporary framework of
atomic and nuclear physics. Their discoveries famously hinged on the
macroscopic lifetimes of both particles, which could be resolved beautifully
with the cloud chamber technology of the early 20${}^{\text{th}}$ century. As
the old cloud chambers gave way to modern day silicon trackers and time
projection chambers, the ability to accurately measure a decay length has
remained an important tool in particle spectroscopy, on par with accurate
energy and momentum measurements. In the era of large particle accelerators,
the focus shifted towards achieving higher center-of-mass energies and
luminosities, while maintaining good energy and momentum resolution. This
approach led to the discovery of the top quark and the $Z$, $W$ and Higgs
bosons, all of which decay promptly. Measuring particle lifetimes nevertheless
remained critical, in particular for $b$-tagging purposes, and a number of
important searches for exotic long-lived particles were carried out both at
LEP (see e.g. [3, 4]) and the Tevatron (see e.g. [5, 6]).

Given these historical precedents, exotic long-lived particles (LLPs) were
always on the radar as a prime discovery mode for beyond the Standard Model
physics at the LHC. In the pre- and early-LHC era, this primarily manifested
itself in the form of signals of supersymmetry, the dominant theoretical
paradigm at the time. It is indeed remarkable how wide of a range of long-
lived signatures even “vanilla” supersymmetry can generate. One feature
supersymmetric signals have in common however, is that they tend to be
relatively _hard_ , in the sense that they come with one or more high energy
photons, jets or leptons. While there certainly existed models early on which
generated much softer displaced signatures [7], this possibility was not yet
as mainstream as it is today, both for practical reasons and because of the
aforementioned theory priors before Run-1 of the LHC.111 As of today, the
LHC’s data taking schedule is divided in runs with in between long shutdowns
for maintenance and upgrades: Run-1 (2010-2012, 7 and 8 TeV), Run-2
(2015-2018, 13 TeV) and Run-3 (2022-2025, 13.6 TeV). From 2029 onwards, the
high-luminosity phase of the LHC project will start, the HL-LHC. The second
major evolution within the theory community has been an increased awareness of
experimental subtleties, as recasting existing LLP searches or proposing new
ones required a much more detailed understanding of the capabilities and
limitations of modern trigger and reconstruction algorithms, as well as an
intuition for the often very subtle backgrounds associated with the searches.

Since the start of the LHC, the ATLAS, CMS and LHCb collaborations have risen
to the challenge and have produced an impressive set of new LLP results, often
relying on highly innovative strategies. This was made possible because the
lessons of the past have not been forgotten, despite the fact that the primary
focus during the design phase of the ATLAS and CMS detectors was on the Higgs
and SUSY discovery potential. Due to their hermeticity and excellent tracking
capabilities, both experiments have proven to be powerful multipurpose
detectors for long-lived signatures. They are further complemented by the
superb tracking and vertexing capabilities of the LHCb detector in the forward
regime. In those instances where the detector design appeared suboptimal for
the signature of interest, the ingenuity of the analysis teams has enabled the
collaborations to greatly surpass their design sensitivity. In some cases this
was done by extending techniques originally developed as probes of the SM,
such as $b$-tagging. Today, the LLP program is a major component of the LHC
program as a whole, and experimental techniques that were once niche, such as
$dE/dx$, displaced tracking or time-delayed signals, are now powerful and
common tools in the experimentalist’s arsenal.

In this review we aim to equip graduate students and postdocs entering the
fascinating field of LLPs with the tools of the trade, both on the theory and
experimental side. By reading our review, we hope young theorists will learn
the many new variables and experimental techniques that are needed in searches
for long-lived particles, as well as develop a good intuition for what is and
is not reasonable experimentally in terms of triggers, reconstruction and
backgrounds. Experimentalists getting started on LLPs will hopefully gain a
better understanding of the modern theory priors, which types of models are
currently on the market and which features they can and cannot predict.
Finally, we hope that this text may also serve as a convenient bridge point
for more senior colleagues on both sides of the LLP effort, by summarizing the
main physics points within a common and mutually intelligible vocabulary.

##  2 Theory perspectives

In this section, we briefly review some general theory priors and lessons from
the Standard Model (SM), followed by some explicit examples. A more
comprehensive overview of models featuring LLPs can be found in [8] and [9].

###  2.1 General theory priors

As argued in the introduction, our (recent) historical focus on prompt
signatures has an excellent theoretical and phenomenological justification: In
all known examples, the natural width of a particle with mass $m$ can be
estimated by $\Gamma\sim m/8\pi$, unless one or more of the following is true:

  1. 1.

The decay occurs through a heavy, off-shell particle, which implies that the
width of the LLP is suppressed by a factor of $(m/M)^{2\\#}$ with $M$ the mass
of the heavy, off-shell state and $\\#$ a positive integer that depends on the
symmetries of the theory. In the SM, the role of the heavy scale $M$ is
usually played by the $W$ mass ($m_{W}$). The $\pi^{\pm}$, muon and kaons are
clear examples.

  2. 2.

The decay is subject to a severe phase space suppression, because the sum of
the masses of the final states is very close to the mass of the LLP. The most
spectacular example in the SM is that of the neutron.

  3. 3.

The decay width is suppressed by a very small dimensionless coupling constant,
associated with a high quality, approximate symmetry. While the smallness of
$V_{cb}$ in particular plays a role in some decays, there is no example in the
SM for which a macroscopic lifetime can be understood solely in terms of a
small coupling constant or mixing angle.

We can summarize these points in the schematic formula

$\Gamma\sim\frac{\epsilon^{2}}{(8\pi)^{a-1}}\frac{m^{n}}{M^{n-1}}$ (1)

where $n$ is always an odd, positive integer. The parameter $a$ is also a
positive integer and indicates the number of final state particles, while
$\epsilon$ represents a potentially small, dimensionless parameter, such as a
Yukawa coupling, CKM matrix element or mixing angle. The toy formula clearly
reflects the three suppression factors discussed above, if we interpret $m$ as
the scale characterizing the volume of the phase space. Usually, this means
identifying $m$ with the mass of LLP. For e.g. the neutron decay one would
instead identify $m\equiv m_{n}-m_{p}-m_{e}\ll m_{W}$, which explains its
extra-ordinary lifetime. Equation (1) therefore allows us to group the
particles in the SM into _equivalence classes_ , indexed by the integer
parameter $n$. The higher $n$, the more long-lived the particle and the
steeper the mass-lifetime dependence. This is illustrated in the left-hand
panel of Fig. 1 for a handful of example particles in the SM. The dashed trend
lines are fits to the data, where we held $n$ fixed but floated the intercept.

For the $\rho$, $J/\psi$, top, Higgs, $W$ and $Z$, none of the above
suppression factors apply and they naturally decay promptly. The $\pi^{0}$ and
$\eta$ decay to photons through the chiral anomaly, which induces the scaling
$\Gamma\sim m^{3}/f^{2}$ with $f$ the scale of chiral symmetry breaking.
Finally, the $\mu$, $\tau$ and flavored mesons all decay through an off-shell
$W$, hence the $\Gamma\sim m^{5}/m^{4}_{W}$ trend. The neutron is the clear
outlier due to its enormous phase space suppression.

Figure 1: Left: Proper lifetime versus mass for selected SM particles, grouped
according to the index $n$ in equation (1). The neutron is the clear outlier
due to its extreme phase space suppression; to keep the axis range manageable,
the neutron point was brought down and labeled with an arrow to indicate that
its true $c\tau$ is much larger than depicted here. It was also excluded from
the fit that resulted in the blue curve. Right: Four examples of BSM models,
where the dashed lines indicate the approximate scaling with their mass.
(Small steps due to kinematic thresholds were neglected.) The overall
normalization for each line was chosen to roughly correspond to the _smallest_
$c\tau$ allowed by current constraints, but is otherwise arbitrary. The figure
is therefore _not_ meant to illustrate that large $c\tau$ at high $m_{LLP}$
are disfavored.

Moving on to extensions of the SM, equation (1) still applies. We can thus
deploy it to critically examine a few pieces of “conventional wisdom” which
are sometimes applied to LLP models. When inspecting the left-hand panel of
Fig. 1 without accounting for the color coding, one may be tempted to conclude
that lifetime simply correlates with mass. This is the origin of a commonly
held intuition:  Heavy particles prefer to decay promptly, while light
particles can have long lifetimes. Heavy LLPs therefore require “special”
models and/or parameter choices. While the proper lifetime ($c\tau$)
undoubtable scales inversely with the particle mass, it is immediately obvious
from equation (1) that the above statement is too hasty: Since $n=5$ is pretty
typical in the SM and beyond, even a moderate hierarchy of scales in our BSM
model suffices to suppress the decay width by many orders of magnitude. The
most famous example is presumably the long-lived gluino in a split-SUSY
scenario [10], which arises from the $m_{\tilde{g}}\ll m_{\tilde{q}}$
hierarchy.222As is the case in the SM, new LLP’s may be charged or neutral, or
even colored in the case of long-lived gluinos. Charged and/or colored LLPs
are prevalent in supersymmetry in particular, while neutral LLPs tend so show
up in models of dark matter, baryogenesis and certain non-SUSY solutions the
hierarchy problem.  Heavy neutral leptons (HNLs) [11] with mass $m\ll m_{W}$
are another example in this same equivalence class, as shown in the right-hand
panel of Fig. 1. In fact, just as for the SM, _all_ BSM LLPs can be classified
according to the index $n$. In Fig. 1 we show the dark photon ($n=1$), the
axion-like particle (ALP, $n=3$), HNL ($n=5$) and dark sector $0^{++}$
glueballs ($n=7$) [12].

