∎

<EMAIL_ADDRESS>
11institutetext: Université de Strasbourg, CNRS, IPHC UMR7178, F-67000
Strasbourg, France 22institutetext: Université de Haute-Alsace, 68100
Mulhouse, France 33institutetext: University of Tennessee, Knoxville,
Tennessee 37996, USA

# Probing displaced top quark signature at the LHC Run 3

Jeremy Andrea addr1  0000-0002-8298-7560 Daniel Bloch addr1
0000-0002-4535-5273 Éric Conte addr1,addr2  0000-0003-3154-3123 Douja
Darejaddr1  Robin Ducrocq e1,addr1  0000-0002-1245-7257 Emery Nibigira addr3
0000-0001-5821-291X

(Received: date / Accepted: date)

###### Abstract

In the context of prospective studies for searches of new physics at the LHC
Run 3, this paper investigates the relevance of using top quarks produced from
new long-lived particles, and detected in the tracker volume of the ATLAS and
CMS experiments. Such a signature, referred to as displaced top quarks, leads
to final states containing displaced vertices and a high multiplicity of
displaced jets and tracks, thanks to the top quark decays. Therefore, it is a
possible powerful tool for searching for new long-lived particles. Three
simplified models based on supersymmetry are explicitly designed for the study
of this signature. They differ according to the nature of the long-lived heavy
particle which produces at least one top quark: electrically neutral or
charged, coloured or non-coloured long-lived particle. For each model, a wide
region of parameter space, consistent with a reasonable number of displaced
top quarks decaying in a typical tracker volume has been probed. From this
study, promising benchmarks are defined and experimental guidelines are
suggested.

###### Keywords:

long-lived particle displaced top quark LHC Run 3 benchmark points

††journal: Eur. Phys. J. C

##  1 Introduction

After a long shutdown for technical maintenance and improvement, the LHC
Evans:2008zzb  Run 3 officially started the 5th of July 2022 and currently
provides proton-proton collisions with an energy in the center-of-mass of 13.6
TeV (14 TeV in the future). Experimentalists explore the TeV physics through
data acquired by the ATLAS ATLAS:2008xda , CMS CMS:2008xjf  and LHCb
LHCb:2008vvz  detectors. The research program is extremely rich and two
particular topics, relevant to this paper, can be highlighted:

  * $\bullet$

The top quark physics relies on the measurement of the production, the decays,
and the properties of the top quark Wuchterl:2022iwh . Its high mass, and the
fact that its decay occurs before its hadronisation, confers to the top quark
a unique place in the Standard Model (SM) of particle physics. Thus, it is
experimentally relevant to use the top quark to explore several BSM (Beyond
Standard Model) physics scenarios involving new heavy particles that couple
preferentially to the top quark.

  * $\bullet$

Among the potential BSM experimental signatures, several recent investigations
focus on exotic particles with a lifetime long enough to allow them to decay
beyond the first layer of detection Alimena:2019zri . While these event
topologies benefit from less background sources, the reconstruction is usually
difficult with traditional algorithms. It can lead to different kinds of
reconstructed signatures such as displaced leptons ATLAS:2020wjh ; CMS:2021kdm
, displaced jets CMS:2020iwv ; ATLAS:2022zhj , displaced vertices
ATLAS:2019fwx ; Bondarenko:2019tss  or disappearing tracks CMS:2020atg ;
ATLAS:2022rme . Analyses devoted to the search of Heavy Stable Charged
Particles (HSCP) can also be considered if the long-lived particle decays
outside the entire volume of the detector Khachatryan_2016 ;
https://doi.org/10.48550/arxiv.2205.06013 .

This paper combines these two topics (top quark physics and long-lived
particles) and proposes the investigation of scenarios at LHC Run 3 with
$\sqrt{s}=14\ \text{TeV}$ where an exotic heavy long-lived particle decays
into at least one top quark. Such a scenario benefits from the possibility to
use trajectories of charged particles (tracks), to reconstruct and identify
displaced vertices. In our study, the displacement of the top quark decay
vertex with respect to the primary vertex must be consistent with the (ATLAS
or CMS) tracker volume. Therefore, we take under consideration scenarios for
which the long-lived particles decay within the tracker volume. In this case,
the tracking algorithm must reconstruct tracks with missing hits in the first
layers of the tracker detector. If the displacement in the transverse plane to
the beam axis is lower than about $4~{}\textrm{cm}$111The first tracker
detection layer is located at 3.3 cm for the ATLAS experiment ATL-INDET-
PUB-2016-001  (2.9 cm for the CMS experiment Adam_2021 ) away from the center
of the detector., the good quality tracks coming from the top decay
(electrically charged) final states should have been reconstructed by
considering all the hits in the first tracker detection layer. If it is even
smaller, the SM top quark production becomes an important and irreducible
background for a potential analysis. At the opposite, if the transverse
displacement is above $1~{}\textrm{m}$, the particle decays outside the
tracker. The present signature will be referred to as displaced top quark in
the following. As top quarks are produced only at the primary vertex in the
SM, the observation of displaced top quarks would constitute a clear evidence
of new physics.

Non-prompt top quark studies have already been considered by CMS and ATLAS
collaborations in an inclusive approach RPVllp1 ATLAS:2022fag
PhysRevD.104.052011  through objects like displaced jets or displaced
vertices. Exclusive research based on displaced top quarks still needs to be
done. If the new physics is top-philic, exclusive research could offer a
better sensitivity than inclusive research. Moreover, a displaced top quark is
also a challenging signature on which the performance of traditional
identification and reconstruction algorithms can be evaluated, such as the
track reconstruction or the identification of b quark jets (although the
latter algorithm is not required for building an experimental analysis
dedicated to the search of displaced top quarks). Naturally, the detection of
this signature involves a specific tune or improvement of the reconstruction
of tracks which are initiating inside the tracker volume.

In the first stage, the study of the displaced top quarks signature requires
simplified models, benchmark points, physics processes, and distributions of
typical observables that this paper will provide. The different models are
classified according to the charges of the long-lived particle, in particular,
the electric charge and the colour. Three cases will be considered: mediation
by an electrically neutral and uncoloured long-lived particle, an electrically
charged and coloured long-lived particle (R-hadron based on a squark), and an
electrically neutral and coloured long-lived particle (R-hadron based on a
gluino).

This paper is organised as follows. The first section is devoted to the common
methodology followed in the presentation of each long-lived particle scenario.
Then, models and benchmarks will be described in a second section. For each
scenario, the relevant region of the parameters space and key experimental
observables will be discussed. Afterwards, a discussion will be given on
several points of attention related to the online selection (achieved by the
trigger system to reduce the data flow to acquire) and offline selections
(built by the experimentalists on top of the reconstructed information).
Finally, the main results will be summarised in the last section.

##  2 Methodology

###  2.1 Targeted strategy

The strategy developed for this paper consists in finding topologies of events
in accordance with the definition of displaced top quarks. One additional
requirement is also imposed: the set of parameters handling the production
cross section of the long-lived mediator must differ from the set of
parameters handling its decay. Such a configuration allows us to decouple the
cross section from the flight distance of the long-lived particles, in order
to get a large variety of top quark displacement values with a production
cross section not too low. These particular topologies lead to the design of
specific simplified models which will be classified according to the charges
of the long-lived particle. Proposing benchmarks and experimental guidelines
require to follow a common and explicit methodology:

  * $\bullet$

A simplified model is defined for describing the displaced top quark
signatures.

  * $\bullet$

The production of the long-lived particle is studied by choosing one or
several physical processes and by calculating the production cross section at
LHC with an energy in the center-of-mass of 14 TeV.

  * $\bullet$

The average flight distance of the long-lived particle, which will be denoted
$\langle d^{\textrm{flight}}\rangle$ subsequently, is determined by computing
the proper time $\tau$ from the decay width of the particle, and the average
kinetic factor $\langle\beta\gamma\rangle$, required to move from the long-
lived particle rest frame to the laboratory frame according to $\langle
d^{\textrm{flight}}\rangle=\langle\beta\gamma\rangle c\tau$.

  * $\bullet$

The most interesting parameter region allowing a displaced top quark is
highlighted, considering that it decays within the tracker volume.

  * $\bullet$

Benchmarks are chosen by selecting points in the parameter space of the model
for which the production cross section gives enough statistics with LHC Run 3
(assuming an integrated luminosity of $300~{}\textrm{fb}^{-1}$). A flight
distance for the long-lived particle consistent with the tracker volume is
also required, as well as particle masses leading to sufficiently high
production cross section and compatible with the range of mass scrutinised by
LHC Runs 1 and 2 analyses. These benchmarks could be the foundation stone of
potential future experimental analyses dedicated to displaced top quarks.

  * $\bullet$

Distributions of key observables are shown and discussed, in order to
highlight the properties of the benchmark points and their differences.

###  2.2 Event generation

Cross section or width calculations and event generation require the
application of several high-energy physics packages. For implementing the
theoretical simplified models, the framework FeynRules feynrules  is used in
order to extract in an automated way the Feynman rules and to store them in
terms of couplings and parameters into the UFO (Universal FeynRules Output)
format ufo . This model is then interfaced to the Monte Carlo (MC) generator
Madgraph_aMC@NLO v.2.7.0 Alwall_2014  in order to generate the hard process at
the leading-order (LO) of the quantum chromodynamics (QCD). No kinematics or
geometrical cut is applied at this stage. The NNPDF30 Ball_2015  parton
distribution functions (PDF) at LO QCD is used for the generation, and is
embedded in the LHAPDF package lhapdf . At this point, the produced events
contain only leptons and partons. The next step of the simulation is to apply
a shower program, in our case Pythia 8.306 Pythia , to handle initial and
final states radiations, hadronisation, hadron decays and underlying events.
All the allowed decay channels of the top quark (hadronic and leptonic) are
accounted for. In order to describe best the LHC data, the Pythia 8 process
must be tuned with experimental measurements gathered into a MC tune including
beam remnants and multiple parton interactions. As no MC tune is available for
LHC at 14 TeV, the MC tune chosen for our analysis is CUETP8M2T4 mctunes
which has been prepared with LHC data taken at a $\sqrt{s}=13\ \text{TeV}$ and
is consistent with the PDF choice made in this article. For coloured long-
lived particles, the R-hadronisation is also achieved by Pythia 8 with the
default parameter values. In particular, the probability of producing a
gluinoball $R^{0}(\tilde{g}g)$ is set to 10% Mackeprang:2007bqa . Online and
offline effects due to additional proton-proton interactions within the same
or nearby bunch crossings have not been considered in this simulation.

Furthermore, the MadAnalysis 5 (MA5) framework Conte_2013 ; Conte_2014 ;
Conte_2018  has been used not only to analyse the event samples but also to
take into account a very simplified detector simulation, thanks to the SFS
module Araz_2021  embedded in the MA5 package. In this detector simulation,
the fiducial acceptance of the tracker has been implemented by modelling the
tracker volume by a hollow cylinder with an inner radius of 4 cm and an outer
radius of 100 cm, extending up to $\pm 300\ \text{cm}$ along the beam axis.
This implementation describes approximately the geometry of a tracker such as
the ATLAS or CMS ones. The magnetic field has also been taken into
consideration by taking a uniform value of B=3.8 T aligned with the beam axis
(z-axis), which corresponds to the one of the CMS detector222Note that even
though the magnetic field of the ATLAS detector is different, the tracking
specifications between both experiments can be considered similar at the
accuracy level of this phenomenological work.. The simulation also benefits
from one of the last developments Araz_2022  of MadAnalysis 5 which allows us
to apply the magnetic field to all electrically charged particles, including
charged long-lived particles. Then, the jets are reconstructed using the
FastJet package Cacciari_2012  with an anti-$k_{T}$ algorithm Cacciari_2008
according to the following parameters: $p_{T}>30\ \text{GeV}$ with $p_{T}$ the
transverse momentum, $\big{|}\eta\big{|}<2.5$ and $\Delta
R=\sqrt{\Delta\eta^{2}+\Delta\phi^{2}}<0.4$ (where $\Delta\eta$ is the
difference in pseudorapidity and $\Delta\phi$ is the difference in azimuthal
angle between two particles).

