aainstitutetext: Department of Theoretical Physics, Tata Institute of
Fundamental Research, Mumbai, India 400005 bbinstitutetext: Bethe Center for
Theoretical Physics & Physikalisches Institut der Universität Bonn,
Nußallee 12, D–53115 Bonn, Germanyccinstitutetext: National Institute for
Theoretical Physics and
School of Physics, University of the Witwatersrand, Johannesburg, South
Africaddinstitutetext: Instituto de Física Corpuscular, IFIC-UV/CSIC,
Valencia, Spaineeinstitutetext: Faculty of Physics, University of Warsaw,
Polandffinstitutetext: Institute for High Energy Physics, Austrian Academy of
Sciences, Austriagginstitutetext: Department of Physics, National Tsing Hua
University, Hsinchu 300, Taiwanhhinstitutetext: Asia Pacific Center for
Theoretical Physics (APCTP) \- Headquarters San 31,
Hyoja-dong, Nam-gu, Pohang 790-784, Korea

# Constraining electroweak and strongly charged long-lived particles with
CheckMATE

Nishita Desai b , Florian Domingo c , Jong Soo Kim d , Roberto Ruiz de Austri
Bazan e , Krzysztof Rolbiecki f , Mangesh Sonawane gh , Zeren Simon Wang
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Long-lived particles have become a new frontier in the exploration of physics
beyond the Standard Model. In this paper, we present the implementation of
four types of long-lived particle searches, viz. displaced leptons,
disappearing track, displaced vertex with either muons or with missing
transverse energy, and heavy charged tracks. These four categories cover the
signatures of a large range of physics models. We illustrate their potential
for exclusion and discuss their mutual overlaps in mass-lifetime space for two
simple phenomenological models involving either a $U(1)$-charged or a coloured
scalar.

††preprint:  APCTP Pre2020-029 BONN-TH-2021-02 TIFR/TH/20-41

##  1 Introduction

The Large Hadron Collider (LHC) started operations over a decade ago and a
large number of searches for physics beyond the Standard Model (BSM) have been
performed. In particular, the general-purpose experiments ATLAS and CMS have
published over a thousand papers each, minutely testing the predictions of the
Standard Model and of more exotic theories. For the simplest cases of new
physics consisting of new strongly charged particles, corresponding searches
already place limits on the BSM masses at about 2-3 TeV Aad:2020aze , closing
in on the maximum reach achievable at a 13 TeV proton collider. We turn our
attention therefore to the equally well-motivated but experimentally more
challenging cases of new physics where BSM particles are long-lived. A long
lifetime may be the natural consequence of a compressed phase space (e.g. in
particular dark matter models Ibe:2006de ; Cirelli:2005uq ;
FileviezPerez:2008bj ; Cirelli:2009uv ; Buckley:2009kv ; Johansen:2010ac ;
Mahbubani:2017gjh ; Khoze:2017ixx ), a suppressed connection to light SM decay
products caused by heavy mediators Giudice:2004tc , or direct but feeble
couplings to SM particles (possibly resulting from an approximate symmetry)
Hall:2009bx ; ArkaniHamed:2008qp ; Graham:2012th ; Zwane:2015bra ;
Belanger:2018sti . These new kinds of scenarios have raised considerable
interest in the community Alimena:2019zri  and making the results of
associated searches available to reinterpretation in terms of other theory
models has been a frequent request to the CheckMATE collaboration.

With the end of Run 2 of the LHC, both ATLAS and CMS have turned their
attention to searches for exotic physics based on new particles with long
lifetimes. The signatures of a long-lived particle (LLP) depend on its charges
as well as its decay modes and actual lifetime, and they can be fairly
complicated to systematically characterize. Several possibilities for LLP
signatures indeed exist, based on the LLP decay modes:

  * •

neutral LLP $\rightarrow$ invisible or neutral stable particles $\Rightarrow$
missing momentum;

  * •

neutral LLP $\rightarrow$ charged leptons $\Rightarrow$ leptons with large
impact parameter (i.e. “displaced” leptons);

  * •

neutral LLP $\rightarrow$ coloured particles $\Rightarrow$ displaced vertices,
or emerging jets;

  * •

stable, charged LLP $\Rightarrow$ charged track (with its time of flight
dependent on mass and boost);

  * •

charged LLP $\rightarrow$ invisible $\Rightarrow$ “disappearing” track;

  * •

charged LLP $\rightarrow$ other charged stable object(s) $\Rightarrow$ kink-
track or displaced vertex.