This brings us to the second piece of common wisdom:  Theorists are not so
good at predicting $c\tau$. While there are important exceptions in very
simple models such as HNLs or dark photons, this is largely true, but not
because of lack of perseverance or cleverness: The heavy mass scale $M$ can
easily be outside the reach of the LHC and can be difficult to pin down
theoretically. In some models it can be estimated by using additional inputs,
such as the dark matter relic density, but even in those cases, even modest
uncertainties on $M$ can get amplified greatly in equation (1). This gets
progressively worse for larger $n$, with some composite particles such as dark
sector glueballs as the most extreme examples. Fortunately, we will see in
Sec. 4.1 that our inability to predict $c\tau$ more precisely is actually not
so relevant experimentally.

All that said, there is a universal and relatively model-independent upper
bound on $c\tau$ from potentially spoiling Big Bang Nucleosynthesis (BBN)
[13]. The argument goes as follows: Any particle with an observable direct
production at the LHC should at some point in time be in thermal equilibrium
with the cosmic plasma, as long as the universe was hot enough to produce it.
As the universe cools, the LLP freezes out from the SM thermal bath and
subsequently decays. If this decay occurs while BBN is happening, the
injection of additional particles and energy tends to modify the primordial
abundances, which tends to be in conflict with observation. This gives us a
range of upper bounds between $\tau\lesssim 0.1$ \- $10^{4}$ s, depending on
the dominant decay modes and the abundance at the onset of BBN. While
interesting for its universality, even 0.1 s is an absurdly long time scale
for any collider experiment, and this bound is therefore almost never
experimentally relevant. Lower bounds on $c\tau$ also exist, but they are
always dependent on the model. This does _not_ mean however they are always
trivial to evade, and they must be investigated on a case-by-case basis.
Following equation (1) and Fig. 1, such lower bounds tend to exist for low $m$
($m\lesssim 5$ GeV) and $n\geq 5$. Concretely, a smaller $c\tau$ would require
a larger $(m/M)^{n}$, while model-dependent lower bounds on the scale $M$
exist from direct searches for other particles in the model. Important
examples are HNLs and any light LLP that is a composite particle, e.g. in
hidden valley models [7, 14].

###  2.2 Example models

Rather than supplying a comprehensive overview of models, which exists
elsewhere [8, 9], here we touch upon three example models, one for each of the
three general principles laid out in the previous section. We deliberately
picked examples which may be a little bit less well-known than the classic
examples (e.g. split SUSY), which are discussed abundantly elsewhere.

The most minimal way of obtaining a macroscopic lifetime is to have the SM
$W$-boson serve as the heavy, off-shell particle. This is precisely what
happens with heavy neutral leptons, which mix with the SM neutrinos and
undergo a 3-body decay to a SM neutrino plus two other SM fermions. It is
therefore no surprise that its lifetime obeys the $c\tau\sim m^{-5}$ scaling
law in Fig. 1, similar to the SM flavorful mesons and heavy leptons. It also
explains why with similar values for their mixing angles and masses, the
lifetime of a low mass HNL is much longer than that of a dark photon. For a
recent and comprehensive overview of HNL phenomenology we refer to [11].

A strong phase space suppression can be the result of an approximate symmetry
[15], as is the case in the long-lived chargino models that generate the
famous disappearing track signature. It can also be motivated from
cosmological considerations however: Suppose we have two dark sector particles
$\chi_{1}$ and $\chi_{2}$ with $m_{\chi_{1}}<m_{\chi_{2}}$, which can
annihilate to SM fermions $(f)$. Since dark matter ($\chi_{1}$) freeze out
tends to happen at a temperature $T\lesssim m_{\chi_{1}}/10$, the
$\chi_{1}\chi_{1}\to f\bar{f}$ freeze-out process receives an
$\sim\exp(-2m_{\chi_{1}}/T)$ Boltzmann suppression in the early universe.
However, if a co-annihilation process $\chi_{1}f\to\chi_{2}f$ exists, it is
only suppressed by $\sim\exp(-m_{\chi_{2}}/T)$ and could therefore easily
dominate if $m_{\chi_{2}}-m_{\chi_{1}}\ll m_{\chi_{1}}$ [16]. The presence of
a second particle $\chi_{2}$ that is close in mass to $\chi_{1}$ can therefore
radically change the relic abundance of $\chi_{1}$. Moreover, if $\chi_{2}$
can be produced at a particle collider, its decay
($\chi_{2}\to\chi_{1}f\bar{f}$) can be very slow, due to the small splitting
between $\chi_{1}$ and $\chi_{2}$ [17, 18].

Though this possibility is not realized in the SM, there are many BSM models
in which a particle acquires a macroscopic lifetime because of a very small
coupling. WIMP baryogenesis is such a model [19, 20]. The idea is that a WIMP
($\chi$) freezes out in the early universe, but instead of being the dark
matter, it is allowed to decay to SM baryons and the actual dark matter ($X$).
If baryon number, C and CP are all violated in the decay, we expect
$\text{Br}(\chi\to BX)\neq\text{Br}(\chi\to\bar{B}\bar{X})$, where $B$
($\bar{B}$) represent a SM final state carrying (anti)-baryon number. If the
decay is fast as compared to the expansion rate of the universe, the inverse
process where the $B$ and $X$ fuse back into $\chi$ is also efficient, and we
end up with no net asymmetry. We must therefore assume that $\chi$ is a long-
lived particle in the context of the early universe. It turns out that for
$m_{\chi}$ in the hundreds of GeV range, this implies that $\chi$ is also
long-lived at the LHC.

##  3 Experimental signatures

The experimental signatures we can use to distinguish LLPs from backgrounds
strongly depends on the properties of the LLP: the $c\tau$, the decay modes,
the LLP mass, charge, etc. At the same time, the experimental capabilities of
the detectors provide the boundary conditions under which these signatures can
be used for signal selection. Luckily, the LHC detectors were instrumented for
heavy flavor identification, have accurate timing capabilities, and possess
other features useful for LLP detection - they are thus very well suited to
search for LLPs.  The LHC detectors were not equipped for LLP detection. This
interplay of the characteristics of the LLPs with the experimental context
drives the selection choices made in data analysis. In the following, we
overview the various experimental tools at the disposal of the intrepid
explorer hunting for new LLPs.

###  3.1 Displaced tracks and vertices

Charged particles from the decay of an LLP are excellent experimental probes
that can be used to discriminate a potential long-lived new-physics signal
from the SM debris of the collisions, for which the charged particles mostly
emerge promptly from the collision point. One of the experimental limitations
restricting the identification of LLPs is the capability for the
reconstruction of displaced charged particles. This reconstruction of
displaced tracks is a trade-off challenge between efficiency and purity, with
computing capabilities restricting the ultimate performance. We discuss
displaced track reconstruction in more detail in Section 4.3. Starting from
the displaced tracks, there are several handles at our disposal for LLP
identification.

Figure 2: Drawing of the transverse (left) and longitudinal (right) cross
sections of the current CMS inner tracker, with an example displaced vertex
and the variables defined in the text. The energy loss ($dE/dx$) in individual
layers can fluctuate significantly, as indicated schematically by the varying
sizes of the yellow stars.

Track impact parameters

The impact parameters of a track are defined by the vector from the collision
point, called the primary vertex (PV), to the closest approach of the track to
that point. Often this is split in its transverse component, the length of
which is referred to as $d_{xy}$, and similarly $d_{z}$ for the longitudinal
component. In Fig. 2, a graphical representation is given of $d_{xy}$ and
$d_{z}$ for a displaced track, depicting the extrapolation towards the primary
vertex.

The impact parameter is a simple yet powerful discriminator between prompt and
displaced tracks. Before the advent of more powerful multivariate techniques,
$d_{xy}$ was a main ingredient in the identification of jets from heavy
flavor, targeting the identification of displaced tracks coming from $B$ and
$D$ meson decays. For this reason, the inner tracking detectors of modern
particle physics detectors are designed for excellent $b$-jet identification,
with a spatial segmentation in the inner silicon pixel detectors as small as
$50\,\mu m$.