###  2.3 Reconstruction of physical observables

Different observables are used to characterise the benchmark points.
Consistently with the MA5 Conte_2013  implementations, the MET (Missing
Transverse Energy) is defined as the modulus of the opposite of the vector sum
of the transverse momenta of the isolated reconstructed objects. Similarly,
the TET (Total Transverse Energy) observable is defined as the sum of the
magnitude of the transverse momenta of the isolated reconstructed objects.
From these two observables, are excluded the invisible particles which are the
neutrinos but also the neutral long-lived mediator if it decays beyond the
volume of the calorimeter. This last statement is based on the assumption that
the long-lived mediator, especially in the R-hadron case, do not interact with
the calorimeter matter or its potential interaction is negligible. In
particular, some expected properties of the R-hadrons interaction with matter
Kraan_2004  such as R-mesons changing into R-baryons, R-hadrons electric
charge changing or R-hadrons stopped in the calorimeter and decaying at some
later time, are not considered in this study.

The THT (Total Transverse Hadronic Energy) or MHT (Missing Transverse Hadronic
Energy) observables are reconstructed on the same basis as the TET and MET
observables, but they are restricted to hadronic particles.

Some specific transverse observables are also investigated such as
$\alpha_{T}$ alphaT . This observable is defined in the case of a dijet-event
topology as:

$\alpha_{T}=\frac{{E_{T_{2}}}}{m_{T_{jj}}}$ (1)

with $E_{T_{2}}$ the transverse energy of the second hardest jet and
$m_{T_{jj}}$ the transverse mass of the two hardest jets. Supersymmetric (and
more generically BSM) events may be identified through high values of this
observable. In the case of multijet events, all possible configurations of
dijets are tested by combining all reconstructed jets into two pseudo-jets
$pj_{1}$ and $pj_{2}$. The configuration which minimises the quantity $\Delta
H_{T}=E_{T}^{pj_{1}}-E_{T}^{pj_{2}}$ is chosen, with $E_{T}$ the transverse
energy. The $\alpha_{T}$-observable in Eq. 1 can then be rewritten as a
function of $\Delta H_{T}$ like:

$\alpha_{T}=\frac{1}{2}\frac{1-\Delta
H_{T}/\text{THT}}{\sqrt{1-(\text{MHT}/\text{THT})^{2}}}\ .$

Finally, the transverse and longitudinal impact parameters $d_{0}$ and $d_{z}$
of the top quarks are also calculated for each benchmark, taking into account
the magnitude of the magnetic field. The MA5 definitions Araz_2022  are
reminded:

$\displaystyle d_{0}=$ $\displaystyle\
\textrm{sgn}\left(r\right)\cdot\left(\sqrt{x^{2}+y^{2}}-\left|r\right|\right)\
,$ $\displaystyle d_{z}=$ $\displaystyle\
z_{v}+\frac{p_{z}}{qB}\arctan\left(\frac{xp_{x}+yp_{y}}{yp_{x}-xp_{y}}\right)$

(with $r=p_{T}/(qB)$, $x=x_{v}+p_{y}/qB$ and $y=y_{v}-p_{x}/qB$) where $q$ is
the electric charge of the particle and $(p_{x},p_{y},p_{z})$ is the momentum
at the position of the displaced vertex $(x_{v},y_{v},z_{v})$.

In this paper, displaced tracks are defined as charged particles with a
production vertex located inside the tracker volume. Such tracks must have a
transverse momentum $p_{T}$ greater than $1~{}\text{GeV}$.

##  3 Models and benchmarks devoted to displaced top quarks

###  3.1 Mediation by an electrically neutral and uncoloured long-lived
particle

In this section, we define a simplified model based on the R-parity violated
MSSM (Minimal Supersymmetric Standard Model) RPV_MSSM . A long-lived
neutralino is postulated, which is an electrically neutral and uncoloured
particle. Two different processes at LHC are investigated in this context: the
pair production of neutralinos and the pair production of smuons.

####  3.1.1 Model description

The simplified model is based on the MSSM density Lagrangian, where the
superpotential $W$ also contains R-parity violated couplings encapsulated in a
term noted $W_{RPV}$:

$W=W_{MSSM}+W_{RPV}$

with:

$\displaystyle W_{RPV}=$ $\displaystyle\ \epsilon_{i}\left(H_{2}\cdot
L_{i}\right)\ +\ \frac{1}{2}\lambda_{ijk}\left(L_{i}\cdot
L_{j}\right)E_{k}^{c}\ $ $\displaystyle+\lambda^{\prime}_{ijk}\left(L_{i}\cdot
Q_{j}\right)D_{k}^{c}+\
\frac{1}{2}\lambda^{\prime\prime}_{ijk}U_{i}^{c}D_{f}^{c}D^{c}_{k}$

where $L_{i}$, $Q_{i}$ and $H_{2}$ are respectively the lepton, the quark and
the up-Higgs SU(2) doublets superfields. The “ $\cdot$ ” binary operation is
the SU(2) inner product. $E_{i}$, $U_{i}$ and $D_{i}$ are respectively the
lepton, the up and down quark SU(2) singlets. The upper index c indicates the
charge conjugate. The lower indices $i$, $j$ and $k$ can be equal to 1, 2 and
3 and match the lepton/quark generation. Finally, the trilinear couplings
$\lambda_{ijk}$, $\lambda^{\prime}_{ijk}$ and $\lambda^{\prime\prime}_{ijk}$
are dimensionless R-parity violated couplings, whereas the bilinear couplings
$\epsilon_{i}$ have the dimension of energy. These couplings are taken real
for simplification. Concerning the trilinear couplings:

  * $\bullet$

$\lambda_{ijk}$ leads to leptonic number violation. Due to the gauge symmetry,
the coupling is antisymmetric according to the two first indices:
$\lambda_{ijk}=-\lambda_{jik}$;

  * $\bullet$

$\lambda^{\prime}_{ijk}$ leads to leptonic and baryonic number violation;

  * $\bullet$

$\lambda^{\prime\prime}_{ijk}$ leads to baryonic number violation. Due to the
gauge symmetry, the coupling is antisymmetric according to the two first
indices: $\lambda^{\prime\prime}_{ijk}=-\lambda^{\prime\prime}_{jik}$.

These couplings are assumed to be small as expected from the experimental
limits RPV_MSSM . Their contributions to the production of supersymmetric
particles at LHC are therefore negligible. However, they allow the LSP
(Lightest Supersymmetric Particle) to decay into standard particles and to fly
in the detector volume according to the low value of the trilinear coupling.
Defining the neutralino as the LSP, only a non-null value for the couplings
$\lambda^{\prime}_{i3k}$ or
$\lambda^{\prime\prime}_{312}=-\lambda^{\prime\prime}_{321}$ can lead to one
top quark in the decay products of the neutralino and only a non-null value
for the coupling $\lambda^{\prime\prime}_{313}=-\lambda^{\prime\prime}_{331}$
or $\lambda^{\prime\prime}_{323}=-\lambda^{\prime\prime}_{332}$ can give two
top quarks. Figure 1 illustrates this decay channel in the case of a non-null
coupling $\lambda^{\prime\prime}_{312}$. Note that since the LSP is not
stable, there is no dark matter candidate in this model.

Figure 1: Neutralino decay diagrams involving the studied R-parity violated
couplings.

The first simplified model is built by taking all R-parity violated coupling
as zero except $\lambda^{\prime\prime}_{312}$ (noted in the following
$\lambda^{\prime\prime}$). We only study extreme cases where the neutralino is
a pure state, i.e. it can be a bino $\tilde{B}$, wino $\tilde{W}^{3}$ or a
higgsino ($\tilde{h}_{1}$ or $\tilde{h}_{2}$). Mixing matrices for sleptons
and squarks are chosen such that there is no mixing between particles of
different generations, and the mass states are in an equal proportion of left-
handed and right-handed states. Finally, the stop squark mass is fixed to
$1.0\ \text{TeV}$ (from typical LHC constraints, see Section 3.2 for more
informations), while the other squarks are set at higher masses and so
neglected.

####  3.1.2 Neutralino production and cross section

To produce the long-lived neutralinos, two hard processes are presupposed at
LO QCD: the direct neutralino-pair production or the smuon-pair production
decaying into neutralino and muon. The additional supersymmetric Higgs bosons
contributions have been ignored, assuming a high mass for such particles.

  * $\bullet$

Figure 2 displays a smuon production
($pp\rightarrow\tilde{\mu}^{+}\tilde{\mu}^{-}$) where the smuon decays into a
muon and a neutralino
($\tilde{\mu}^{\pm}\rightarrow\tilde{\chi}^{0}_{1}\mu^{\pm}$). We set a
branching fraction of 100% by assuming that the smuon $\tilde{\mu}$ is the
Next-to-Lightest-Supersymmetric-Particle (NLSP). The muons in the final states
are very convenient for triggering the events and facilitating the
identification of such a signature. Concerning the definition of the
neutralino, the pure $\tilde{h}_{2}$ state is forbidden to make the smuon
decay possible. Consequently, only a bino-like neutralino $\tilde{B}$ can
decay via the RPV-coupling $\lambda^{\prime\prime}$. The smuon mass is then
the only free parameter for the cross section computation. The coupling
between the Z boson and the smuons is fixed by the choice of mixing between
left and right smuons. Note also that this analysis can be easily extended to
the case
$pp\rightarrow\tilde{\ell}\bar{\tilde{\ell}}\rightarrow\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}\ell\bar{\ell}$,
with the lepton $\ell$ including electrons and taus.

Figure 2: Feynman diagram (at LO QCD) related to the indirect production of
neutralinos by smuons.

  * $\bullet$

The second process is the direct neutralino production
($pp\rightarrow\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}$) in the s-channel and
the t-channel (see Fig. 3). The squark exchange only contributes to a few
percent of the production cross section. The only types of neutralino which
allow such processes are $\tilde{h}_{1}$ and $\tilde{h}_{2}$. Due to the form
of the couplings, a pure $\tilde{h}_{1}$ or $\tilde{h}_{2}$ state leads to the
same cross section. Thus the cross section depends only on the neutralino
mass.

Figure 3: Feynman diagrams (at LO QCD) related to the direct production of
neutralinos.

Another related signal can be investigated, the production of an LSP
neutralino and an NLSP chargino leading to a higher production cross section
and one lepton in the final state
($\tilde{\chi}^{0}_{1}\tilde{\chi}^{\pm}_{1}\rightarrow\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}\ell^{\pm}$).
However, we do not consider such signal contributions since the chargino mass
and mixing matrix will provide supplementary free parameters to fix, which
makes the benchmarks more complicated. Figures 4 and 5 illustrate the
dependence of the neutralino-pair production cross section as a function of
the smuon mass (for the bino case) and the neutralino mass (for the higgsino
case). These cross sections decrease with the mass of the supersymmetric
particles. A more complete analysis of a neutralino pair production at
$\sqrt{s}=14\ \text{TeV}$ including QCD-corrections can be found in
NLOdirectneutralino . Concerning the smuon production, the parameter region is
limited to a smuon mass of $350~{}\textrm{GeV}$ in order to collect and
analyse a sufficient amount of data with a Run 3 integrated luminosity of
$\mathcal{L}=300~{}\textrm{fb}^{-1}$. A more precise theoretical Next-to-
Leading Order (NLO) calculation of the cross section can be found in Baer_1998
. The K-factor, $K=\sigma_{NLO}/\sigma_{LO}$, for both processes is around
1.4. For the direct neutralino production, we expect to probe a neutralino
mass range up to $500~{}\textrm{GeV}$. The associated uncertainties reflect
variations on the PDF, the factorisation and the renormalisation scales at
leading order.