There is a built-in complementarity in different searches simply because
particle decay follows an exponential distribution. For example, charged LLPs
with intermediate lifetimes will be visible in both disappearing track and
heavy charged track searches. Similarly, a neutral particle decaying into
quarks mostly in the electromagnetic or hadronic calorimeter will also likely
appear as a smaller, simultaneous signal in the displaced vertex (decay in the
tracker) and emerging jet (decay in the hadronic calorimeter) searches.
Furthermore, the lifetime of a particle in the lab-frame also depends on its
boost, which means that the production mechanism can also significantly alter
where in the detector the particle decays. The same particle may accordingly
result in different signal distributions depending on whether it is produced
“directly” or in the decays of a much heavier particle (i.e. with a higher
boost). Finally, several decay modes may be open to a single new particle
resulting in sensitivity in multiple searches. The identification of the
underlying physics therefore requires a full coverage in terms of the lifetime
of new particles.

As we can see, the identification of an LLP is highly complicated and so far,
there are no standard algorithms like those available for the identification
of standard objects, such as leptons, $b$\- or $\tau$-tagged jets, etc.
Consequently, it is not always clear how the results of a dedicated LLP search
can be “reinterpreted” for a physics model that differs from the tested one,
though it displays a priori similar signatures. A detailed study of models
capable of LLP signatures, the reinterpretation struggles and recommendations
have been detailed in the community study Alimena:2019zri . In this present
work, we use the signal efficiencies published by the experiments in order to
implement five searches in the CheckMATE reinterpretation package. The current
searches use the existing respective detector implementations for ATLAS and
CMS experiments in Delphes. It should also be possible to implement dedicated
searches from experiments like FASER Feng:2017uoz , CODEX-b Gligorov:2017nwh
or even proposed experiments like MATHUSLA Chou:2016lxi  if a corresponding
Delphes module or efficiency parametrisations become available.

CheckMATE Drees:2013wra ; Dercks:2016npn  is a public tool that allows the
reinterpretation of a wide variety of ATLAS and CMS results for new physics
models in a coherent and cohesive manner. It consists of an engine written in
C++ that runs each analysis cut-by-cut in order to assess the final number of
expected events satisfying the requirements of the corresponding analysis. The
engine is also capable of using external libraries like Madgraph
Alwall:2014hca  and Pythia 8 Sj_strand_2015 111Internal running of Monte Carlo
processes or use of Madgraph interface in CheckMATE is currently compatible
only with the 8.2 series of Pythia 8. in order to generate events, while the
detector simulation is performed by Delphes de_Favereau_2014 . The User
Interface and the statistical analyses are provided by a collection of Python
scripts (including the AnalysisManager Kim:2015wza  that guides the users
through the implementation of their own analyses).

In section 2, we briefly summarize the main ingredients of the LLP recast,
referring the reader to the appendix for a more complete description. Then, in
section 3, we illustrate the performance of the implemented searches in two
simple models with LLPs, and discuss their complementarity. Conclusions and a
brief outlook are proposed in section 4. The appendix consists of a short
guide for the user, as well as a more detailed presentation of the implemented
LLP searches.

##  2 Implementation of long-lived particle searches

Below we offer a brief description of the implemented LLP searches and a
comparison with experimentally published results. This includes the 8 and 13
TeV versions of the CMS displaced lepton search Khachatryan_2015 ; CMS-PAS-
EXO-16-022 , two different displaced vertex searches Aaboud:2017iio ;
Aad:2020srt , the 13 TeV ATLAS disappearing track Aaboud:2017mpt  and heavy
charged particle track Aaboud:2019trc  searches. Together, these searches are
capable of probing a wide range of parameter space. Details of the
implementation are available in the appendix. Technically, each analysis is
encapsulated in a detector-specific “analysis handler” class which provides
special functions and efficiencies specific to the detector in question. We
deliberately separate the analysis handlers for long-lived particle searches
from those used in prompt searches: this accounts for the fact that the
implemented prompt searches do not in fact use any decay length information
and all particles denoted stable by the Monte Carlo generator are then
clustered into jets based on their kinematics only.222For this reason, we
advise user discretion when applying prompt search limits to models with LLPs.
It may be possible to find a conservative limit from prompt searches by e.g.
removing decay products of LLPs from the Monte Carlo events beforehand.
However each case needs to be evaluated separately and we do not provide a
built-in solution for that reason.