The resolution of the $d_{xy}$ and $d_{z}$ observables is driven by the
detector granularity of the detector layers the charged particle crosses
first, but also by the distance of the extrapolation to the beamline and by
the number of hits on the track. Thus, the $d_{xy}$ resolution of a track that
emerges far from the collision vertex may be rather poor. This is in
particular the case for tracks reconstructed in the muon spectrometer only. To
account for this track-dependent varying resolution, some analyses employ the
significance of the impact parameter, $d_{xy}/\sigma_{xy}$ and
$d_{z}/\sigma_{z}$, where $\sigma_{xy}$ and $\sigma_{z}$ are estimates of the
impact parameter uncertainties obtained from the track fit. These significance
quantities are more performing, at the expense of requiring more care when
modeling or reproducing their efficiencies and related systematic
uncertainties.

For prompt background tracks, the impact parameter resolution $\sigma_{xy}$
can reach below $20\,\mathrm{\mu m}$ for tracks with transverse momenta above
$10\,\mathrm{GeV}$, but is more of the order of $100\,\mathrm{\mu m}$ for
tracks with $p_{\rm T}\sim 1\,\mathrm{GeV}$ or large $\eta$ [21, 22]. In the
$z$ direction, where the pixels are elongated, the resolution is somewhat
worse, reaching at best $\sigma_{z}\sim 30\,\mathrm{\mu m}$ for high-momentum
tracks. Another complication along the $z$-direction is the spread of the
collision vertices with a standard deviation of about $11\,\mathrm{cm}$, while
the transverse beam size is at the level of $25\,\mathrm{\mu m}$. An accurate
determination of $d_{z}$ thus relies on an unambiguous identification of the
primary vertex in the presence of many simultaneous proton collisions, known
as _pileup_. Often analyses will not make use of $d_{z}$ for these reasons,
but only use the already very powerful displaced track identification handles
$d_{xy}$ and $d_{xy}/\sigma_{xy}$.

Track and vertex displacement

If the LLP is light and has a high momentum, the tracks may emerge well
displaced but still have a small impact parameter and low significance. At
that point, the transverse ($L_{xy}$) and longitudinal ($L_{z}$) displacements
of the decay vertex, shown in Fig. 2, may be better discriminators. For a
displaced, isolated single track, the starting point cannot be unambiguously
determined, but the missing inner hits still hold valuable information. In the
case of two or more tracks emerging from the same LLP decay, a vertex can be
fit from the tracks, yielding an often accurate 3-dimensional estimate of the
decay point, and thus of $L_{xy}$ and $L_{z}$. Also here the dimensionless
significance is sometimes preferred. In particular in the case of a decay with
collimated displaced tracks, the position of the vertex in the direction of
flight may be difficult to estimate, and the normalization of the displacement
to its uncertainty can provide improved discrimination against poorly measured
backgrounds.

Track multiplicity and vertex mass

Another discriminator that may be of use to select particular signals such as
displaced hadron jets is the track multiplicity at a displaced vertex. The
inefficiency of displaced tracking reconstruction, increasing with the
displacement of the decay vertex, makes this variable sensitive to mismodeling
in simulation. As a result, a requirement of a minimum or maximum number of
tracks can induce a significant signal selection uncertainty. Nevertheless, a
minimum number of tracks is often imposed as a preselection (having a vertex
already implies at least two) as it strongly suppresses some of the
backgrounds (see Sec. 5). Alternatively, also vertex mass (i.e. the invariant
mass of the tracks that form a vertex) is a related observable that can be
used for this purpose. Examples of the use of both variables can be found in
Refs. [23, 24, 25, 26].

Decay direction

For new-physics scenarios where the LLP decays to charged as well as neutral
particles, e.g. in the case of displaced tau leptons, the jet clustered from
the charged particles may point off-axis with respect to the LLP flight
direction. In such cases, both the impact parameters ($\Delta_{xy}$ and
$\Delta_{z}$ in Fig. 2) and the displacement of the jet can be used for signal
selection, and in case of discovery the direction information from the
displaced tracks can be used to also characterize the neutral component. The
usefulness of this observable is expected to be strongly decreasing for larger
displacements, as tracking inefficiencies will wash out possible sensitivity
from the decay direction. For signals without neutral particles in the final
state, $\Delta_{xy}$ and $\Delta_{z}$ are expected to be zero, within the
resolution. For those signals they can therefore be useful variables to
suppress backgrounds from fake vertices (see Section 5.3), as those tend to
produce a flat distribution in $\Delta_{xy}$ and $\Delta_{z}$ (see e.g. [27]).

Ionization loss

Tracking detectors such as time projection chambers or silicon trackers can
also measure ionization energy deposits per unit length along the track
($dE/dx$).333Though the units of $dE/dx$ are MeV/cm, it is usually reported in
units of MeV$\times\text{cm}^{2}/g$ in the Particle Data Book [28]. In these
cases one has divided out by the mass density of the target, as this allows
for a more consistent definition for gas targets. For new charged particles
with masses well beyond the charged pion, kaon or proton mass, or with
electric charge different from $1e$, the ionization deposits provide an
additional handle to discriminate such signals from regular tracks from
SMbackgrounds.

The strongest discrimination may be achieved for particles with a boost
($\beta\gamma$) that places them below the usual minimum ionizing plateau of
the Bethe-Bloch curve. It is possible to discriminate different SM hadron or
ion species in this manner, but only at very low momenta, of the order of
$1\,\text{GeV}$ (see e.g. supplementary material of [29]). New heavy stable
charged particles (HSCP’s), such as R-hadrons, can have masses in the multi-
TeV regime and have therefore a low enough boost to allow discrimination with
$dE/dx$ measurements [30, 31]. For new particles with non-unit electric
charge, on the other hand, the ionization mean free path dependence on the
square of the charge makes the ionization loss a strong discriminator for a
much broader mass range [32]. Searches for ultra-highly ionizing particles
such as magnetic monopoles are the most extreme examples in this category
[33].

While ionization of material by particles is a very well understood phenomenon
[28], the accurate simulation of the background particles that are produced in
the collisions, as well as the description of the detector geometry, material,
aging with radiation, electronics saturation, etc, render a precise simulation
of ionization loss in the detector very difficult for backgrounds as well as
signals. Analyses using ionization loss as a discriminator between signal and
background thus need to carefully calibrate energy loss using e.g. hadron
tracks or muons from $Z$ bosons, often combined with background predictions
extracted directly using data.

Simulations of $dE/dx$, whenever possible, are performed with very
sophisticated software packages such as GEANT4 [34] and may therefore not be
practical for theorists seeking to perform a reinterpretation. It is therefore
tempting to just rely on the Bethe-Bloch curve, which provides the _mean_
energy loss of a particle through matter. This leads to an important pitfall
however, as the energy loss is a stochastic process with a highly skewed
distribution, such that the mean energy loss is dominated by rare, high energy
collisions. The _most probable_ energy loss is often a more useful estimator
[28], especially for thin detectors such as the tracking layers.444Since
energy loss in materials is a very subtle matter, we advise theorists seeking
to use analytic formulas to model $dE/dx$ to verify all the limitations of the
formulas provided in the relevant chapter in Ref. [28].

Track timing

Precise time measurements of the energy deposits in the detectors provide
another source of information on charged particles. Inner tracking detectors,
close to the beamline, do not provide a direct estimate of the peak time of
the hits, and are typically read out in a narrow time window around the
expected arrival time for particles traveling at or close to the speed of
light ($\beta=v/c=1$). For moderate $\beta$, this implies that an apparent
smaller ionization is recorded, though estimating a time delay from these
stochastic measurements is typically imprecise and further confounded by the
unknown LLP mass and charge. Track timing is more relevant for particles
traversing the outer muon chambers, where the individual cluster timing
measurements are controlled to the level of about $2\,\mathrm{ns}$ [35, 36].
Furthermore, the track fit can be improved by having the hit position
estimates from the gas-ionization signals to be dependent on $\beta$. The
accurate measurement of the particle’s speed is leveraged by the long travel
distance in the detector, which makes it possible to reach a resolution on
$\beta$ as small as 5% [37, 31].

###  3.2 Calorimeter signals

The reconstruction of calorimetric signatures of LLPs is not hampered by a low
efficiency, as can be the case for the displaced tracking, but the lack of
tracks means that identifying LLP-induced energy deposits is very non-trivial.
There are however a number of powerful, advanced experimental handles, as
described in the following.

Delayed calorimeter signals

Calorimeters measure particle energies by observing scintillation light
arising from electromagnetic or hadronic showers induced by the interactions
of the incoming particle with an absorber. Depending on the calorimeter
design, accurate measurements of the signal timing are available. In ATLAS and
CMS, the electromagnetic calorimeters (ECAL) have an ideal, intrinsic time
resolution as low as about $70\,\mathrm{ps}$ for energies larger than several
tens of $\mathrm{GeV}$ [38, 39]. In practise, the time resolution for
measurements of energetic photons in the barrel is at best of the order of
$200\,\mathrm{ps}$ [39, 40], dominated by a component arising from the
longitudinal spread of the LHC beams. For the hadron calorimeters (HCAL) the
timing resolution is also rather accurate. The ATLAS Tile Calorimeter achieves
a resolution as low as $0.4\,\mathrm{ns}$ for high energy deposits [41], while
CMS reports a time resolution of its HCAL of $1.2\,\mathrm{ns}$ for jet
energies above $100\,\mathrm{GeV}$ [42].