Figure 4: Cross section of the smuon production at leading order as a function
of the smuon mass $m_{\tilde{\mu}}$. The uncertainties are related to PDF
variations and the factorisation and renormalisation scales. Figure 5: Cross
section of the direct neutralino production at leading order as a function of
the neutralino mass $m_{\tilde{\chi}_{1}^{0}}$. The uncertainties are related
to PDF variations and the factorisation and renormalisation scales.

####  3.1.3 Neutralino decay and flight distance

Together with the long-lived particle lifetime, the Lorentz factor
$\beta\gamma$ of the neutralino in the two processes has been taken into
account as a requirement to estimate the flight distance in the detector
frame. Figure 6 illustrates the variation of the average of the mean value of
$\beta\gamma$ as a function of the mass of the neutralino. Several values for
the smuon mass $m_{\tilde{\mu}}$ have been investigated for the slepton
production. For $m_{\tilde{\chi}^{0}_{1}}\gtrsim 250\ \text{GeV}$, the mean
value $\langle\beta\gamma\rangle$ in the bino case is higher for a lower smuon
mass $m_{\tilde{\mu}}$. It has to be noted that we have the explicit
constraint $m_{\tilde{\mu}}>m_{\tilde{\chi}_{1}^{0}}$. The mean Lorentz factor
$\langle\beta\gamma\rangle_{\tilde{\chi}_{1}^{0}}$ varies from 1.8 to 3.0 and
so affects the flight distance $\langle
d^{\text{flight}}_{\tilde{\chi}^{0}_{1}}\rangle$.

Figure 6: Mean value of the Lorentz factor $\langle\beta\gamma\rangle$ for the
flying neutralino in both production cases as a function of the neutralino
mass $m_{\tilde{\chi}^{0}_{1}}$.

The mean flight distance of the neutralino $\langle
d^{\textrm{flight}}_{\tilde{\chi}_{1}^{0}}\rangle$ in the laboratory frame can
be then computed for different values of the R-parity violated coupling
$\lambda^{\prime\prime}$ and neutralino masses, as shown in Figs. 7 and 8,
where we fix $m_{\tilde{\mu}}=350\ \text{GeV}$ in the case of the smuon pair
production. The dependency in $m_{\tilde{\mu}}$ comes only from the kinetic
factor $\langle\beta\gamma\rangle$. Thus, a lower mass $m_{\tilde{\mu}}$ does
not highly modify the flight distance, see Fig. 6. As expected, the flight
distance is inversely proportional to $\lambda^{\prime\prime 2}$ and, for
$\lambda^{\prime\prime}$ fixed, it decreases with the mass of the neutralino.
We also notice that an RPV coupling higher than $10^{-2}$ is not allowed to
avoid a long-lived particle from decaying outside the tracker volume.
Nonetheless, all neutralino masses in the range $[200,475]\ \text{GeV}$ can be
scrutinised with ATLAS or CMS detectors at LHC Run 3.

Figure 7: Average flight distance $\langle
d^{\text{flight}}_{\tilde{B}}\rangle$ of the neutralino $\tilde{B}$-like in
the laboratory frame as a function of $\lambda^{\prime\prime}$ and the
neutralino mass $m_{\tilde{\chi}^{0}_{1}}$. We assume a smuon mass of
$m_{\tilde{\mu}}=350\ \text{GeV}$. The black lines represent the geometrical
limits of the tracker volume. Figure 8: Average flight distance $\langle
d^{\text{flight}}_{\tilde{h}}\rangle$ of the neutralino $\tilde{h}_{2}$-like
in the laboratory frame as a function of $\lambda^{\prime\prime}$ and the
neutralino mass $m_{\tilde{\chi}^{0}_{1}}$. The black lines represent the
geometrical limits of the tracker volume.

####  3.1.4 Benchmarks definition

Name | $\tilde{B}_{0.2}^{10}$ | $\tilde{B}_{0.2}^{30}$ | $\tilde{B}_{0.2}^{50}$ | $\tilde{B}_{0.2}^{70}$ | $\tilde{B}_{0.3}^{10}$ | $\tilde{B}_{0.3}^{30}$ | $\tilde{B}_{0.3}^{50}$ | $\tilde{B}_{0.3}^{70}$
---|---|---|---|---|---|---|---|---
$\langle d^{\textrm{flight}}_{\tilde{B}}\rangle~{}[\textrm{cm}]$ | 10 | 30 | 50 | 70 | 10 | 30 | 50 | 70
$m_{\tilde{\chi}_{1}^{0}}~{}[\textrm{TeV}]$ | 0.20 | 0.20 | 0.20 | 0.20 | 0.30 | 0.30 | 0.30 | 0.30
$m_{\tilde{t}}~{}[\textrm{TeV}]$ | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00
$m_{\tilde{\mu}}~{}[\textrm{TeV}]$ | 0.35 | 0.35 | 0.35 | 0.35 | 0.35 | 0.35 | 0.35 | 0.35
$\lambda^{\prime\prime}~{}[10^{-6}]$ | $6407$ | $3699$ | $2865$ | $2422$ | $158$ | $91$ | $71$ | $60$
Table 1: Definition of the eight benchmarks for the smuon production case $pp\rightarrow\tilde{\mu}\bar{\tilde{\mu}}$. Name | $\tilde{h}_{0.2}^{10}$ | $\tilde{h}_{0.2}^{30}$ | $\tilde{h}_{0.2}^{50}$ | $\tilde{h}_{0.2}^{70}$ | $\tilde{h}_{0.4}^{10}$ | $\tilde{h}_{0.4}^{30}$ | $\tilde{h}_{0.4}^{50}$ | $\tilde{h}_{0.4}^{70}$
---|---|---|---|---|---|---|---|---
$\langle d^{\textrm{flight}}_{\tilde{h}}\rangle~{}[\textrm{cm}]$ | 10 | 30 | 50 | 70 | 10 | 30 | 50 | 70
$m_{\tilde{\chi}^{0}_{1}}~{}[\textrm{TeV}]$ | 0.20 | 0.20 | 0.20 | 0.20 | 0.40 | 0.40 | 0.40 | 0.40
$m_{\tilde{t}}~{}[\textrm{TeV}]$ | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00
$\lambda^{\prime\prime}~{}[10^{-6}]$ | $2599$ | $1501$ | $1162$ | $983$ | $15$ | $8.8$ | $6.8$ | $5.8$
Table 2: Definition of the eight benchmarks for the direct neutralino
production $pp\rightarrow\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}$.

From the previous results, we define benchmarks by making a compromise between
a large cross section and a relevant flight distance for the neutralino. The
cross section in Fig. 4 limits the choice of the smuon mass. In order to get a
sufficient amount of data, we fix the mass of the smuon to $350\ \text{GeV}$,
leading to approximately 300 events for $\mathcal{L}_{int.}=300\
\text{fb}^{-1}$. A choice of $\Delta
m=m_{\tilde{\mu}}-m_{\tilde{\chi}^{0}_{1}}$ will mainly impact the particle’s
transverse momentum in the final state, whereas $m_{\tilde{\mu}}$ influences
the global observables such as THT. Note that the cross section of the direct
neutralino production is sufficiently large to not constrain the choice of
$m_{\tilde{\chi}^{0}_{1}}$ in the range $[200,500]\ \text{GeV}$. Two values of
neutralino mass are chosen for both processes, and four mean flight distances
(10, 30, 50 and $70\ \text{cm}$) are considered to cover all the tracker
volume geometry. A neutralino mass of $200$ and $400\ \text{GeV}$ has been
taken in the direct neutralino production. For the slepton channel, we assume
$m_{\tilde{\chi}^{0}_{1}}=200$ and $300\ \text{GeV}$. The RPV coupling
$\lambda^{\prime\prime}$ is fixed through the choice of the two mentioned
parameters and varies between $10^{-3}$ and $10^{-6}$. We summarise the
definition of these benchmarks in Table 1 and Table 2.

No exclusive analysis has yet been designed for the signal studied here.
Indeed related signals have already been investigated by the CMS and ATLAS
collaborations: some processes including R-Parity conserved (RPC) slepton and
direct neutralino production (see RPCslepton1 ; RPCslepton2 ; RPCslepton3 ;
RPCslepton4 ; RPCslepton5 ; RPCslepton6 ; RPCslepton7  with exclusion from
$m_{\tilde{\mu}}>200\ \text{GeV}$ to $m_{\tilde{\mu}}>700\ \text{GeV}$ with
massless neutralino by usually considering pure left or right smuon states),
and other RPV decays (see RPVprompt1 ; RPVprompt2 ; RPVprompt3 ; RPVprompt4 ;
RPVprompt5 ; RPVprompt6  for prompt and PhysRevD.104.052011 ; RPVllp1 ;
RPVllp2 ; RPVllp3 ; RPVllp4  for non-prompt decays) with constraints like
$m_{\tilde{\mu}}>1.2\ \text{TeV}$, $m_{\tilde{\chi}^{0}_{1}}$ excluded between
200 and 300 GeV or other limits on RPV couplings, gluino and stop masses. It
can be noticed that another analysis PhysRevD.102.032006  achieved by the
ATLAS collaboration has similarities with the sleptons case. Indeed, the
analysis deals the production of non-prompt top quark by a different RPV
couplings and takes advantage of the presence of two non-prompt muons in the
final state.

####  3.1.5 Event kinematics

Figure 9: Transverse momentum $p_{T}$ of the leading muon of the event in the
$\tilde{B}$-like case.

Figure 10: Transverse momentum $p_{T}$ of the leading jet of the event in the
$\tilde{B}$-like case.

Figure 11: Total transverse hadronic energy THT in the $\tilde{B}$-like case.

Figure 12: Track multiplicity associated with displaced vertices in the
$\tilde{B}$-like case.

Figure 13: Transverse impact parameter $d_{0}$ of the top quarks in the
$\tilde{B}$-like case with $m_{\tilde{\chi}^{0}_{1}}=200\ \text{GeV}$.

Figure 14: Longitudinal impact parameter $d_{z}$ of the top quarks in the
$\tilde{B}$-like case with $m_{\tilde{\chi}^{0}_{1}}=200\ \text{GeV}$.

Figure 15: Transverse impact parameter $d_{0}$ of the top quarks in the
$\tilde{B}$-like case with $m_{\tilde{\chi}^{0}_{1}}=300\ \text{GeV}$.

Figure 16: Longitudinal impact parameter $d_{z}$ of the top quarks in the
$\tilde{B}$-like case with $m_{\tilde{\chi}^{0}_{1}}=300\ \text{GeV}$.

Figure 17: Missing transverse energy MET in the $\tilde{B}$-like case.

Figure 18: Total transverse energy TET in the $\tilde{B}$-like case.

Figure 19: Missing transverse hadronic energy MHT in the $\tilde{B}$-like
case.

Figure 20: Transverse momentum $p_{T}$ of the second leading jet in the
$\tilde{B}$-like case.

Figure 21: Transverse momentum of the leading lepton in the $\tilde{B}$-like
case.

Figure 22: Transverse momentum of the second leading lepton in the
$\tilde{B}$-like case.

Figure 23: Jet multiplicity $N_{J}$ in the $\tilde{B}$-like case.

Figure 24: $\alpha_{T}$ distributions in the $\tilde{B}$-like case.