###  2.1 Displaced Lepton searches

The displaced lepton searches Khachatryan_2015 ; CMS-PAS-EXO-16-022  look for
two high-$p_{T}$, isolated leptons ($\ell$) with large impact parameter
relative to the primary vertex. The benchmark used for this search is
motivated by R-parity violating (RPV) supersymmetry Barbier:2004ez ;
Graham:2012th  where a top-squark ($\tilde{t}_{1}$) decays via the lepton-
number-violating LQD operator as $\tilde{t}_{1}\rightarrow\ell b$. The leptons
thus produced have large $p_{T}$ and are well isolated. The two searches
implemented here correspond to 8 TeV Khachatryan_2015  and 13 TeV CMS-PAS-
EXO-16-022  versions of the CMS displaced supersymmetry search. The
identification and fiducial acceptances are provided on generator-level
events. We therefore reproduce the Monte-Carlo production process for
validation of the search. Corresponding details are provided in appendix B.

(a) 8 TeV

(b) 13 TeV

Figure 1: A Comparison of the exclusion limits on the Displaced Lepton search
provided by CMS with those obtained from CheckMATE (left: 8 TeV, 19.7 fb-1;
right: 13 TeV, 2.6 fb-1).

The event selection for both 8 TeV and 13 TeV analyses was performed in two
stages. The first stage (i.e. preselection) selects events with exactly one
electron and one muon with opposite electric charges, each expected from the
decay of a different top squark. Further selection cuts and isolation
requirements are then applied. In the second stage, the events are classified
into three signal regions (SR) corresponding to increasing ranges of the
leptonic impact parameter $d_{0}$.

The validation results are shown in Fig. 1, with the experimental exclusion
limits displayed in blue, while the recast produces the black exclusion bound.
A reasonable agreement is observed at $8$ TeV in Fig. 1(a). The situation at
$13$ TeV is somewhat more subtle.

Indeed, as efficiencies have been provided by the experimental collaboration
for the $8$ TeV, but not for the $13$ TeV search—in particular, the considered
ranges of $p_{T}$ and $d_{0}$ do not match and the modelling of efficiencies
in this latter case appears as an important assumption in the recast. An
attempt to validate the 13 TeV search using the 8 TeV efficiencies results in
poor agreement with the numbers in the signal region at large impact
parameters, and therefore in a much stronger expected limit for high values of
the LLP lifetime $c\tau$ (see the dashed black curve in Fig. 1(b)). A simple
linear interpolation performs rather poorly as well. We therefore make a
conservative estimate of this detector effect by adding a single bin for the
$d_{0}$ range (20 mm - 100 mm) and determine the associated efficiency via a
$\chi^{2}$-fit of the expected number of events in all three signal regions.
The outcome of this procedure is displayed in Fig. 1(b) as a solid black line
and shows a considerably improved agreement with the expected exclusion
limits. Exact numbers in each of the signal regions are produced in Appendix
B.

###  2.2 Displaced Vertex searches — DV + MET

This ATLAS search Aaboud:2017iio  looks for high-mass displaced vertices
(DVs), reconstructed from five or more tracks. Large missing transverse
momentum is also required. The outcome of $32.8~{}\text{fb}^{-1}$ of $13$ TeV
collision data is a yield consistent with the expected background.

The template considered by the ATLAS collaboration consists in the (strong)
production of a pair of long-lived gluinos ($\tilde{g}$), then decaying into
light quarks and stable neutralinos. Heavy squark mediators result in
suppressed gluino decay widths. Recast instructions were provided in DVrecast
and include a preselection at generator-level, followed by the application of
parametrized efficiencies. Previously to our implementation, this strategy has
been applied with success by the publicly available codes Cottincode ;
Lessacode  (see also Contribution $22$ in Brooijmans:2018xbu ). Details on the
implementation are provided in appendix C.