These rather precise time resolutions make it possible to use also the
calorimeters to distinguish background low-mass ultrarelativistic particles
from particles with low $\beta$ or a delayed signal from an increased path
length from a displaced decay. While a low $\beta$ easily induces
multi-$\mathrm{ns}$ delays for new particles with large masses, e.g. in the
$\mathrm{TeV}$ range, also the path length can bring several $\mathrm{ns}$
delay, well above the timing resolution [43].

Displaced calorimeter signals

The identification of signals unusually directed or displaced inside a
calorimeter is another powerful experimental handle on LLPs decaying in the
detector. Although calorimeters are often sampling the developing shower in
several alternating layers of absorber and scintillator, a detailed
segmentation of the shower is not always available for offline analysis. This
is particularly the case for the CMS HCAL, because the detector design
aggregates measurements along the shower depth. For most purposes, the shower
energy estimate suffices, but for LLP identification 3-dimensional information
on the shower profile is desired. It can then be used to search for decays
happening deep inside the calorimeter using e.g. the absence of an ECAL energy
component in a hadronic shower, or the detailed calorimeter cluster depth
position [44, 45]. Furthermore, fine-grained calorimeter shower information
may help identify photons that hit the calorimeter under an angle due to a
displaced decay, using the photon’s direction measurement [46] or its
elliptical shower shape [47].

###  3.3 Other experimental handles

Beyond the basic functionalities described above, detectors also start being
used well beyond their initial design. The recent CMS search for hadronic
decays in the muon detectors is an excellent example [48]: Thanks to the thick
steel of the magnet’s return yoke, the CMS muon system can be used as a
calorimeter, clustering low-level muon chamber hits in the muon detectors. A
related search also exists in ATLAS [49], which has a larger fiducial volume
in its muon detector, but has air as opposed to steel in between its tracking
detectors. As a result, ATLAS relies on reconstructing displaced tracks and a
displaced vertex, as opposed to searching for a calorimetric shower. This
means that the efficiency of the ATLAS search is more sensitive to the mass of
the LLP, while the CMS search primarily depends on its energy. Combining
information across subdetectors can also shed light on the individual
measurements, as recently exemplified in Ref. [30].

More detailed particle identification can furthermore be a powerful tool in
specific exclusive searches, but is harder to achieve without dedicated
detectors. An example here is the use of the LHCb RICH detector to identify
slow moving HSCPs [50] and charged kaons coming from displaced low-mass scalar
decays [51]. With the advent of the HL-LHC, we will see an expansion of the
experimenter’s portfolio with additional sub-detectors and detector
capabilities. Most prominently, dedicated precise timing detectors with
resolutions down to a few tens of ps are being added in front of the
calorimeters [52, 53]. But also many of the other subdetectors are getting
upgrades that provide opportunities for inventive new approaches to establish
LLP particle signatures (see e.g. [54, 55]).

##  4 Signal selection

Quantifying the signal selection efficiency is more complicated for LLPs than
for prompt particles, and can be very difficult to simulate and parametrize.
This leads to a number of important pitfalls for theorists in particular when
trying to model an existing or proposed analysis. For experimentalists it is
moreover important to simulate the desired signals as efficiently as possible,
due to the high computational complexity of the full detector simulation. In
this section we summarize a few tricks, and point out possible subtleties.

###  4.1 Signal reweighting and geometric acceptance

For prompt particles, the geometric acceptance is typically estimated by
requiring that the final states all satisfy a set of relatively simple $\eta$
and $p_{T}$ cuts associated with the detector geometry. For unstable LLPs this
is more complicated, as the trigger and reconstruction efficiencies depend
strongly on the location of the decay vertex. Crucially, these efficiencies
however do _not_ depend on the LLP’s proper lifetime ($c\tau$). We can use
this to our advantage in a simple reweighting algorithm, as described below.

For simplicity, let us assume that we have a single LLP with momentum
$\vec{P}$ and denote the momenta of its decay products by the set of momenta
$\vec{p}_{i}$. For a specific model, a sample of $(\vec{P},\vec{p}_{i})$ can
be generated with standard Monte Carlo simulation codes, such as
MadGraph5_aMC@NLO [56] or PYTHIA 8 [57]. To calculate the efficiencies for low
$c\tau$ correctly, it is very important to model the tail of the LLP’s
$p_{T}$-distribution carefully and with enough statistics. Depending on the
signal, this can mean including hard initial state radiation (ISR) or
simulating events which are weighted based on the LLP’s $p_{T}$. The latter is
particularly important for light LLPs which are produced non-resonantly, e.g.
in Drell-Yan or in exotic $B$-meson decays (see e.g. [58]). The reason is that
the most boosted LLPs will dominate the efficiency for small $c\tau$.

When generating LLP events with a simulation tool, it is always most efficient
to ignore the displacements of LLP decay vertices provided by the event
generator, and instead manually generate them along the halfline defined by
$\vec{P}$. In other words, we draw a positive real number $a$ from a
probability distribution $f(a)$ and relate it to the distance the LLP traveled
before decaying. The choice of $f(a)$ is a priori arbitrary, but should be
such that the resulting vertex locations efficiently sample the detector
volume of interest. For example, suppose that our sensitive detector element
is a co-axial cylinder, e.g. a calorimeter, and we care only about LLPs for
which $L_{xy}^{-}<L_{xy}<L_{xy}^{+}$ with $L_{xy}^{\pm}$ the inner and outer
radii of the detector. It then makes sense to choose e.g. the uniform
probability distribution

$f(a)=\frac{1}{L_{xy}^{+}-L_{xy}^{-}}\quad\mathrm{with}\quad
L_{xy}^{-}<a<L_{xy}^{+}$ (2)

and define the vertex location as

$\vec{x}=(a\cos\phi,a\sin\phi,a\sinh\eta)$ (3)

with $\phi$ and $\eta$ the azimuthal angle and pseudorapidity associated with
$\vec{P}$. This ensures that as few events as possible are wasted on decays
outside the fiducial volume. These events can then be passed through the
detector simulation to calculate the combined trigger and reconstruction
efficiency $\epsilon_{r}(\vec{x},\vec{p}_{i})$ depends only on the vertex
location $\vec{x}$ and final state kinematics $\vec{p}_{i}$. (More on this in
Sec. 4.2 and Sec. 4.3.) This calculation is usually very computationally
expensive with a full-fledged detector simulation, but we only need to do it
once. With $\epsilon_{r}(\vec{x},\vec{p}_{i})$ in hand, we can obtain the
efficiency as a function of $c\tau$ by defining the following weight for each
event

$w(\vec{P},\vec{p}_{i},\vec{x},c\tau)\equiv\frac{e^{-|\vec{x}|/\beta\gamma
c\tau}/\beta\gamma c\tau}{f(a)}\times\epsilon_{r}(\vec{x},\vec{p}_{i})\times
w_{p_{T}}(\vec{P})$ (4)

where $\beta\gamma$ is the boost factor of the LLP and $w_{p_{T}}(\vec{P})$
the $p_{T}$-dependent weight provided by the event generator. The total signal
efficiency for a particular $c\tau$ is then defined by simply averaging over
the weights

$\epsilon(c\tau)=\frac{1}{N}\sum_{\text{events}}w(\vec{P},\vec{p}_{i},\vec{x},c\tau)$
(5)

with $N$ the number of events.555To generalize the algorithm to multiple DV
per event, it suffices to define a weight like equation (4) for each DV and
combine them in equation (5) with the appropriate combinatorics. We thus are
able to recycle the same set of events for any value of $c\tau$ we want,
greatly reducing the computation cost of the simulation.