Figure 25: Jet multiplicity $N_{j}$ in the $\tilde{h}_{2}$-like case.

Figure 26: Transverse momentum of the leading jet of the event in the
$\tilde{h}_{2}$-like case.

Figure 27: Total transverse hadronic energy $THT$ in the $\tilde{h}_{2}$-like
case.

Figure 28: Track multiplicity associated with displaced vertices in the
$\tilde{h}_{2}$-like case.

Figure 29: Transverse impact parameter $d_{0}$ of the top quarks in the
$\tilde{h}_{2}$-like case with $m_{\tilde{\chi}^{0}_{1}}=200\ \text{GeV}$.

Figure 30: Longitudinal impact parameter $d_{z}$ of the top quarks in the
$\tilde{h}_{2}$-like case with $m_{\tilde{\chi}^{0}_{1}}=200\ \text{GeV}$.

Figure 31: Transverse impact parameter $d_{0}$ of the top quarks in the
$\tilde{h}_{2}$-like case with $m_{\tilde{\chi}^{0}_{1}}=400\ \text{GeV}$.

Figure 32: Longitudinal impact parameter $d_{z}$ of the top quarks in the
$\tilde{h}_{2}$-like case with $m_{\tilde{\chi}^{0}_{1}}=400\ \text{GeV}$.

Figure 33: Missing transverse energy MET in the $\tilde{h}_{2}$-like case.

Figure 34: Total transverse energy TET in the $\tilde{h}_{2}$-like case.

Figure 35: Transverse momentum $p_{T}$ of the leading lepton in the
$\tilde{h}_{2}$-like case.

Figure 36: Transverse momentum $p_{T}$ of the second leading lepton in the
$\tilde{h}_{2}$-like case.

Figure 37: Missing transverse hadronic energy MHT in the $\tilde{h}_{2}$-like
case.

Figure 38: Transverse momentum $p_{T}$ of the second leading jet in the
$\tilde{h}_{2}$-like case.

Figure 39: $\alpha_{T}$ distributions in the $\tilde{h}_{2}$-like case.

Figure 40: $\Delta R$ distributions in the $\tilde{h}_{2}$-like case.

| $\tilde{B}_{0.2}^{10}$ | $\tilde{B}_{0.2}^{30}$ | $\tilde{B}_{0.2}^{50}$ | $\tilde{B}_{0.2}^{70}$ | $\tilde{B}_{0.3}^{10}$ | $\tilde{B}_{0.3}^{30}$ | $\tilde{B}_{0.3}^{50}$ | $\tilde{B}_{0.3}^{70}$ | $t\bar{t}$
---|---|---|---|---|---|---|---|---|---
$\sigma_{LO}(pp\rightarrow\tilde{\mu}\bar{\tilde{\mu}})\ [\text{fb}]$ | 1.12 | -
$R_{\ =\ 2\ \textrm{displaced t}}$ [%] | 38.1 | 87.7 | 94.6 | 94.4 | 47.5 | 95.2 | 98.6 | 98.0 | -
$R_{\ \geq\ 1\ \textrm{displaced t}}$ [%] | 72.7 | 99.0 | 99.8 | 99.3 | 63.6 | 99.0 | 99.8 | 99.4 | -
$\langle\text{MET}\rangle$ [GeV] | 63 | 65 | 64 | 64 | 58 | 59 | 58 | 58 | 43
$\langle\text{TET}\rangle$ [GeV] | 989 | 986 | 990 | 988 | 980 | 982 | 983 | 982 | 465
$\langle\text{THT}\rangle$ [GeV] | 464 | 461 | 464 | 463 | 611 | 612 | 611 | 613 | 327
$\langle\text{MHT}\rangle$ [GeV] | 126 | 126 | 127 | 126 | 90 | 92 | 90 | 91 | 52
$\langle\alpha_{T}\rangle$ | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.48
$\langle p_{T}(j_{1})\rangle$ [GeV] | 219 | 218 | 218 | 218 | 230 | 233 | 231 | 231 | 118
$\langle p_{T}(j_{2})\rangle$ [GeV] | 19 | 15 | 15 | 16 | 41 | 36 | 42 | 56 | 23
$\langle p_{T}(\ell_{1})\rangle$ [GeV] | 210 | 211 | 211 | 210 | 89 | 90 | 90 | 90 | 32
$\langle p_{T}(\ell_{2})\rangle$ [GeV] | 210 | 211 | 211 | 210 | 88 | 88 | 89 | 90 | 31
$\langle|d_{0}(t)|\rangle$ [cm] | 0.17 | 0.50 | 0.83 | 1.16 | 0.73 | 2.19 | 3.65 | 5.10 | -
$\langle|d_{z}(t)|\rangle$ [cm] | 0.47 | 1.37 | 2.26 | 3.11 | 1.91 | 5.64 | 9.30 | 12.9 | -
$N_{\text{jets}}$ | 4.6 | 4.6 | 4.6 | 4.6 | 6.6 | 6.6 | 6.6 | 6.6 | 4.3
$N_{\text{tracks from displaced t}}$ | 35.6 | 35.4 | 35.4 | 35.4 | 47.0 | 46.9 | 46.8 | 46.8 | -
Table 3: Mean values of some observables and number of events with top quarks decaying in the tracker volume for the eight benchmarks in the bino $\tilde{B}$ case. We remind that $R_{\geq\textrm{1 displaced t}}$ corresponds to the number of events with at least one top quark decaying in the tracker volume. | $\tilde{h}_{0.2}^{10}$ | $\tilde{h}_{0.2}^{30}$ | $\tilde{h}_{0.2}^{50}$ | $\tilde{h}_{0.2}^{70}$ | $\tilde{h}_{0.4}^{10}$ | $\tilde{h}_{0.4}^{30}$ | $\tilde{h}_{0.4}^{50}$ | $\tilde{h}_{0.4}^{70}$ | $t\bar{t}$
---|---|---|---|---|---|---|---|---|---
$\sigma_{LO}(pp\rightarrow\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1})\ [\text{fb}]$ | 39.7 | 2.52 | -
$R_{\ =\ 2\ \textrm{displaced t}}$ [%] | 25.1 | 81.8 | 92.3 | 93.5 | 44.5 | 90.4 | 96.3 | 96.8 | -
$R_{\ \geq\ 1\ \textrm{displaced t}}$ [%] | 33.1 | 89.7 | 96.7 | 98.3 | 53.1 | 94.7 | 98.5 | 98.7 | -
$\langle\text{MET}\rangle$ [GeV] | 51 | 51 | 51 | 51 | 56 | 56 | 56 | 56 | 43
$\langle\text{TET}\rangle$ [GeV] | 549 | 549 | 551 | 548 | 995 | 992 | 996 | 995 | 465
$\langle\text{THT}\rangle$ [GeV] | 384 | 384 | 385 | 384 | 718 | 713 | 718 | 716 | 327
$\langle\text{MHT}\rangle$ [GeV] | 63 | 64 | 64 | 64 | 84 | 85 | 85 | 85 | 52
$\langle\alpha_{T}\rangle$ | 0.48 | 0.48 | 0.48 | 0.48 | 0.50 | 0.50 | 0.50 | 0.50 | 0.48
$\langle p_{T}(j_{1})\rangle$ [GeV] | 156 | 155 | 156 | 155 | 257 | 256 | 258 | 258 | 118
$\langle p_{T}(j_{2})\rangle$ [GeV] | 32 | 25 | 14 | 20 | 46 | 37 | 39 | 44 | 23
$\langle p_{T}(\ell_{1})\rangle$ [GeV] | 50 | 51 | 52 | 52 | 53 | 56 | 54 | 55 | 32
$\langle p_{T}(\ell_{2})\rangle$ [GeV] | 19 | 19 | 19 | 19 | 21 | 21 | 19 | 21 | 31
$\langle|d_{0}(t)|\rangle$ [cm] | 0.15 | 0.44 | 0.74 | 1.03 | 1.19 | 3.57 | 5.94 | 8.34 | -
$\langle|d_{z}(t)|\rangle$ [cm] | 0.69 | 2.04 | 3.40 | 4.62 | 3.85 | 11.12 | 17.88 | 24.04 | -
$N_{\text{jets}}$ | 4.3 | 4.3 | 4.3 | 4.3 | 7.3 | 7.3 | 7.3 | 7.3 | 4.3
$N_{\text{tracks from displaced t}}$ | 35.6 | 35.5 | 35.5 | 35.4 | 53.5 | 53.5 | 53.4 | 53.4 | -
Table 4: Mean values of some observables and number of events with top quarks
decaying in the tracker volume for the eight benchmarks in the higgsino
$\tilde{h}_{2}$ case. We remind that $R_{\geq\textrm{1 displaced t}}$
corresponds to the number of events with at least one top quark decaying in
the tracker volume.

Distributions of some relevant observables have been computed for each
benchmark. The results of the analysis are summarised in Table 3 and Table
4333Despite the fact that all the distributions are not Gaussian, we still
present the mean values to summarise the information of all observables. This
remark applies to each analysis.. Some distributions are presented in Figs. 14
to 24 for the $\tilde{B}$-case and in Figs. 30 to 40, for the
$\tilde{h}_{2}$-case. Standard top quark pair (prompt) production has also
been added to the distributions to compare our signals with a typical source
of background. The distributions are all normalised to unity.

First, considering the bino-like configuration, we expect at least two muons
in the final state with jets coming from the top quarks. The distributions of
transverse momentum of muons are shown in Fig. 14. On average, the muons
coming from the smuons are more energetic in the signal than in the $t\bar{t}$
background events (the muons coming essentially from the top quark leptonic
decay). The same comment remains valid for the transverse momentum of the
leading jets $p_{T}(j_{1})$ according to Figure 14 as the top quarks in the
signal are harder.

One of the main differences between the benchmarks and the $t\bar{t}$ prompt
signal is the presence of down and strange quarks coming from the neutralino
decay which increases the jet multiplicity in the events. An analysis at the
partonic level showed that the d and s quarks coming from the neutralino decay
do not have a large $p_{T}$, thus not passing the requirement of the
anti-$k_{T}$ algorithm: $p_{T}>30\ \text{GeV}$.

Unlike the RPC studies mentioned in Section 3.1.4, only a moderate amount of
MET is expected from the leptonic W boson decays. However, with a mean value
of $65$ GeV, this observable may still be of interest to reduce the QCD
background. Conversely, the mean values of THT (see Fig. 14) are approximately
$460$ and $610\ \text{GeV}$ for $m_{\tilde{\chi}^{0}_{1}}=200$ and $300\
\text{GeV}$, respectively.

Finally, more than 90% of the events include at least one top quark decaying
inside the simplified-tracker acceptance for $\langle
d^{\textrm{flight}}_{\tilde{\chi}^{0}_{1}}\rangle\geq 30\ \text{cm}$ (see
Table 3) producing on average between 35 and 47 displaced potential tracks
(Fig. 14). As shown Figures 14 to 20, the transverse and longitudinal impact
parameters $d_{0}$ and $d_{z}$ of the top quarks depends strongly on the
neutralino mass. Their mean values do not exceed 13 cm in absolute although
the $d_{z}$ parameter can reach values between -40 cm and 40 cm.

Concerning the higgsino-like process, the experimental signature is similar to
the bino case other than the absence of a muon with high $p_{T}$ coming from
the smuon decay in the final state. It can be noted in Figs. 30 and 30 (jet
multiplicity and THT) that distributions of the prompt $t\bar{t}$ and the
higgsino direct production with $m_{\tilde{\chi}^{0}_{1}}=200\ \text{GeV}$ are
quite similar and so could make difficult the identification of the signal
with such observables.