For the validation, we considered the limits on the gluino production cross-
sections presented in the ATLAS paper. In a first scenario, the long-lived
gluino and the neutralino LLP are separated by a wide mass-gap, with the
neutralino fixed at $100$ GeV while the gluino takes mass of
$m_{\tilde{g}}=1.4$ TeV or $2$ TeV. The LLP lifetime is varied between
$\tau_{\tilde{g}}=0.003$ ns and $50$ ns. The results from the recast search
are shown in Fig. 2, using two statistical approaches: the simplified
evaluation of CheckMATE defining a ratio $r$ (blue curve)—see Eq. (1) of
Dercks:2016npn —and the full $p$-value analysis (red curve); both return very
similar limits. The statistical uncertainty in the simulation is below percent
level ($10^{6}$ events are generated at each point), except for the end point
$\tau_{\tilde{g}}=0.003$ ns, where it reaches $3-4\%$. The $95\%$ CL limits
from the experimental analysis is shown in black. We observe a general
qualitative agreement. In the lower row of plots of Fig. 2, the experimental
observed limits are normalized to the limiting cross-sections of the recast
procedure (with $r$-approach in blue and $p$-values in red). Quantitatively,
we find that the bounds agree within $20\%$ of the cross-section value, with
outliers at up to $50\%$ discrepancy for small lifetimes.

Nevertheless, this apparent success of the recast strategy with efficiencies
applied on truth-level objects needs to be tempered as it seems to perform
worse in the case of a compressed spectrum. This was confirmed to us by the
authors of Cottincode ; Lessacode . A more detailed comparison is provided in
the appendix.

Figure 2: Validation of the DV+MET search in the scenario with large mass-
splitting for two different benchmarks (left $m_{\tilde{g}}=1.4$ TeV, right:
$m_{\tilde{g}}=2$ TeV.). The bottom panel in both cases shows the ratio of the
published ATLAS exclusion to the one obtained by CheckMATE. There is no
observable difference between using the full CLs method and an exclusion based
on the ratio ($r$) of cross section from CheckMATE of events passing all cuts
to the 95% upper limits on cross sections published by ATLAS.

###  2.3 Displaced Vertex searches — DV + $\mu$

In this section we discuss a search for massive, long-lived particles decaying
to final states with a DV and an energetic muon Aad:2020srt . The search
analyzed $139~{}\text{fb}^{-1}$ of data collected by ATLAS at the centre of
mass energy 13 TeV.

The benchmark process considered by the experiment was pair production of top
squarks followed by the RPV decay into a light quark and a muon. Other physics
scenarios, for example, long-lived lepto-quarks, right-handed neutrinos or
long-lived electroweakinos in RPV, could result in similar signals including a
DV and a muon. In Section 3 we apply this search to sbottom pair-production
followed by the RPV decay.

The event selection defines two mutually exclusive trigger-based signal
regions: $E_{\mathrm{T}}^{\text{miss}}$ Trigger SR and Muon Trigger SR. The
former requires significant missing transverse momentum ($>180$ GeV), while
the latter is recorded with the muon trigger and has low ($<180$ GeV )
transverse momentum. Additionally, at least one displaced vertex is required
to be present in the fiducial region. There is no explicit requirement for a
signal muon to originate from the reconstructed vertex.

The search was validated using a benchmark RPV-supersymmetric (SUSY) model for
the process $pp\to\tilde{t}_{1}\tilde{t}_{1}$, $\tilde{t}_{1}\to\mu\,q$. In
Figure 3 we show a comparison of the ATLAS result and CheckMATE recasting in
the stop lifetime-mass plane, $\tau_{\tilde{t}}$–$m_{\tilde{t}}$. The yellow
band shows a 2-sigma range of the ATLAS expected exclusion limit, the blue
solid line is the ATLAS observed exclusion while the blue dashed the ATLAS
expected exclusion, and the black solid line shows an exclusion line obtained
with CheckMATE. Generally a good agreement is observed, however in a range of
lifetimes $0.01$–$0.1$ ns, the recast exclusion is significantly weaker,
though within the 2-sigma band. Further details can be found in Appendix D.

Figure 3: Comparison of exclusion limits in the stop mass and stop lifetime
plane, $m_{\tilde{t}}$–$\tau_{\tilde{t}}$, reported by ATLAS: expected –
dashed blue; observed – solid blue and CheckMATE – black solid. The yellow
band shows a 2-sigma range of the ATLAS expected exclusion limit.

###  2.4 Heavy Charged Particles searches

In this section we focus on the search for heavy charged long-lived particles
performed by the ATLAS experiment using a data sample of
$36.1~{}\text{fb}^{-1}$ of collisions at 13 TeV Aaboud:2019trc . In our
implementation we cover searches for long-lived charginos and sleptons.