To gain an intuition for equation (4), let us return to our simplified
calorimeter example of the co-axial cylinder, such that
$\epsilon_{r}(\vec{x},\vec{p}_{i})=0$ unless $L^{-}_{xy}<L_{xy}<L_{xy}^{+}$.
For $\beta\gamma c\tau\ll L_{xy}^{-}$ we then recover the expected exponential
suppression of $\epsilon(c\tau)$, regardless of our choice for $f(a)$. The
events with the largest $\beta\gamma$ carry an exponentially larger weight,
hence the need to generate $p_{T}$-weighted events.666This means that for low
enough $c\tau$, the efficiency will always be dominated by a handful of
events, rendering the simulation unreliable. A plot of the cumulative
distribution of the weights is a good way to verify whether one is in this
regime. For $\beta\gamma c\tau\gg L_{xy}^{+}>|\vec{x}|/(1+\sinh^{2}\eta)$, we
can approximate $e^{-|\vec{x}|/\beta\gamma c\tau}\approx 1$ which implies that
the dependence on $c\tau$ factorizes from the efficiency. In other words

$\epsilon(c\tau)\approx\epsilon(c\tau^{\prime})\times\frac{c\tau^{\prime}}{c\tau}\quad\text{if}\quad\beta\gamma
c\tau^{\prime},\beta\gamma c\tau\gg L_{xy}^{+}.$ (6)

This is the reason that the limits from _all_ searches for a single displaced
vertex scale as $1/c\tau$ in the large lifetime limit, _regardless of the
detector geometry and analysis details_. Similarly, one can show that in the
long lifetime regime the sensitivity for searches for two independent
displaced particles should always scale as $1/(c\tau)^{2}$. It moreover
disproves the following common misconception:  If the detector is close to the
interaction point (IP), it is exponentially difficult to detect LLPs with
large $c\tau$. Placing the detector further away therefore leads to a higher
signal efficiency. Indeed, equation (6) shows that it is only _linearly_
difficult to detect LLPs with long lifetimes. This is why some LLP searches
can be sensitive to $c\tau$ values as large as a kilometer [48]. The scaling
law in (6) is moreover independent of the detector location, provided that the
typical lab frame lifetime ($\beta\gamma c\tau$) is much larger than the
distance between the detector and the IP. For low $c\tau$ on the other hand,
we saw that the efficiency depends exponentially on the distance between the
detector and the IP, such that _a detector as close as possible to the IP
always covers the largest range in $c\tau$._ Of course, in practice there may
be engineering and/or background considerations which can force one to place
detectors further out, as discussed in Sec. 5 and Sec. 6.

###  4.2 Triggering on long-lived particles

The trigger infrastructure at all LHC experiments is both a critical and very
complex component of the experiment. It is responsible for reducing the data
intake from 40 MHz sampling frequency down to roughly a kHz, all in very
limited time with finite read-out and computing resources. As a result, it is
in constant development, through system upgrades, improved algorithms and
calibrations, as well as shifting physics priorities. All of this especially
affects searches for LLPs, as the event reconstruction is much more complex
than for prompt particles. It is therefore important to recognize the
trigger’s capabilities and limitations before starting an analysis or theory
study.

The trigger at ATLAS and CMS consists of two stages: The “Level 1” (L1 or
LVL1) or “hardware” trigger and the “High Level Trigger” or “software” trigger
(HLT). The L1 trigger is responsible for reducing the rate from 40 MHz to
maximum 100 kHz, using limited detector information. In particular, there is
no time to read out and process the inner tracker data at this stage, and the
calorimeters and muon systems are read out at reduced granularity. The L1
trigger therefore makes a decision based on relatively simple observables such
as missing transverse energy (MET), the scalar sum of all reconstructed
transverse energy ($H_{T}$), a high-momentum lepton, etc. Events passing the
L1 selection are handed over to the HLT, which reduces the rate further from
about 100 kHz to the order of 1 kHz. It can perform a fairly faithful
reconstruction of the full event, including tracking information. This allows
for more sophisticated selection criteria and there are thus many more trigger
paths. Nevertheless, the HLT is currently limited to an average latency of
roughly 250 ms, which means that a very resource-intensive task such as
reconstructing displaced tracks cannot be taken for granted. A good example is
that of a purely hadronic displaced vertex in the tracker, for which displaced
tracks currently cannot be efficiently reconstructed at HLT. When combined
with a moderate $H_{T}$ requirement, it is however possible to trigger on a
jet with an anomalously low number of prompt tracks [25]. If such a prompt-
veto strategy is not viable, then one must rely on the traditional MET,
$H_{T}$, lepton, etc. triggers, depending on the signature sought.

Because of these complications, it could be tempting to conclude the
following:  You cannot trigger on LLPs, so you always need to use MET or an
associated hard lepton etc. As we will see, this is however not (anymore) the
case, as there have been many exciting new developments where dedicated
triggers are being developed for LLPs. A nice example is that of displaced
muons, for which the muon system can reconstruct muons which do not point to
the beamline and/or do not have a matching track in the detector [59, 60]. The
absence of an inner detector track in particular implies that the momentum
resolution for displaced muons at the HLT is substantially worse than for
prompt muons.

The ATLAS calorimeter ratio trigger is another successful example [61]: It
takes advantage of the fact that for an LLP decay in the hadronic calorimeter,
the ratio of the energy deposited in the ECAL over that in the HCAL is much
lower than that for a typical jet. This trigger has good efficiency for the
range of $L_{xy}$ that corresponds to the HCAL extent, as is nicely
illustrated in the left-hand panel of Fig. 3.777Fig. 3 is an excellent example
of a “high value” plot for theorists seeking to understand or reinterpret an
analysis: It presents a very important but detector-specific quantity, the
trigger efficiency, in terms of a model-independent, truth-level variable
(truth-level $L_{xy}$), which can easily be simulated by theorists using the
recipe in Sec 4.1. An analogous plot in terms of $c\tau$ would have been much
less useful, as one would need to reproduce the exact simulation settings of
the collaboration to unfold the $L_{xy}$ dependence of the efficiency, a
process which is prone to potential pitfalls. Another very interesting option
is the possibility to trigger on LLPs decaying in the muon system, mainly
because the muon systems have a large fiducial volume, are well shielded by
the hadronic calorimeter and are well suited to implement L1 trigger
strategies. ATLAS already has such a trigger in place [62, 49], while CMS will
during Run-3 [63]. A more complete overview of existing and upcoming trigger
strategies dedicated to LLPs can be found in the recent LLP working group
report [63].

Figure 3: Left: Trigger efficiency as a function of the truth-level $L_{xy}$
for the calorimeter ratio trigger at ATLAS [61]. Right: Inclusive dimuon
spectra for multiple $L_{xy}$ bins and $p_{T}^{\mu}>3$ GeV, as measured in the
CMS dimuon scouting analysis [27]. Displaced $\omega/\rho$, $\phi$ and
$J/\psi$ mesons can be produced in the decays of boosted $B$ hadrons, hence
the persistence of those resonance peaks in the high $L_{xy}$ bins. The
$K_{S}$ resonance peak is due to $K_{S}\to\pi^{+}\pi^{-}$, where both pions
were misidentified as muons.

In some cases it is possible to circumvent the normal bandwidth limitations by
only committing a reduced event format to storage, at high rate. CMS and LHCb
recently employed this “data scouting” technique (called “Trigger-Level
Analysis” in ATLAS and “Turbo Stream” at LHCb) to search for displaced dimuon
pairs with very low $p_{T}$ threshold [27, 64, 65], as shown in the righthand
panel of Fig. 3. For Run-3, LHCb moreover eliminates their hardware trigger
entirely, such that the data scouting concept can in principle be applied to
their entire dataset [66]. This is expected to be a major boost to LHCb’s
sensitivity for low mass LLPs [67].

In addition to the trigger’s computing farms, the experiments also have
extensive computing infrastructure which promptly performs the more precise
(and more demanding) offline reconstruction for all the events that pass the
trigger (see Sec. 4.3). There are certain cases where a higher trigger rate is
desired to be recorded than what this offline reconstruction can keep up with.
To achieve this, a fraction of the data can be “parked” in raw format until
more computing resources are available, e.g. during LHC downtime. As an
interesting example, in 2018 the CMS experiment registered a unique dataset of
about $1.2\times 10^{10}$ events containing a soft, displaced muon, the
signature of a semi-leptonic $B$-hadron decay [68]. This data was recorded
during the end of LHC fills, when the collision rates and thus the demand on
the trigger are lower. Several LLP analyses are ongoing using this dataset, as
it is indeed a very interesting opportunity for low-mass LLP searches, by
searching for new physics in the decays of the about $10^{10}$ inclusive
$B$-hadron decays in this sample.

Unlocking the full potential of the data scouting and parking techniques is
still an actively developing and exciting area of research during Run-3, as
well as towards the HL-LHC. In general, for the HL-LHC, the detectors will be
upgraded with capabilities that will also vastly boost the prospects to
trigger on LLPs: Track reconstruction in the Level-1 hardware trigger; ultra-
precise timing with dedicated timing layers in front of the calorimeters;
impressively improved calorimetry in terms of e.g. depth segmentation; etc.
[54, 55]. As was the case for Run-1 and Run-2, we fully expect that the
ingenuity of the analysis teams will leverage these new hardware capabilities
into often unexpected sensitivity gains for LLPs.

###  4.3 Offline signal reconstruction and selection

Once data events are selected by the trigger and the raw data saved to
storage, a more detailed offline event reconstruction takes place, including
more precise calibrations and using algorithms with higher complexity and thus
more time-consuming than what is possible online. This reconstruction starts
from the electronic signals, local hit reconstruction, clustering in higher-
level objects such as tracks and jets, and high-level object identification
algorithms such as $b$-tagging. While standard reconstruction is described in
detail in numerous references on the detectors and their physics objects,
several of the reconstruction steps have interesting features that apply
specifically to long-lived particles.