The transverse momentum of jets is presented in Fig. 30 with a mean value of
$160$ and $260\ \text{GeV}$ for $m_{\tilde{\chi}^{0}_{1}}=200$ and $400\
\text{GeV}$, respectively. Higher neutralino masses naturally lead to a higher
$p_{T}$ of the jets, and it affects the jet multiplicity (see Fig. 30). The
multiplicity for $m_{\tilde{\chi}^{0}_{1}}=200\ \text{GeV}$ is equivalent to
the prompt $t\bar{t}$ signal with a mean value of $4.3$, whereas for
$m_{\tilde{\chi}^{0}_{1}}=400\ \text{GeV}$ the number of jets is higher with a
mean value of $7.3$. The first value remains low compared to the expected one
(between 6 and 10 quarks in the final state) but is explained by the selection
cut of 30 GeV on the jet transverse momentum.

Considering the global variables, the MET in the event (with a mean value of
$50$ and $55\ \text{GeV}$ for $m_{\tilde{\chi}^{0}_{1}}=200$ and $400\
\text{GeV}$, respectively, see Table 4) originates from the neutrino in the W
decay. The total hadronic energy THT is presented in Fig. 30. The
$m_{\tilde{\chi}^{0}_{1}}=400\ \text{GeV}$ case stands out with a mean value
of $710\ \text{GeV}$ whereas at $m_{\tilde{\chi}^{0}_{1}}=200\ \text{GeV}$,
the THT distribution is close to the prompt $t\bar{t}$ one.

One main difference with the bino-like process is the number of events with at
least one top quark decaying inside the “tracker volume” (see $R_{\ \geq\ 1\
\textrm{displaced t}}$ observable in Table 4). For $\langle
d^{\textrm{flight}}_{\tilde{h}}\rangle=10\ \text{cm}$, only $33\%$ and $53\%$
(for $m_{\tilde{\chi}^{0}_{1}}=200$ and $400\ \text{GeV}$, respectively) of
the events match such a configuration. These values correspond to $73\%$ and
$64\%$ for the bino case. This difference can be explained by the fact that
the top quarks are produced closer to the beam in the bino case, leading to
fewer top quarks in the tracker volume for small flight distances.

###  3.2 Mediation by a coloured long-lived particle (R-hadron based on a
squark)

A particular simplified model where the long-lived particle is a R-hadron
containing a squark is designed, inspired by the R-parity conserved MSSM where
supersymmetry is broken by the GMSB mechanism (for Gauge Mediated
Supersymmetry Breaking GaugeMediated1 ; GaugeMediated2 ). This model
introduces a top squark, and a gravitino as an electrically neutral LSP.

####  3.2.1 Model description

The model is based on the Lagrangian density of the MSSM to which is added the
kinetic part of the gravitino as well as its interactions with the fermions
and bosons in the MSSM. These interactions can be expressed as:

$\displaystyle\mathcal{L}_{3/2}^{int}=-\frac{\sqrt{4\pi}}{m_{p}}\Big{(}\mathcal{D}_{\nu}\phi^{i}$
$\displaystyle\bar{\chi}_{Li}\gamma^{\mu}\gamma^{\nu}\Psi_{\mu}^{(M)}$ (2)
$\displaystyle+\mathcal{D}_{\nu}\phi^{\dagger}_{i}\bar{\Psi}_{\nu}^{(M)}\gamma^{\nu}\gamma^{\mu}\chi_{R}^{i}\Big{)}$

where $\mathcal{D}_{\nu}\phi^{i}$ is the covariant derivative of the scalar
field $\phi$, $\Psi_{\mu}^{(M)}$ the gravitino field, $\bar{\chi}_{L}$ and
$\chi_{R}$ the left and right-handed fermions corresponding to the
superpartner of $\phi$, and $m_{p}$ the Planck mass. In this context, the
gravitino mass is naturally suppressed by the Planck scale, as shown in the
equation below, and leads to a natural LSP gravitino:

$\displaystyle m_{3/2}\propto\frac{F_{0}}{m_{p}}$

with $\sqrt{F_{0}}$ the fundamental scale of supersymmetry breaking.

In order to build a simplified model, only the superpartners of the third
generation of quarks are involved. Like in the previous section, we define the
stop squark as an equal mixing of the left-handed and right-handed stop
states. This stop squark is taken as the NLSP and therefore the decay mode
$\tilde{t}\rightarrow t\psi_{\mu}$ is the only possible. Removing this
assumption would merely consist in specifying a branching fraction to the
latter decay mode.

####  3.2.2 Cross sections and decays

At the LO QCD, the stop squarks can be produced at the LHC by quark
annihilation (Fig. 41) or by gluon fusion (Fig. 42). The cross section depends
only on the stop mass $m_{\tilde{t}}$ and has already been calculated in more
extensive studies (up to NLO calculation, with
$K=\sigma_{NLO}/\sigma_{LO}\approx 1.3$, see Beenakker:1996ch ;
Beenakker:1997ut ). Figures 44 and 44 display the variation of the production
cross section and the mean Lorentz factor $\langle\beta\gamma\rangle$ of the
stop squark according to its mass. The uncertainties in Fig. 44 are related to
PDF and factorisation and renormalisation scale variations. The cross section
decreases with the stop mass and, assuming a Run 3 integrated luminosity of
$\mathcal{L}=300~{}\textrm{fb}^{-1}$, gives an upper limit on the stop mass
that we can reach with a reasonable number of events. Concerning the Lorentz
factor $\langle\beta\gamma\rangle$ (see Fig. 44), the mean value is between
0.8 and 1.9 for the investigated masses, so the flight distance of the stop is
not highly modified for a stop mass higher than approximately $600\
\text{GeV}$.

Figure 41: Feynman diagram of a stop squark pair production by quark
annihilation. Figure 42: Feynman diagrams of a stop squark pair production by
gluon fusion.

Figure 43: Leading order cross section
$\sigma(pp\rightarrow\tilde{t}\tilde{\bar{t}})$ at $\sqrt{s}=14\ \text{TeV}$
as a function of the stop squark mass $m_{\tilde{t}}$. The uncertainties are
related to PDF variations and the factorisation and renormalisation scales.

Figure 44: Average value of the Lorentz factor $\beta\gamma$ of the stop
squark.

The total decay width of the stop can be calculated analytically from the
interaction terms present in the Lagrangian density (see Eq. 2) and is
expressed as:

$\displaystyle\Gamma(\tilde{t}\rightarrow\psi_{\mu}t)=$
$\displaystyle\frac{1}{6m_{p}^{2}m_{\tilde{t}}^{3}m_{3/2}^{2}}\lambda^{3/2}\left(m^{2}_{\tilde{t}},m^{2}_{t},m^{2}_{3/2}\right)$
$\displaystyle\times\left(m_{\tilde{t}}^{2}-(m_{t}-m_{3/2})^{2}\right)$

with the Källén function $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2xz-2yz$.

Taking into account the mean Lorentz factor $\langle\beta\gamma\rangle$, the
mean flight distance of the stop in the laboratory frame $\langle
d^{\text{flight}}_{\tilde{t}}\rangle$ is shown in Fig. 45. This observable is
totally driven by the stop and the gravitino masses. If we focus on the
geometrical boundaries of $4\ \textrm{cm}$ and $100\ \textrm{cm}$, a long-
lived stop can be observed for a wide range of values of stop mass.

Figure 45: Mean flight distance of the stop in the laboratory frame as a
function of the stop and the gravitino masses. The black lines represent the
geometrical limits of the tracker volume, i.e., $4\ \text{cm}$ and $100\
\textrm{cm}$.

####  3.2.3 Benchmarks definition

With the help of Figs. 44-45, we have identified eight different benchmarks
given in Table 5. Limits on stop squark decaying to a top quark and a
gravitino have not been achieved yet at the LHC. Nevertheless, the equivalent
process $\tilde{t}\rightarrow t\tilde{\chi}^{0}_{1}$ is widely investigated
and can be compared with our signal in the limit
$m_{\tilde{\chi}^{0}_{1}}\rightarrow 0$ (in the RPC case). The constraints
coming from ATLAS and CMS collaborations with prompt stop squarks (see
ATLAS:2020syg ; CMS:2021beq ; Aaboud_2017 ; aad:hal-03143576 ; Aad:2746797 ;
sirunyan:hal-02934073 ) give lower bounds between $m_{\tilde{t}}>0.9$ to $1.9\
\text{TeV}$ in the most restrictive case. Thus two masses are considered in
the following benchmarks: $m_{\tilde{t}}=1.0$ and $1.4\ \textrm{TeV}$ with a
production cross section of $4.7$ and $0.4\ \text{fb}$, respectively. We also
assume a flight distance of $\langle
d^{\text{flight}}_{\tilde{t}}\rangle\in[10,30,50,70]\ \text{cm}$. For each
benchmark, the gravitino mass $m_{3/2}$ is totally constrained and its value
is in a range between $20$ and $140~{}\textrm{keV}$. Note that these values
are consistent with astrophysics and cosmology in some scenarios, see cosmo1 ;
cosmo2 .

It is important to highlight that a long-lived stop, as a coloured particle,
must hadronise before decaying. More precisely, two kinds of R-hadrons can be
produced: the R-meson $R(\tilde{t}\bar{q})$ and the R-baryon $R(\tilde{t}qq)$
which can be electrically neutral or charged according to the flavour of the
constituent quarks. The case of a charged R-hadron offers the possibility to
reconstruct its hits in the tracker sensors and thus its track. In the
following, we do not take into account this potential signature and, by misuse
of language, we refer the R-hadron as the stop squark.

Name | $\tilde{t}_{1.0}^{10}$ | $\tilde{t}_{1.0}^{30}$ | $\tilde{t}_{1.0}^{50}$ | $\tilde{t}_{1.0}^{70}$ | $\tilde{t}_{1.4}^{10}$ | $\tilde{t}_{1.4}^{30}$ | $\tilde{t}_{1.4}^{50}$ | $\tilde{t}_{1.4}^{70}$
---|---|---|---|---|---|---|---|---
$\langle d^{\textrm{flight}}_{\tilde{t}}\rangle~{}[\textrm{cm}]$ | 10 | 30 | 50 | 70 | 10 | 30 | 50 | 70
$m_{\tilde{t}}~{}[\textrm{TeV}]$ | 1.0 | 1.0 | 1.0 | 1.0 | 1.4 | 1.4 | 1.4 | 1.4
$m_{3/2}~{}[\textrm{keV}]$ | 22 | 39 | 50 | 59 | 53 | 93 | 120 | 142
Table 5: Definition of the eight benchmarks for the stop squark pair
production $pp\rightarrow\tilde{t}\bar{\tilde{t}}$ analysis.

####  3.2.4 Event kinematics

Classical geometric and kinematic observables have been studied for each
benchmark. Figs. 51 to 59 are a selection of some relevant observables for
this model. A summary of the results can be found in Table 6.

The first point to take into account is that the flight distance of the stop
squark has no impact on several observables (see Table 6) because the
difference in gravitino masses in the benchmarks is negligible. The kinematics
study will be mainly based on only two sets of distributions: a stop mass of
$1.0\ \textrm{TeV}$ and $1.4\ \textrm{TeV}$ regardless of the mass of the
gravitino.444This simplification is valid for benchmarks with $\langle
d^{\text{flight}}_{\tilde{t}}\rangle$ not too close to the upper limit of the
tracker volume, i.e., 100 cm. If we assume a benchmark with $\langle
d^{\text{flight}}_{\tilde{t}}\rangle=90\ \text{cm}$ or more, a non-negligible
amount of stops will decay after the tracker volume which will impact the
global observables reconstruction. Distributions of prompt top quark pair-
production ($pp\rightarrow t\bar{t}$) have also been added, which is one of
the possible background sources. Moreover, the multiplicities of displaced
tracks corresponding to one displaced signature are close for the two stop
squark mass scenarios, with a mean value between 34 and 38.