The ATLAS collaboration reported no significant excess of observed data events
above the expected background in this search. Thus, the collaboration have
published upper limits at 95% confidence level on the cross-sections for stau
and chargino production for specific benchmark models. These limits have been
obtained applying the CLs prescription Read:2002hq .

For the validation of our implementation in CheckMATE we have employed
HistFitter Baak:2014wma  to estimate the CLs while running $10^{5}$ toy-
experiments, given the low backgrounds for the considered signal regions and
assuming a 10% signal uncertainty.

The validation has been performed through the comparison with the observed
upper cross-section limits reported by the ATLAS collaboration, as is depicted
in Fig. 4. A very good qualitative agreement is visible between the ATLAS
(red) and the CheckMATE-derived (dashed blue) limits for both chargino and
stau scenarios. More details can be found in Appendix E.

Figure 4: Validation of the HCP search – observed upper cross-section limits
using the best expected signal region for chargino pair production (left) and
stau pair production (right). ATLAS results are shown as a solid red line
while CheckMATE predictions are in form of a dashed blue line. The theory
prediction is displayed as a dot-dashed black line.

###  2.5 Disappearing track searches

The ATLAS collaboration presented a search333HEPData accessed from
https://doi.org/10.17182/hepdata.78375.v5 Aaboud:2017mpt  for direct
electroweak (EW) gaugino or gluino pair production with wino-like
electroweakinos (hence near-degenerate charged and neutral $SU(2)$-triplet
fermions). The chargino decays via
$\tilde{\chi}^{+}\rightarrow\pi^{+}\tilde{\chi}^{0}$; the neutralino is
stable. The experimental collaboration analysed data based on the integrated
luminosity of 36.1 fb-1 recorded between 2015 and 2016. Due to the small mass
difference between the two states (which is of the order of the pion mass),
the chargino is long-lived and the decay products are entirely invisible to
the detector. Thus the chargino appears as a “disappearing” track, i.e. a
track that does not reach the outer edges of the tracker detector but stops
before. In order to define a trigger isolating the signal from large SM
backgrounds, the search additionally demands a large momentum jet from
initial-state radiation or four jets originating from the gluino decay. The
observed number of events is consistent with the SM expectations and
constraints for wino-like charginos with a lifetime of 0.2 ns, and mass up to
460 GeV are derived. In the strong production channel, where the chargino
emerges from the decay of a gluino (colour octet fermion), limits on gluino
masses up to 1.65 TeV are reported, under the assumption of a chargino mass of
460 GeV and lifetime of 0.2 ns.

In order to reinterpret this search, we follow all procedures regarding
production of signal events described in the original paper as closely as
possible. A detailed description can be found in appendix F. The kinematic
cuts for both signal regions are summarised in Table 1. ATLAS further applies
quality requirements, as that the tracklet is required to have hits in all
four pixel layers, and a disappearance condition is demanded for each event,
as the number of SCT hits associated with the tracklet must be zero. Although,
the two latter cuts are impossible to simulate in a phenomenological study,
ATLAS provides efficiency maps for the tracklets for EW SR and strong SR,
respectively. In addition, the collaboration provides a transverse momentum
smearing function for the chargino. We also use the benchmark SLHA files for
the EW and strong scenarios, the pseudo analysis code, and all relevant data
made publicly available at HEPData DisSearchData .