Displaced tracks:

The arguably most important impact on LLP searches arises from reconstruction
of the tracks of displaced charged particles. As mentioned before,
optimization of the reconstruction of displaced tracks is a trade-off between
efficiency and purity, and in practice it is limited by the available
computing resources. The reason is the huge number of hits per event in the
inner tracker, which makes track reconstruction a major combinatorical
challenge. In the standard track reconstruction, one requires each track to
have hits on the innermost layers, which have the best spatial resolution, and
to originate from near the beamline. These conditions provide an enormous
reduction of the number of possibilities and therefore the complexity of the
computational task.

After the prompt tracks are completed, further reconstruction of displaced
tracks is attempted with the remaining, unused hits in the tracker. These
unused hits are largely from particles produced in nuclear interactions of
primary tracks in the detector material, and from particles that were too soft
to be considered in the standard reconstruction. For the LHC’s current pileup
conditions, those unused hits still range in the several thousands per event.
The further reconstruction of displaced tracks with looser constraints, in
particular on $d_{xy}$, thus still brings a high level of combinatorial
complexity, which in addition scales non-linearly with the amount of pileup.
As a consequence, the reconstruction of displaced tracks can only be partially
efficient, to avoid picking up too many “fake” tracks from nuclear
interactions or from combining unrelated hits into a displaced track. Such
fake tracks can be a background in certain LLP searches, as is discussed in
Section 5.3.

Concretely, the inner parts of the ATLAS and CMS detectors are equipped with
several layers of high-precision silicon pixel detectors, starting from a
transverse distance to the beam of about $3\,\text{cm}$, and yielding
typically at least three precise measurements for tracks emerging at a
transverse displacement of about $10\,\text{cm}$ [69, 70]. Current offline
reconstruction algorithms operate at nearly full efficiency for tracks
emerging before the first pixel layer – essential for their superb $B$-hadron
identification – and still show above 50% efficiency for tracks produced just
outside the pixel detector outer radius in LHC Run-2 pileup conditions with
$p_{\text{T}}\gtrsim 1\,\text{GeV}$. For larger transverse displacements, an
efficiency of about 40% is still achieved at a track production radius of
$30\,\text{cm}$ [21, 71], as shown in Fig. 4. For LHCb, the inner VELO
detector [72] is specifically constructed to be highly efficient for forward
displaced charged particles arising in the VELO detector [73]. During Run 3,
LHCb will moreover have access to a new class of highly displaced tracks (“T
tracks”) [74], which will enhance their signal efficiency for LLPs with larger
$c\tau$ in particular.

Figure 4: Left: Track reconstruction efficiency in $t\bar{t}$ events as a
function of simulated track production vertex radius (corresponding to
$L_{xy}$ in Figure 2) for the CMS tracker in its 2017 data-taking
configuration. The various colors represent the successive iterations in the
reconstruction algorithm, where ever loser criteria are applied to increase
the efficiency [75]. The green histograms represent tracks reconstructed with
algorithms specialized for displaced tracks. Right: Track reconstruction
efficiency in ATLAS for displaced charged hadrons produced by the decay of
long-lived gluino $R$-hadrons, as a function of the displaced decay radius
($r_{\text{prod}}$, corresponding to $L_{xy}$ in Figure 2) [71]. The additive
efficiencies of the standard and dedicated displaced tracking algorithms are
shown.

The performance calibration of displaced tracking in data is a challenge. The
SM $K_{S}$ and $\Lambda$ hadrons, commonly referred to as $V^{0}$’s, are
luckily well modeled in simulation [76, 77, 78], and can thus be used as a
source of displaced tracks on which displaced tracking performance can be
tested.

Short tracks:

Short tracks are of great interest as an experimental signature of displaced
decays of new, charged particles. Often, such particles decay into a nearly
mass-degenerate neutral particle that escapes detection, along with a very
soft track [79, 80], or may continue into another charged track with a kinked
signature [81].

In the former case, to reduce backgrounds, a track reconstruction is employed
that—unlike standard tracking that is more permissive—requires all consecutive
hits on the track to be recorded, with the absence of hits on several outer
layers confirming the track has stopped in combination with a quiet
calorimetric environment [82, 83]. The shortest tracks thus reconstructed are
promptly produced and have four consecutive hits in the pixel detector. A
strong confirmation of such a signal would come from a matching of a soft pion
from the short-track’s endpoint. However, such a pion would not only appear
displaced, it would also circle on a helix in the tracker detector due to its
low momentum. This reconstruction is in principle possible, but a costly
investment. At this time it is unnecessary, since the background can be
suppressed sufficiently in other ways.

Regarding kinked tracks, there is currently no dedicated analysis for this
signature. That said, the existing disappearing track searches likely have
excellent sensitivity already [84]. This is because the outer part of the
kinked track typically fails to be reconstructed, such that the signature is
effectively identical to that of a disappearing track.

Displaced muons:

For muons with a matched inner tracker track, the efficiency is set by the
inner track reconstruction, as described above. Muons can also be
reconstructed up to very large displacements beyond the inner tracker using
the muon system only [60, 85], albeit with a lower position and momentum
resolution. This can be done very efficiently up to track transverse
displacements as large as $400\,\mathrm{cm}$, imposing quality requirements
that suppress backgrounds from hadrons punching through the calorimeters and
other non-beam backgrounds (see Section 5). For these muons, it is important
to not use the collision vertex as a constraint, as it biases the
reconstructed momentum to lower values, and induces inefficiencies at large
impact parameter values.

Displaced electrons, taus and photons:

For displaced electrons originating in the first part of the tracker, the
efficiency will mostly mirror the displaced track reconstruction performance.
The reconstruction efficiency of such displaced electrons can be estimated in
data by looking at photon conversions in the detector material [86]. If its
track is too displaced to be reconstructed but emerges before the EM
calorimeter, the electron can still be reconstructed as a photon. While this
comes at the cost of a higher background, it is particularly useful in the
trigger [87, 88] prior to a more detailed offline selection.

Genuine photons from LLP decays benefit from their own dedicated treatment,
with adapted identification requirements on timing or direction [46, 47]. Also
tau leptons emerging from displaced decays are an interesting target, either
as a displaced electron or muon, or a displaced jet-like signature. Dedicated
displaced hadronic tau identification is the most complex final state, which
we expect to see develop strongly during LHC Run-3.

Displaced vertices:

Once the displaced tracks have been reconstructed, the reconstruction of their
corresponding displaced vertex is virtually 100% efficient. Nevertheless, in
cases where the tracking efficiency is reasonably inefficient, it may still be
better to forgo vertex reconstruction, as the overall efficiency will scale as
the tracking efficiency raised to the number of tracks required to reconstruct
the vertex. In particular, if one expects multiple displaced decays in the
event, it may therefore be more beneficial to ask for a number of displaced
tracks, without insisting that they belong to a single vertex [89].

Exotic objects:

In the above, we discussed aspects of offline reconstruction of the most
common final state objects. More specialized reconstruction furthermore
targets specific experimental signatures, some of which, like showers in the
muon system, were touched upon in Section 3.3. The potential for future
development of very advanced tracking for exotic tracks from monopoles is also
exciting, as they are bent along the magnetic field direction. Quirks
oscillating in pairs when traversing the detector [90, 91] are another
example.

A recent avenue of significant progress is the development of more advanced
techniques which aggregate various sources of experimental input into a
multivariate discriminator. This can significantly boost the sensitivity of a
search [92, 93]. The potential downside of less detailed control of a
selection cut can be offset by using the neural network as a discriminator to
select physics objects, coupled with more robust background predictions from
the data. The simulation description of the multivariate discriminator, on the
other hand, is difficult to assess because of the limited types of control
samples with genuine displaced tracks available. Here, techniques like domain
adaptation [92] and adversarial networks [93] alleviate this challenge.

As ever more advanced novel reconstruction algorithms are developed, also
lower-level detector features come in focus. This is at odds with the
increasing computing complexity, which requires one to use as high-level and
thus compact objects as possible. Advanced offline reconstruction thus
benefits from being integrated in the standard reconstruction chain (see e.g.
Ref. [94] for the integration of ATLAS Large Radius Tracking into the main
reconstruction from Run-3 onwards), or must run on dedicated data streams
where low-level objects are being kept availabe for such dedicated analysis
purposes. Either way, such efforts require sometimes heroic, long-term
investments.