Nevertheless, the benchmarks differ for a fixed stop mass in terms of impact
parameter distributions (see Figs. 51 and 51 where the $d_{0}$ and $d_{z}$
distributions get wider for larger flight distance values). The choice of the
flight distance parameter also leads to a different multiplicity of displaced
top quark signatures in the same event: two, one or zero. The fraction of
events corresponding to each configuration is displayed in Table 6. Except for
the $\langle d^{\text{flight}}_{\tilde{t}}\rangle=10\ \text{cm}$ case, more
than $90\%$ of the events have at least one stop squark decaying inside the
tracker volume.

| $\tilde{t}_{1.0}^{10}$ | $\tilde{t}_{1.0}^{30}$ | $\tilde{t}_{1.0}^{50}$ | $\tilde{t}_{1.0}^{70}$ | $\tilde{t}_{1.4}^{10}$ | $\tilde{t}_{1.4}^{30}$ | $\tilde{t}_{1.4}^{50}$ | $\tilde{t}_{1.4}^{70}$ | $t\bar{t}$
---|---|---|---|---|---|---|---|---|---
$\sigma_{LO}(pp\rightarrow\tilde{t}\bar{\tilde{t}})~{}[\text{fb}]$ | 4.7 | 0.4 | -
$R_{\ =\ 2\ \textrm{displaced t}}$ [%] | 60.0 | 92.5 | 96.3 | 93.9 | 67.8 | 94.7 | 97.3 | 94.7 | -
$R_{\ \geq\ 1\ \textrm{displaced t}}$ [%] | 71.8 | 97.4 | 99.0 | 96.6 | 77.2 | 98.1 | 99.0 | 96.7 | -
$\langle\text{MET}\rangle$ [GeV] | 495 | 494 | 494 | 493 | 696 | 692 | 698 | 698 | 43
$\langle\text{TET}\rangle$ [GeV] | 1144 | 1138 | 1142 | 1141 | 1454 | 1447 | 1453 | 1451 | 465
$\langle\text{THT}\rangle$ [GeV] | 785 | 778 | 781 | 781 | 1000 | 993 | 1001 | 1003 | 327
$\langle\text{MHT}\rangle$ [GeV] | 351 | 348 | 349 | 348 | 491 | 486 | 491 | 488 | 52
$\langle\alpha_{T}\rangle$ | 0.55 | 0.55 | 0.55 | 0.55 | 0.53 | 0.53 | 0.54 | 0.54 | 0.48
$\langle p_{T}(j_{1})\rangle$ [GeV] | 459 | 459 | 460 | 460 | 668 | 664 | 667 | 663 | 118
$\langle p_{T}(j_{2})\rangle$ [GeV] | 30 | 30 | 56 | 47 | 38 | 55 | 70 | 55 | 23
$\langle p_{T}(\ell_{1})\rangle$ [GeV] | 105 | 105 | 106 | 105 | 133 | 134 | 136 | 136 | 32
$\langle p_{T}(\ell_{2})\rangle$ [GeV] | 31 | 32 | 32 | 32 | 40 | 40 | 35 | 37 | 31
$\langle\Delta R(bq_{W})\rangle$ | 1.0 | 1.0 | 1.0 | 1.0 | 0.9 | 0.9 | 0.9 | 0.9 | 1.5
$\langle|d_{0}(t)|\rangle$ [cm] | 3.2 | 9.6 | 15.9 | 22.3 | 3.6 | 10.7 | 17.9 | 25.0 | -
$\langle|d_{z}(t)|\rangle$ [cm] | 7.2 | 21.1 | 34.0 | 46.4 | 7.1 | 20.8 | 33.9 | 46.2 | -
$N_{\text{jets}}$ | 5.3 | 5.3 | 5.3 | 5.3 | 5.1 | 5.2 | 5.2 | 5.2 | 4.3
$N_{\text{tracks from displaced t}}$ | 34.3 | 34.0 | 34.0 | 34.1 | 37.6 | 37.4 | 37.6 | 37.6 | -
Table 6: Mean values of some observables and number of events with top quarks
decaying in the tracker volume for the eight benchmarks of the stop squark
pair production. We remind that $R_{\geq\textrm{1 displaced t}}$ corresponds
to the number of events with at least one top quark decaying in the tracker
volume.

As expected, the heavier stop squark events induce a larger mean value on the
leading jet $p_{T}$, the THT, and the MET (around $460$, $780$ and $500\
\text{GeV}$, respectively, for $m_{\tilde{t}}=1.0\ \text{TeV}$ and $660$,
$1000$ and $690\ \text{GeV}$, respectively, for $m_{\tilde{t}}=1.4\
\text{TeV}$). Two particles contribute to the MET in our events: the LSP
gravitino and the neutrinos coming from the leptonic decay of the W bosons
(this latter contribution is, however, negligible in comparison to the
gravitino). The MET in the prompt-top background (which only comes from the
neutrinos) is relatively small (around 40 GeV, see Table 6). The leptons
coming from the decay of the W bosons also get a high transverse momentum with
mean value of $105$ and $135\ \text{GeV}$ for $m_{\tilde{t}}=1.0$ and $1.4\
\text{TeV}$, respectively.

Figure 46: Missing transverse energy MET in the stop squark $\tilde{t}$ case.

Figure 47: $\alpha_{T}$ distributions in the stop squark $\tilde{t}$ case.

Figure 48: $\Delta R$ distributions in the stop squark $\tilde{t}$ case.

Figure 49: Multiplicity of tracks generated by the long-lived particle for the
eight benchmarks in the stop squark $\tilde{t}$ case.

Figure 50: Transverse impact parameter $d_{0}$ distributions for the eight
benchmarks for the long-lived stop squark with a magnetic field of $B=3.8\
\text{T}$.

Figure 51: $d_{z}$ distributions for the eight benchmarks for the long-lived
stop squark with a magnetic field of $B=3.8\ \text{T}$.

Figure 52: Total transverse energy TET distributions in the stop squark
$\tilde{t}$ case.

Figure 53: Missing transverse hadronic energy MHT in the stop squark
$\tilde{t}$ case.

Figure 54: Total transverse hadronic energy in the stop squark $\tilde{t}$
case.

Figure 55: Transverse momentum $p_{T}$ of the leading jet in the stop squark
$\tilde{t}$ case.

Figure 56: Transverse momentum $p_{T}$ of the second leading jet in the stop
squark $\tilde{t}$ case.

Figure 57: Transverse momentum $p_{T}$ of the leading lepton in the stop
squark $\tilde{t}$ case.

Figure 58: Transverse momentum $p_{T}$ of the second leading lepton in the
stop squark $\tilde{t}$ case.

Figure 59: Jet multiplicity $N_{j}$ in the stop squark $\tilde{t}$ case.

Figure 51 presents the distributions of $\alpha_{T}$ (as defined in Section
2.3). The two distributions of signals are symmetric with a central value of
about $0.5$, which is the typical value from which the QCD contribution
decreases fiercely and vanishes. This can be seen on the top-prompt
distribution where the majority of the events correspond to $\alpha_{T}<0.5$.
An offline selection on this observable may therefore be useful to increase
the signal over background ratio.

From the large mean values of the transverse momentum of the hardest jet
$\langle p_{T}(j_{1})\rangle$ in Table 6, we may identify boosted topologies
in our events. It may impact the jet reconstruction in our signal. Indeed, as
the b quark and the quarks from the W boson decay (coming from the decay of
the top quark) get closer, they appear as one fat jet instead of several jets.
To investigate this effect, the distributions of $\Delta
R(bq)=\sqrt{\Delta\eta(bq)^{2}+\Delta\phi(bq)^{2}}$ have been displayed (see
Fig. 51). This variable characterises the angular distance between b quarks
and the closest light quark coming from the W boson decay, which also arises
from $t\rightarrow bW$. Through the anti-$k_{T}$ algorithm with $\Delta R=0.4$
and $p_{T}^{min.}=30\ \text{GeV}$, a non-negligible portion of jets merge into
one single jet. Therefore, an analysis of the jets substructure may be
relevant to identify the signal. In this case, a larger cone $\Delta R$ must
be used, leading to the reconstruction of a fat jet allowing us to study its
substructure.

###  3.3 Mediation by a coloured long-lived particle (R-hadron based on a
gluino)

The review of scenarios involving displaced top quarks signatures cannot be
complete without taking into account the production of long-lived gluinos.
Although this event topology is already studied at the LHC, a devoted
simplified model has been designed based on the split supersymmetry framework.

####  3.3.1 Model description

The split supersymmetry model Giudice_2005 ; Arkani_Hamed_2005  offers a
superpartner particle spectrum split into two categories. From one side, the
scalar particles, i.e., squarks and sleptons, acquire a mass close to the
energy scale of supersymmetry breaking which can be as large as the GUT scale.
From the other side, the fermions, the neutralinos and charginos have a mass
near the electroweak scale and can potentially solve the gauge-coupling
unification and dark matter puzzles of the Standard Model. In this context,
assuming that the lightest neutralino is the LSP and the gluino is the NLSP,
the allowed decay channels for the gluino infer a tiny decay width because
they involve a heavy-massive mediator as shown in Fig. 60. Therefore the
gluino (or more precisely the R-hadron containing the gluino) is expected to
be long-lived.

Figure 60: Feynman diagram describing the gluino decay in a top quark pair and
a neutralino

This simplified model is based on the R-parity conserved MSSM where the
spectrum of masses satisfies the two typical scales of the split supersymmetry
framework. As in the previous models, we define the stop squark as an equal
combination of left-handed and right-handed states. Concerning the neutralino,
we only study the gauginos and the higgsinos, without any combinations.

####  3.3.2 Cross sections and decays

Gluino production at the LHC is very similar to the stop production described
in the previous section. At LO QCD, the gluinos can be produced by quark
annihilation (Fig. 61) or by gluon fusion (Fig. 62). Diagrams involving a
squark particle mediator can be discarded by requiring squark masses greater
than 1 TeV, which is consistent with the split supersymmetry assumption. Such
diagrams should give a negligible contribution to the gluino production.
Therefore, the corresponding cross section depends only on the gluino mass.

Figures 63 and 64 show the evolution of the cross section production and the
mean Lorentz factor $\langle\beta\gamma\rangle$ of the gluino as a function of
the gluino mass $m_{\tilde{g}}$. The calculations have been achieved at the LO
QCD and more accurate values of the cross section can be found in the
literature Beenakker:1996ch ; Kauth:2011vg . With a gluino mass of $1\
\text{TeV}$, and assuming some theoretical assumptions, the K-factor
$K=\sigma_{NLO}/\sigma_{LO}$ for the gluino production is higher than 2. We
consider here only uncertainties on the PDF and the renormalisation and
factorisation scales. As expected, both observables decrease with the gluino
mass. The evaluation of the cross section production will help us to fix the
gluino mass in the benchmarks for a reasonable number of events at the Run 3.
The mean Lorentz factor $\langle\beta\gamma\rangle$ (see Fig. 64) varies
between 0.6 and 1.3 for the chosen masses, so the flight distance of the
gluino is not enhanced, as it was the case for the stop production.

Figure 61: Feynman diagram of gluino pair production by quark annihilation
Figure 62: Feynman diagrams of gluino pair production by gluon fusion Figure
63: Leading order cross section $\sigma(pp\rightarrow\tilde{g}\tilde{g})$ at
$\sqrt{s}=14\ \text{TeV}$ as a function of the gluino mass $m_{\tilde{g}}$.
The uncertainties are related to PDF variations and the factorisation and
renormalisation scales. Figure 64: Average value of the Lorentz factor
$\beta\gamma$ of gluinos as a function of the gluino mass $m_{\tilde{g}}$.