##  5 Backgrounds

Searches for LLPs make use of non-standard experimental signatures in both
online and offline selection to bring the prompt backgrounds down dramatically
at a modest cost in signal efficiency. This has led to the following, general
assumption in many LLP theory studies  LLP searches are background free.
Though in many analyses the backgrounds can indeed be reduced to be
negligible, this is only possible after extensive and subtle analysis efforts
on the experimental side. Even after such extensive background reduction
efforts, often unusual irreducible backgrounds from instrumental, algorithmic,
or other origins remain. In the following, we review various sources of
backgrounds in searches for LLPs and some methods employed to eliminate them.
Often, simulations are not reliable or robust enough to estimate the remaining
background, even if very small, and data-driven techniques are needed to
quantify them reliably.

###  5.1 Standard Model long-lived particles

Charm and bottom flavored hadrons are the most ubiquitous SM background for
LLP searches, as they can easily produce a multi-track displaced vertex. A
priori they can be removed effectively with a cut on the vertex mass of
$\gtrsim 5$ GeV (see e.g. [95]), though this severely limits the sensitivity
to low mass LLPs. Their proper decay length is also only $\mathcal{O}(0.1)$
mm, such that cuts on $d_{xy}$ and/or $L_{xy}$ are also very effective, see
e.g. [27, 96] and the right-hand panel of Fig. 3. One must be aware however
that there are still many events with highly boosted $b$ or $c$-jets, which
could still leak into the signal region in some cases. When using simulation
to estimate these backgrounds, it is therefore essential to use
$p_{T}$-weighted events. Whenever possible it is moreover advisable to also
reweight the events in terms of the long-lived meson’s decay vertex location,
as described in Sec. 4.1.

The charged pions have a proper decay length of 780 m, and the vast majority
of them therefore reaches the calorimeter without decaying. Bearing in mind
Sec. 4.1 however, we can ballpark the probability for a $\pi^{\pm}$ to decay
in the tracker. If we suppose a $\pi^{\pm}$ with $\beta\gamma\sim 10$, we find
$1\,\text{m}/(10\times 780\,\text{m})\approx 10^{-4}$. This seems very small,
but we must keep in mind that _every_ collision produces tens to hundreds of
$\pi^{\pm}$, though most of them are soft. Multiplying this with the
$\mathcal{O}(100)$ for the number of pileup collisions per event, we see that
an $\mathcal{O}(1)$ fraction of all events will have a $\pi^{\pm}$ which
decayed “early”. Similar considerations apply for the $K^{\pm}$, $K_{L}$ and
especially the $K_{S}$. Both pions and charged kaons are therefore a huge
potential source of individual displaced muons in particular. Fortunately,
this background drops quite rapidly if isolation is imposed and if the $p_{T}$
cut on the muon is tightened. Kaons can moreover produce a genuine displaced
vertex. For example, the right-hand panel of Fig. 3 shows a clear peak in the
kaon mass range, which is due to $K_{S}\to\pi^{+}\pi^{-}$, where both pions
are misidentified as muons. In general such displaced vertices can easily be
eliminated by requiring vertex mass well above the kaon mass and/or demand a
larger number of tracks.

Jets with an anomalously low number of tracks are an interesting handle for
displaced decays near the back of the tracker or in the calorimeter [93] or
for strongly interacting dark matter candidates [97]. A priori, QCD can
produce jets primarily with $K_{L}$ and neutrons, which don’t leave tracks but
do deposit energy in the HCAL. While the corresponding probability per jet is
very low, this background can nevertheless be important due to the huge QCD
multi-jet cross section.

For all the above backgrounds, the most important point is perhaps to be very
mindful of the limitations of all simulation codes, as they are not designed
to model very rare effects in very specific corners of phase space. This can
to some extent be offset with the weighting procedures described in Sec. 4.1,
but more often than not a data-driven validation is needed.

###  5.2 Material interactions

The details of the detector material, its density and geometry are critical
for LLP searches, as it both reduces and generates more backgrounds. First,
the trackers are always designed to have as a little material as possible, as
particles scattering in sensors or support structures adversely affect the
momentum resolution of the tracks. This design driver is also helpful for LLP
searches, as inelastic collisions or photon conversions in detector material
can produce secondary vertices. This is _not_ a rare effect by any means:
About 5% of all $\pi^{\pm}$ with $p_{T}\gtrsim 5$ GeV create a secondary
vertex within the CMS inner tracker [98]. Extrapolated to HL-LHC conditions,
this implies secondary vertices at a rate of $\sim 30$ MHz, essentially in
every event! Fortunately, this background is very strongly dependent on the
track $p_{T}$, the vertex mass and the number of tracks per vertex [98]. The
exquisite vertex resolutions of their trackers moreover allow ATLAS, CMS and
LHCb to make beautiful maps such as Fig. 5, pinpointing precisely where the
vertices are produced. This can then be interpreted as a radiography of the
detector material and be used as a veto map to suppresses backgrounds
extremely efficiently in a real LLP analysis. To avoid masking too much of the
detector volume, which reduces signal efficiency, further track and vertex
selections can be made to improve the vertex resolution, e.g. avoiding
vertices from very collimated tracks. Kinematic cuts may also prove useful
against SM LLPs produced in material interactions, which may decay in the
unmasked region.

Figure 5: Left: Number of reconstructed secondary vertices in the ATLAS inner
tracker [24, 23]. Right: Same, for the LHCb Vertex Locator (VELO) [99].

Unlike the trackers, the calorimeters are designed to stop as many particles
as possible. Especially the hadronic calorimeters are made out of heavy
elements and are made as thick as the engineering constraints allow. For
example, the ATLAS HCAL in the barrel is made up of $\sim$ 2 m of steel plus
scintillator tiles. At $\eta\approx 0$, this corresponds to roughly 7.4
nuclear interaction lengths or 9.7 nuclear interaction lengths when we include
the liquid argon ECAL. We can therefore estimate the probability for a hard
$\pi^{\pm}$ to punch through the calorimeter into the muon system to be
$e^{-9.7}\approx 5\times 10^{-5}$. This is small, but not very small, given
the huge number of $\pi^{\pm}$ that hit the calorimeter during a typical run.
This means that large numbers of hadrons still punch through the calorimeters
into the muon chambers, where they may fake a muon or a displaced vertex.
Fortunately, hard hadrons typically live inside a jet, and the calorimeter
itself is therefore a very effective veto for such “punch-through” events.
This background cannot be simulated faithfully and must be estimated from
data. A search along these lines was first performed by ATLAS, using a
dedicated LLP trigger on activity in the muon system [100, 49]. The CMS
calorimeter is significantly thinner, and thus a priori suffers from more
punch-through backgrounds. CMS can however offset this by using some of the
steel in the return yoke of its magnet as an additional shield [48].

###  5.3 Fake tracks and vertices

While material interactions inject real displaced particles in the detector,
tracks may also be reconstructed that do not arise from a real particle. Fake
hits from detector noise, or more likely hits from the cloud of thousands that
come from the pileup particles in the inner detectors may align up to form a
track. For example, assuming on average 50 pileup vertices, the four innermost
tracking layers in the ATLAS barrel collect on average about $\sim 2\times
10^{4}$ hits per bunch crossing [101]. Roughly 25% of these hits will be
associated with reconstructed, prompt tracks, leaving several thousands of
unassociated hits in each event. While it is very unlikely that such unrelated
hits will accidentally line up and have a good fit quality for the hypothesis
of prompt production, this becomes more likely for displaced tracks and when
only a handful of hits are required, as e.g. in searches for disappearing
tracks [82, 83], or in cases of displaced muon reconstruction [85, 60].

Fake vertices are also common, and can be composed out of a set of real but
unrelated tracks. This frequently happens when a vertex from muons with very
large displacements is reconstructed, or vertices with low displaced track
multiplicity are considered in the inner detector. A particular source of
fakes arises from overlapping tracks from different pileup vertices. While in
this particular case, such fakes can be identified from the tracks lining up
in a plane with the beamline, more generally fake vertices may be rejected by
vetoing hits on the tracks upstream of the vertex.

Backgrounds from fake tracks and vertices are best measured from data, though
there have been attempts to estimate them in phenomenological studies [102,
103, 104]. In general they must however be extracted from data, and caution is
needed especially when a very high degree of background rejection is required.

###  5.4 External particle backgrounds

The above sources of background relate directly or indirectly to the proton
collisions. There are however a few additional sources of background that may
enter the signal phase space of long-lived particle searches.

Beam halo

As the protons travel through the LHC in dense bunches, some stray too far
from their ideal trajectory, posing problems to the sensitive equipment of the
LHC and the experiments. Dedicated absorbers clean the proton beam of such
beam halo, during which beam halo muons may be produced that travel along with
the proton bunch parallel to the beam and may traverse the thick shielding in
front of the detectors unhampered. The rate of such muons decreases strongly
as a function of the radial distance to the beam [105].