At LO QCD, the only possible decays have the following topology:
$\tilde{g}\rightarrow\tilde{\chi}^{0}_{1}q\bar{q}$ where $q$ is a quark, and
potentially a top quark if the neutralino mass is sufficiently smaller than
the gluino one (see Fig. 60). For simplification purposes, we assume in the
following that the branching fraction of this decay channel involving top
quarks is 100%. It is worth to note that other channels can appear at the NLO
QCD but they are not taken into account in this study. Therefore, the gluino
decay width (and the gluino flight distance) depends on three masses (the
gluino mass, the stop mass, the neutralino mass) and the nature of the
neutralino, i.e., bino, wino or (up-like only) higgsino. Figures 65 and 66
illustrate this dependence. The first figure shows the average flight distance
of the gluino in the gluino mass - stop mass plane for a fixed value of
neutralino mass equal to 250 GeV and a neutralino defined as a bino state. The
second one shows the variation of the same quantity in the gluino mass -
neutralino mass plane for a fixed value of stop mass equal to 500 TeV and a
neutralino defined as a bino state. The nature of the neutralino has a
negligible impact on the order of magnitude of the gluino flight distance. We
conclude that a region of the parameter space allows us to observe a displaced
top quark signature, especially for a stop mass between 100 TeV and 1000 TeV
when the gluino mass is about 1 - 3 TeV.

Figure 65: Average flight distance in the laboratory frame of the gluino
$\langle d_{\tilde{g}}^{\textrm{flight}}\rangle$ as a function of the gluino
mass $m_{\tilde{g}}$ and the stop mass $m_{\tilde{t}}$ for a fixed value of
the neutralino mass $m_{\tilde{\chi}^{0}_{1}}=250~{}\textrm{GeV}$ and
$\tilde{\chi}^{0}_{1}=\tilde{B}$. The black lines represent the geometrical
limits of the tracker volume. Figure 66: Average flight distance in the
laboratory frame of the gluino $\langle
d_{\tilde{g}}^{\textrm{flight}}\rangle$ as a function of the gluino mass
$m_{\tilde{g}}$ and the neutralino mass $m_{\tilde{\chi}^{0}_{1}}$ for a fixed
value of the stop mass $m_{\tilde{t}}=500~{}\textrm{TeV}$ and
$\tilde{\chi}^{0}_{1}=\tilde{B}$. The black lines represent the geometrical
limits of the tracker volume. The white region is forbidden from the
constraint $m_{\tilde{\chi}^{0}_{1}}+2m_{t}<m_{\tilde{g}}$.

####  3.3.3 Benchmarks definition

Name | $\tilde{g}_{1.8\cdot 1.3}^{10}$ | $\tilde{g}_{1.8\cdot 1.3}^{30}$ | $\tilde{g}_{1.8\cdot 1.3}^{50}$ | $\tilde{g}_{1.8\cdot 1.3}^{70}$ | $\tilde{g}_{2.2\cdot 1.3}^{10}$ | $\tilde{g}_{2.2\cdot 1.3}^{30}$ | $\tilde{g}_{2.2\cdot 1.3}^{50}$ | $\tilde{g}_{2.2\cdot 1.3}^{70}$
---|---|---|---|---|---|---|---|---
$\langle d^{\textrm{flight}}_{\tilde{g}}\rangle~{}[\textrm{cm}]$ | 10 | 30 | 50 | 70 | 10 | 30 | 50 | 70
$m_{\tilde{g}}~{}[\textrm{TeV}]$ | 1.8 | 1.8 | 1.8 | 1.8 | 2.2 | 2.2 | 2.2 | 2.2
$m_{\tilde{\chi}_{1}^{0}}~{}[\textrm{TeV}]$ | 1.3 | 1.3 | 1.3 | 1.3 | 1.3 | 1.3 | 1.3 | 1.3
$m_{\tilde{t}}$ [TeV] | 489 | 643 | 731 | 795 | 1347 | 1770 | 2013 | 2189
$\tilde{\chi}^{0}_{1}$ type | $\tilde{B}$ | $\tilde{B}$ | $\tilde{B}$ | $\tilde{B}$ | $\tilde{B}$ | $\tilde{B}$ | $\tilde{B}$ | $\tilde{B}$
Table 7: Definition of the eight benchmarks for the long-lived gluino analysis

By using the results from Figs. 63 to 66, we have built eight different
benchmark points and their definition is given in Table 7. Two values of cross
section have been chosen by fixing the gluino mass: 1.8 TeV and 2.2 TeV.
Concerning the neutralino, we choose the bino state and its mass is fixed to
1.3 TeV. The mean flight distance of the gluino in the laboratory frame is a
parameter of the benchmark point, and the stop mass is adjusted for this
purpose. These benchmarks take into account the current experimental limits
coming from the search of prompt-gluino production (for ATLAS ATLAS:2021gld ;
ATLAS:2020syg ; ATLAS:2019fag  and for CMS CMS:2019zmd ; CMS:2019ybf ;
CMS:2021beq ; CMS:2020cur ; CMS:2020cpy ; CMS:2017qxu ; CMS:2017okm ;
CMS:2017abv ). The best experimental constraints forbid a gluino with a mass
lower than 2.2 or 2.3 TeV. The limits on the neutralino mass depend on the
gluino mass and can reach 1.2 or 1.3 TeV. Searches for long-lived gluinos have
been also scrutinised at LHC Run 1 and 2 by using the displaced jets signature
CMS:2020iwv ; CMS:2018ncl  or the displaced vertices signature
PhysRevD.104.052011 ; ATLAS:2017tny .

We note that the long-lived particle is not the gluino but the R-hadron made
up of the gluino. Three kinds of R-hadrons can be produced: the gluinoball
$R(\tilde{g}g)$, the R-meson $R(\tilde{g}q\bar{q})$ and the R-baryon
$R(\tilde{g}qqq)$. The gluinoball is always electrically neutral whereas
R-mesons and R-baryons can have a negative, neutral or positive electric
charge according to the flavour of the quarks. When it is electrically
charged, the R-hadron has a trajectory bent by the magnetic field and can
induce a reconstructed track in the tracker. For instance, charged R-hadrons
based on gluino and decaying outside the tracker (and so corresponding to HSCP
signatures) have been also studied by ATLAS and CMS collaborations on the
basis of the dE/dx energy deposit ATLAS:2022pib ; CMS:2016ybj . In the
following, we do not take into account this potential property and, by misuse
of language, we refer to the R-hadron as gluino.

####  3.3.4 Event kinematics

Classical geometrical and kinematics observables have been studied for each
benchmark. Figures 72 to 80 are a selection of some relevant observables for
this model. A summary of the results can be found in Table 8.

First, the main property of this signal topology is to offer 4 produced top
quarks per event instead of 2 in the previous cases. This multiplicity of top
quarks coming from the decay of a heavy R-hadron leads to a high multiplicity
of reconstructed jets (about 8 on average) with a large transverse momentum
(the hardest jet has a mean transverse momentum of about 230 GeV and 410 GeV
respectively for a gluino mass of 1.8 TeV and 2.2 TeV according to Fig. 72)
and a large hadronic activity in the calorimeter (the mean total hadronic
energy, THT, is about 640 GeV and 950 GeV for both gluino masses).

The probability to have a four displaced top quark signature is about 63% for
a mean flight distance of the R-hadron of 10 cm and it can increase to more
than 97% for a mean flight distance of 50 cm. As the Figs. 72 and 72 show, the
distributions of impact parameter $d_{0}$ and $d_{z}$ of the top quarks get
wider when the mean flight distance of the gluino increases and their mean
values are similar to the previous cases of stop benchmarks.

Another key observable is the MET distribution (see Fig. 72) since this
observable is expected to be larger for the gluino events than for the
$t\bar{t}$ events. In the signal sample, the MET is mainly dominated by the
neutralino contribution coming from the decays of the two gluinos. The other
contributions are the neutrinos coming from the top quark decays and the rare
cases of gluino decaying outside the calorimeter. The MET mean value is about
260 GeV and 480 GeV for gluino with a mass of 1.8 TeV and 2.2 TeV,
respectively.

Similarly to the stop production case, Figure 72 shows that the indicator
$\alpha_{T}$, usually used in searches for supersymmetric signatures, can be
also used in this case to discriminate the signal from Standard Model
background sources like the $t\bar{t}$ one. This indicator can be also
relevant for discriminating the signal from the standard four top quarks
production if the selection strategy requires a such multiplicity of quark
tops with a small displacement. Contrary to the stop production case, the b
quark and the W boson coming from the top decay are separated enough in the
transverse plane in order to reconstruct non-merged jets, see Figure 80.
Obviously, the flight distance of the gluino, driven by the stop mass, has a
negligible impact on the angle between the b quark and the W decay products.

Figure 67: Missing transverse energy MET in the gluino $\tilde{g}$ case. The
main contribution is coming from the invisible neutralino.

Figure 68: $\alpha_{T}$ distributions in the gluino $\tilde{g}$ case.

Figure 69: Transverse momentum of the leading jet in the gluino $\tilde{g}$
case.

Figure 70: Multiplicity of displaced tracks for each displaced top quark decay
vertex for a gluino mass of $1.8\ \mathrm{TeV}$ and $2.2\ \mathrm{TeV}$.o

Figure 71: Transverse impact parameter $d_{0}$ distributions for the eight
benchmarks with a magnetic field of $B=3.8\ \text{T}$.

Figure 72: $d_{z}$ distributions for the eight benchmarks with a magnetic
field of $B=3.8\ \text{T}$.

Figure 73: Total transverse energy TET in the gluino $\tilde{g}$ case

Figure 74: Total transverse hadrnoic energy THT in the gluino $\tilde{g}$ case

Figure 75: Missing transverse hadronic energy MHT in the gluino $\tilde{g}$
case.

Figure 76: Transverse momentum $p_{T}$ of the second leading jet in the gluino
$\tilde{g}$ case.

Figure 77: Transverse momentum $p_{T}$ of the leading lepton in the gluino
$\tilde{g}$ case.

Figure 78: Transverse momentum $p_{T}$ of the second leading lepton in the
gluino $\tilde{g}$ case.

Figure 79: Jet multiplicity $N_{j}$ in the gluino $\tilde{g}$ case.

Figure 80: $\Delta R$ distributions in the gluino $\tilde{g}$ case.

Name | $\tilde{g}_{1.8\cdot 1.3}^{10}$ | $\tilde{g}_{1.8\cdot 1.3}^{30}$ | $\tilde{g}_{1.8\cdot 1.3}^{50}$ | $\tilde{g}_{1.8\cdot 1.3}^{70}$ | $\tilde{g}_{2.2\cdot 1.3}^{10}$ | $\tilde{g}_{2.2\cdot 1.3}^{30}$ | $\tilde{g}_{2.2\cdot 1.3}^{50}$ | $\tilde{g}_{2.2\cdot 1.3}^{70}$ | $t\bar{t}$
---|---|---|---|---|---|---|---|---|---
$\sigma_{LO}(pp\rightarrow\tilde{g}\tilde{g})~{}[\text{fb}]$ | 2.0 | 0.3 | -
$R_{\ =\ 4\ \textrm{displaced t}}$ [%] | 63.2 | 94.3 | 97.5 | 96.4 | 65.5 | 94.3 | 97.3 | 96.3 | -
$R_{\ \geq\ 2\ \textrm{displaced t}}$ [%] | 72.1 | 97.2 | 99.0 | 98.1 | 72.4 | 97.1 | 98.7 | 97.8 | -
$\langle MET\rangle$ [GeV] | 254 | 253 | 253 | 257 | 473 | 470 | 470 | 478 | 43
$\langle TET\rangle$ [GeV] | 940 | 937 | 939 | 941 | 1422 | 1417 | 1417 | 1424 | 465
$\langle THT\rangle$ [GeV] | 642 | 639 | 641 | 641 | 962 | 953 | 955 | 955 | 327
$\langle MHT\rangle$ [GeV] | 189 | 188 | 188 | 190 | 340 | 338 | 339 | 335 | 52
$\langle\alpha_{T}\rangle$ | 0.54 | 0.54 | 0.54 | 0.54 | 0.56 | 0.56 | 0.56 | 0.56 | 0.48
$\langle p_{T}(j_{1})\rangle$ [GeV] | 234 | 230 | 231 | 235 | 410 | 409 | 408 | 409 | 118
$\langle p_{T}(j_{2})\rangle$ [GeV] | 40 | 33 | 51 | 57 | 50 | 53 | 80 | 52 | 23
$\langle p_{T}(\ell_{1})\rangle$ [GeV] | 56 | 57 | 56 | 56 | 92 | 94 | 93 | 95 | 32
$\langle p_{T}(\ell_{2})\rangle$ [GeV] | 23 | 23 | 22 | 23 | 32 | 33 | 32 | 33 | 31
$\langle\Delta R(bq_{W})\rangle$ | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.5
$\langle|d_{0}(t)|\rangle$ [cm] | 3.0 | 9.3 | 15.3 | 21.3 | 3.4 | 10.2 | 17.2 | 24.0 | -
$\langle|d_{z}(t)|\rangle$ [cm] | 6.9 | 20.3 | 32.3 | 44.3 | 7.5 | 22.1 | 36.0 | 49.2 | -
$N_{\text{jets}}$ | 7.8 | 7.8 | 7.8 | 7.8 | 8.2 | 8.2 | 8.2 | 8.2 | 4.3
$N_{\text{tracks from displaced t}}$ | 51 | 51 | 51 | 51 | 56 | 56 | 56 | 56 | -
Table 8: Mean values of some observables and number of events with top quarks
decaying in the tracker volume for the eight benchmarks of the long-lived
gluino analysis. We remind that $R_{\geq\textrm{2 displaced t}}$ corresponds
to the number of events with at least two top quarks decaying in the tracker
volume.

##  4 Discussion on the design of experimental analyses devoted to displaced
top quark signature

Using the various properties and distributions of the studied benchmarks, we
discuss possible strategies which can be implemented in future experimental
analyses for extracting the signal from the background. The discussion will be
split in two parts: one will be devoted to the online selection by the trigger
system of the ATLAS and CMS multipurpose detectors, the other will be devoted
to the offline selection achieved on the data.

###  4.1 Selecting online the signal events with the detector trigger system

Due to the high frequency of proton-proton collisions at the LHC ($40\
\text{MHz}$), the CMS and ATLAS collaborations use (hardware and software)
trigger systems to extract events of interest for physics studies and detector
commissioning. It is crucial for experimentalists to define a strategy for
triggering on the signal events. Traditional observables used for prompt
physics can be relevant to the signal topologies studied in this paper.

  * $\bullet$

The main interest of the production of displaced top quarks by a pair of
sleptons remains in the presence of two prompt leptons in the final state. The
transverse momentum of these muons for a small mass of neutralino (mean
$p_{T}$ of 90 GeV for a neutralino mass of 200 GeV) is ideal for using lepton
triggers. This kind of trigger can be also used, to a lesser extent, for
firing events where the top quarks decay leptonically and with a small
displacement. For instance, leptons coming from top quarks produced by a
neutralino pair have an average transverse momentum of $\sim 45-50\
\text{GeV}$.

  * $\bullet$

Displaced top quark events are very energetic. For this reason,
experimentalists have the opportunity to use the total hadronic energy (THT)
in the event or the presence of jets with a large $p_{T}$. For instance, in
the case where the long-lived particle is a R-hadron, THT can reach more than
1 TeV and the hardest jets have a mean $p_{T}$ between 400 and 700 GeV.

  * $\bullet$

A trigger associated with MET may be relevant to suppress the QCD background
and highlight the signal from the SM $t\bar{t}$ production. This MET is the
sum of the contributions of the LSP (except the case where it is the long-
lived particle), the neutrinos coming from the top quark decay and the small
number of events where the long-lived particle decayed outside the detector.
Despite the extremely small LSP mass, the gravitino contributes to most of the
MET in the stop benchmarks (between 500 and 700 GeV, depending on the stop
mass). This allows us to use a trigger based on a MET threshold.

Undeniably, the trigger strategy can be based on only one of the previous
discriminating observables or on the combination of several observables with
lower thresholds (if they are not fully correlated). As an illustration, some
HSCP analyses have a trigger based on the criteria $\text{MET}>170\
\text{GeV}$ and a reconstructed muon with $p_{T}>50\ \text{GeV}$
Khachatryan_2016 . Thus, such selection thresholds on MET can be applied in
the case of the benchmarks containing large MET sources (stop squark
$\tilde{t}$ and gluino $\tilde{g}$ models). Moreover, all studied signal
sources present a large value of THT and large transverse momenta for jets. A
criterion such as $\text{THT}>430\ \text{GeV}$ is relevant, as well as
$p_{T}(j_{1})>150\ \text{GeV}$. For the bino case, a selection rule on the
leading muon may help to trigger the production of a smuon pair with
$p_{T}(\mu_{1})>24\ \text{GeV}$. Assuming such criteria, the $t\bar{t}$
background is decreased by 90% and 99%, without and with a cut on MET,
respectively. According to the considered benchmark, the signal is reduced by
a ratio varying between 60 and 30% for the bino $\tilde{B}$-like case, 70% and
30% for the higgsino $\tilde{h}$-like case, 20% and 10% for the stop squark
$\tilde{t}$ case and between 30% and 10% for the gluino $\tilde{g}$ case.

This discussion cannot be complete without mentioning the alternative to
design and develop a specific trigger path for the displaced top quark
signatures. The main difficulty comes from the difficulty to reconstruct
displaced vertices or tracks in the first level of the trigger system. The
design of a sophisticated discriminator taking into account the displacement
requires the information coming from the tracker and the calorimeters while
satisfying the constraint of processing time budget. A review of ongoing or
scheduled trigger development dedicated to long-lived particles for the LHC
Run 3 can be consulted Alimena:2021mdu .

###  4.2 Selecting the signal after the trigger system

For designing a complete experimental analysis, the main background sources
must be considered. The most important one for signal events with a low
displacement is composed of top quark pairs $t\bar{t}$ and QCD processes in
general with an associated MET coming from an incorrect reconstruction, $tW$
processes or $t\bar{t}+V$ (with $V=Z$ or $W$). Drell-Yan events with
associated jets are also relevant in the case of top quarks produced
indirectly by sleptons, due to the presence of leptons in the final state.
This list of background sources must be completed by taking under
consideration all diboson cases, or even tribosons (associated to jets). For
extracting the signal from the Standard Model contributions, experimentalists
must keep a close eye on several points.

  * $\bullet$

The offline selection should refine the criteria imposed by the trigger
system, i.e. the transverse momenta of the leptons and jets, the total
hadronic energy THT and the missing transverse energy MET. For the slepton
benchmarks, Drell-Yann background can be reduced by requiring a specific range
of values for the dilepton invariant mass or, more safely, by applying
thresholds on hadronic variables.

  * $\bullet$

As an offline selection devoted to the benchmarks introducing a long-lived
R-hadron, cuts on various observables such as $\alpha_{T}$ with
$\alpha_{T}>0.5$ (see Fig. 51) may help to suppress standard background
events.

  * $\bullet$

A special care must be taken with the reconstruction of jets in the context of
stop-based R-hadrons. Indeed, the study of the kinematic properties reveals
that quarks coming, directly or indirectly, from the top quark decay are very
close for a large fraction of events. Collimated jets can be reconstructed and
the jet substructure can be investigated for identifying properly the
displaced top quark signature.

The majority of the displaced vertex background is identified as effects that
are not related to the pp collision, but to secondary interactions and
instrumental noises. For instance, nuclear interactions within the detector
matter can also induce secondary vertices, as well as reconstruction
algorithm-induced fakes (see Alimena:2019zri  for more information).

Moreover, assuming a full simulation for the background process, a lot of
$t\bar{t}$ events will induce secondary interaction with layers of detection
which can be confused with displaced processes. Since the production cross
section of $t\bar{t}$ is much larger than for the studied signals, the
multiplicity of displaced tracks is then not enough to highlight the signal
over the background. Moreover, applying a reconstruction efficiency on the
reconstructed displaced tracks will decrease such multiplicity. Note that the
precise experimental analyse based on the signal-background ratio is out of
the scope of this paper.

###  4.3 Towards a specific algorithm devoted to displaced top quark signature

Currently, there are no specific algorithms devoted to the identification of
displaced top quarks. Nonetheless, such algorithms could be designed by
gathering information from displaced jets, b-tagging, and displaced vertices
(taking advantage of the multiplicity of displaced tracks coming from the top
quark decay). Note also that each method cited here can also be used on its
own. Thus, including such reconstruction in the analysis, it would be possible
to require at least one displaced top quark in the events, and thus increase
the signal/background ratio. The amount of events offering several displaced
quark top signatures raises the question to identify and to reconstruct
several displaced candidates. Defining signal regions sorted according to the
multiplicity of displaced candidates can increase the sensitivity of the
analysis. Moreover, specific Multi-Variate Analysis can also be set up using
techniques such as Boosted Decision Trees or neutral networks in order to
improve the performance of the selection.

##  5 Summary

In this paper, we have addressed the displaced top quark signature which
extends the list of displaced objects (displaced jets, displaced vertex,
displaced leptons, …) already studied by the CMS and the ATLAS collaborations
and which should lead to a complementary approach to the inclusive research of
exotic long-lived particles at the LHC. We have presented a comprehensive
review of event topologies that offer such a signature. For each topology, a
simplified model has been designed, the relevant region of the parameter space
has been highlighted and some promising benchmark points have been identified.
Some of these physics processes are known and studied with other displaced
objects like the long-lived gluino, and others are still unexplored like the
displaced neutralino stemming from slepton pair production. The simplified
models studied here cover also various theoretical scenarios: the presence of
a dark matter candidate with the GMSB mechanism, the non-conservation of
R-parity in the RPV scenario and the non-naturalness in the Split-SUSY model.
The properties of the benchmarks have been illustrated by distributions of key
observables and their relevance has been shown by comparing the obtained
distributions with the distributions of Standard Model top quark pair
production.

This work is the first concrete step toward the exploration of TeV physics
with the displaced top quark signature. It provides practical tools for both
experimentalist and phenomenologist communities for LHC Run 3. First, the
benchmarks can be used for evaluating the performance of reconstruction and
identification algorithms of the ATLAS and CMS detector in extreme and
challenging conditions. Devoted algorithms to displaced top quark can be built
in order to optimise the sensitivity to this signature and to probe data with
it. The benchmarks can also interest phenomenologists working in the
reinterpretation of the LHC experimental limits by using some framework like
Rivet Buckley_2013 , CheckMate Dercks_2017  or MadAnalysis 5 Conte_2014 .

##  6 Acknowledgments

The authors are grateful to the CMS and ATLAS collaborations for the
interesting discussions about this prospective work. We would like to thank
also Gilbert Moultaka and Michel Rausch de Traubenberg for their precious help
on the GMSB and the supergravity models. We cannot conclude without thanking
Benjamin Fuks and Jack Araz for their valuable developments within the
MadAnalysis 5 framework.

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