These muons may create some harmless background hits in the tracking
detectors, but in the calorimeters and muon spectrometer they can lead to
unusual backgrounds. Since the muons do not originate from the proton
collisions, their timing is asynchronous, though in a predictable manner. They
may leave straight muon tracks in the forward muon tracker, as well as
significant energy deposits in the calorimeters. The parallel direction of the
muon tracks, the potential match of these tracks with calorimeter deposits,
the anomalous shapes of such deposits parallel to the beamline, depth
information, and the early or late timing of these signatures may be used to
suppress this background. The azimuthal $\phi$ distribution is another
interesting handle: The beam halo rate spikes in the horizontal plane and is
smallest near the bottom of the detector, as the floor of the LHC tunnel acts
as a shield [105, 106].

Cosmic muons

A fraction of the relativistic muons that are created in cosmic ray showers
reach underground and cross the detectors downward with a rate of roughly
$1\,\mathrm{Hz/m^{2}}$ [105]. Their energy spectrum falls exponentially and
they exhibit a significant spread in direction and a geometric asymmetry,
mainly due to the access shafts above the detectors. Only a small fraction is
reconstructed as a track or a muon, since the cosmic muon only rarely has a
direction that is compatible with the constraints of the track reconstruction
algorithms. In addition, the arrival time of the muon is random, and thus the
timing of the energy deposits in the various subdetectors, including
calorimeters, is not syncronized with the collision it may have an overlap
with. For downgoing tracks in the upper half of the detector, the travel
direction is even opposite of what is expected for particles originating from
collisions. Given these features of cosmic muons, they are only a potential
background in LLP searches, where anomalous timing or track displacement are
selected, see e.g. [60, 107].

Satellite collisions

Though the LHC beam delivers its main bunches in 25 ns intervals, the beam
inevitably also has “satellite bunches”, which follow and precede each main
bunch with a 2.5 ns time gap. These satellite bunches contain about $10^{-5}$
times the number of protons of the main bunches and their collisions can
generate very rare, out-of-time backgrounds. Such backgrounds arrive with
well-defined $5\,\mathrm{ns}$ delays as compared to the primary collision and
can be important particularly for analyses focusing on delayed calorimeter
signals, e.g. [43].

##  6 Dedicated detectors

As we have seen, ATLAS, CMS and LHCb are excellent detectors to hunt for long-
lived particles. There are however a couple important cases where dedicated
long-lived particle detectors can perform qualitatively better. Concretely,
there are three reasons to consider auxiliary detectors at relatively large
distances from the interaction point: _(1)_ to catch LLPs with very forward
kinematics, which are inaccessible to the main detectors, _(2)_ to allow for a
different detector technology and _(3)_ to allow room for additional
shielding. All proposals rely on _(3)_ to some extent.

The FASER experiment [108] makes use of a small service tunnel, 480 m forward
of the ATLAS interaction point. Its extreme forward location allows it to
search for LLPs produced through the huge pion flux that goes down the beam
pipe with each collision. A larger version of FASER would need to be housed in
a proposed, dedicated cavern, the forward physics facility (FPF) [109]. The
FACET experiment [110] would follow a similar philosophy and would be located
roughly 100 m forward from CMS. It would cover a somewhat larger pseudo-
rapidity range than FASER, though beam backgrounds may be more challenging to
mitigate. Finally, the SND$@$LHC detector [111] is meant to detect high
energy, forward neutrinos from the ATLAS interaction point, but may also have
sensitivity to some low mass LLPs [112]. When modeling the acceptance of
forward LLP detectors, it is important to remember that standard simulation
tools such as PYTHIA and MadGraph are typically not appropriate in this
kinematical regime. Instead specialized tools are needed, see e.g. [113].

The MoEDAL [114] and milliQan [115] experiments rely on alternative detector
designs, specialized to stable LLPs which leave anomalous tracks. MoEDAL has
searched for magnetic monopoles which get trapped in a set of aluminum rods,
which were scanned for monopoles with a SQUID sensor [116]. The MoEDAL
collaboration also aims to install an extension which can look for decaying
LLPs at intermediate rapidities [117]. Similar to FASER and SND$@$LHC, the
milliQan experiment is also housed in an existing service tunnel, but at
moderate rapidity above CMS. It is shielded by 17m of rock and designed to
detect fractionally charged particles. It can do so by looking for coincident
hits in 4 aligned bars of plastic scintillator. Currently, a first modular
phase of the detector is under construction [118], with sensitivity opening up
a large new phase space during LHC Run-3 and HL-LHC.

Finally, proposed experiments such as CODEX-b [119, 120] and MATHUSLA [121,
122] would look for displaced vertices in a similar manner as the ATLAS and
CMS muon chambers, but would come with a much thicker shield between the
detector and interaction point. This results in much lower backgrounds, at a
cost in geometric acceptance. MATHUSLA would be constructed on the surface
above CMS, using the rock as shielding. CODEX-b would be installed directly in
the LHCb cavern and a suitable shield would need to be constructed. In either
case the shield must be equipped with an active muon veto to confidently
reject muon-induced secondaries as potential backgrounds [119, 120, 123].

##  7 Closing thoughts

The ability to reconstruct and characterize displaced particle decays has
historically delivered many discoveries and continues to be an essential tool
for particle identification today. It is moreover a misconception that heavy,
beyond the SM particles are supposed to decay promptly: Not only do there
exist many counterexamples, the general conditions for macroscopic lifetimes
are also very simple and likely generic, given our experience with the SM
itself. Searches for displaced signatures moreover benefit from a number of
experimental handles which are unavailable for promptly decaying new particles
and can therefore often maintain very low backgrounds, even with very high
integrated luminosity. Long-lived particles are therefore one of the primary
areas where we may achieve a major discovery at HL-LHC.

Searches for long-lived particles are also fascinating because they are
complicated, subtle and often messy. The signal efficiency depends on the
details of the detector design, trigger capabilities and limitations, as well
as the available offline reconstruction methods. The backgrounds are delicate
and sometimes impossible to simulate; understanding them requires careful and
clever estimation techniques. The detector capabilities moreover continue to
evolve, especially with the upcoming HL-LHC upgrades, all while new trigger
strategies are constantly added, unlocking qualitatively new searches. All of
this means that there is still a lot of room for new ideas which may impact
the direction of the ongoing and future search programs.

The flip-side is that the up-front investment can seem very high for those
seeking to get started in this field, especially as a good deal of the
essential knowledge is often either unwritten or buried deep in fist-thick
technical design reports. In this review we have attempted to collect some of
this knowledge in a manner that aims to be accessible to novices, along with
some practical suggestions. We conclude with a few further words of advice for
both beginning theorists and experimentalists. In particular for theorists,

  * •

It pays off to learn how to read technical design reports and performance
papers. While they may seem daunting at first, many are structured in a
similar manner, and after a while you would be surprised how quickly you can
mine them for the trigger and/or reconstruction efficiencies you need.

  * •

Be realistic when it comes to your ability to accurately model complicated
backgrounds. Rather than producing overly aggressive or overly conservative
(projected) limits, consider plotting signal yields only, and let
experimentalists follow up with a full analysis.

  * •

Consult extensively with experimental colleagues, either by collaborating
directly or by reaching out to the authors of the analysis you are studying.
Figures may rely on important assumptions which may not be obvious from the
published material. The conveners of the analysis groups can put you in touch
with the main analysis authors.

For experimentalists, we suggest to keep the following in mind:

  * •

Connect with the experts in the individual subdetectors, who are often eager
to understand together low-level issues in the detector data or simulation.
Also share your challenges and progress regularly with your colleagues, as
similar problems may find solutions in very different contexts.

  * •

Plan ahead any special trigger or computing needs, including data storage and
access. Also prepare to release publicly efficiencies, cut-flow tables, etc,
to allow theorists to recast your analysis, which greatly amplifies its impact
(see e.g. [27]).

  * •

Keep abreast of the phenomenology literature and community. Theorists are
generally happy to connect and follow up on their work. This may even lead to
fruitful common research.

Long-lived particles provide a compelling window on physics beyond the SM, and
a rich research arena for both experimentalists and theorists. With this
review, we hope to have conveyed this excitement, with links, tools and advice
that may help new physicists make the leap towards long-lived particles.

## DISCLOSURE STATEMENT

The authors are not aware of any affiliations, memberships, funding, or
financial holdings that might be perceived as affecting the objectivity of
this review.

## ACKNOWLEDGMENTS

We would like to sincerely thank Dean Robinson, Simone Pagan Griso, Jessie
Shelton, Matthew Citron, Daniele Del Re and Brian Shuve for their very
valuable inputs to this review. We are particularly grateful to Jessie
Shelton, Federico Leo Redi and Alberto Escalante del Valle for providing
comments on the manuscript. The work of SK was supported by the U.S.
Department of Energy, Office of Science under contract DE-AC02-05CH11231. Part
of this work was performed at the Aspen Center for Physics, which is supported
by National Science Foundation grant PHY-1607611. SL would like to acknowledge
2021 sabbatical leave support from Vrije Universiteit Brussel and support of
FWO Vlaanderen under the “Excellence of Science – EOS” – be.h project n.
3082081.

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