EW SR | strong SR
---|---
at least one jet with $p_{T}>140$ GeV |  $p_{T}(j_{1})>100$ GeV, $p_{T}(j_{2})>50$ GeV, $p_{T}(j_{3})>50$ GeV
$E_{\text{T}}^{\text{miss}}>140$ GeV |  $E_{\text{T}}^{\text{miss}}>150$ GeV
$\Delta\Phi(E_{\text{T}}^{\text{miss}}$, jets($p_{T}>50$ GeV)) $>1.0$ |  $\Delta\Phi(E_{\text{T}}^{\text{miss}}$, jets($p_{T}>50$ GeV)) $>0.4$
cuts on charged LLP with $p_{T}>100$ GeV | cuts on charged LLP with $p_{T}>100$ GeV
Table 1: Summary of the selection criteria for signal events for direct electroweakino production and the strong channel channel where the chargino is produced in gluino decays. | CM $\tilde{\chi}_{1}^{\pm}\tilde{\chi}_{1}^{\pm}$ | CM $\tilde{\chi}_{1}^{+}\tilde{\chi}_{1}^{0}$ | CM $\tilde{\chi}_{1}^{-}\tilde{\chi}_{1}^{0}$ | CM all channels | ATLAS
---|---|---|---|---|---
Trigger | 445.1 | 624.0 | 274,4 | 1343.5 | 1276
Lepton Veto | 423.4 | 608.5 | 267.3 | 1308.2 | 1181
MET and jet requirements | 164.2 | 229.6 | 101.0 | 494.8 | 579
EW SR | 5.2 | 4.4 | 1.6 | 11.2 | 13.5
Table 2: Cutflow comparison for a chargino produced in direct electroweak production with ($m_{\tilde{\chi}^{\pm}_{1}}$, $\tau_{\tilde{\chi}^{\pm}_{1}}$) = (400 GeV, 0.2 ns). | CM | ATLAS
---|---|---
Trigger | 289 | 285
Lepton Veto | 277 | 278
MET and jet requirements | 216 | 202
strong SR | 11 | 11
Table 3: Cutflow comparison for a chargino produced in strong production
channel with ($m_{\tilde{g}}$, $m_{\tilde{\chi}^{\pm}_{1}}$,
$\tau_{\tilde{\chi}^{\pm}_{1}}$) = (1600 GeV, 500 GeV, 0.2 ns) in the high–MET
region.

We use the ATLAS benchmark points as the test case scenarios which correspond
to
$(m_{\tilde{\chi}^{\pm}_{1}},\tau_{\tilde{\chi}^{\pm}_{1}})=(400\,\rm{GeV},\,0.2\,\rm{ns})$
for the EW signal region and
$(m_{\tilde{g}},m_{\tilde{\chi}^{\pm}_{1}},\tau_{\tilde{\chi}^{\pm}_{1}})=(1600\,\rm{GeV},\,500\,\rm{GeV},\,0.2\,\rm{ns})$
in the strong signal region. The results for the EW and strong SR are
summarized in Table 2 and 3, respectively and our recast results show
satisfactory agreement with the public ATLAS results.

We did not validate the disappearing track search in a grid scan with ATLAS
exclusions, since the event generation requires matched events with two
additional partons in the final state in order to reproduce the ATLAS
cutflows. As only a few events would pass all selection cuts, such a scan
would be costly to perform and we thus opted against it.

##  3 Performance and interplay in LLP scenarios

In this section, we consider two simple LLP scenarios that put forward the
complementarity of the implemented searches.

###  3.1 Electroweak LLP

Figure 5: Upper limits in the lifetime-mass plane (left) and coupling-mass
plane (right) of the LLP scalar $\phi$. We do not get limits from the
disappearing track search due to a veto on leptons in the event.

We first consider a model addressing the case of electroweakly produced LLPs.
It extends the SM with a scalar ($\phi$), charged under $U(1)_{Y}$ and sharing
the gauge quantum numbers of a right-handed lepton, and a SM-singlet Dirac
fermion ($\chi$). An extra $Z_{2}$-symmetry under which both $\phi$ and $\chi$
are charged further constrains their interactions. The scalar is produced in
pairs via the Drell-Yan process $pp\rightarrow\phi^{*}\phi$. After that it
decays according to $\phi\rightarrow\ell\chi$, mediated by the Yukawa coupling
$y_{\ell}\phi\bar{\ell}_{R}\chi+\mathrm{h.c}$. The singlet fermion is assumed
to be stable. The model has been implemented in Pythia 8. The Lagrangian is
simply

$\mathcal{L}=\frac{1}{2}D_{\mu}\phi
D^{\mu}\phi-\frac{1}{2}m_{\phi}^{2}\phi^{2}+\bar{\chi}(i\gamma^{\mu}\partial_{\mu})\chi-m\bar{\chi}\chi-\sum_{\ell}(y_{\ell}\phi\bar{\chi}\ell_{R}+\mathrm{h.c.})\,.$
(1)

For small values of the Yukawa coupling, the scalar $\phi$ is long-lived and
could be visible in the charged track searches. In the case where the lifetime
is too short to leave the required hits in the tracker modules, the displaced
lepton search might detect the products of the decay. To examine the
complementarity of these searches, we set $y_{e}=y_{\mu}$, ensuring that the
targets of the displaced lepton search are actually produced. As the lifetime
is determined by the smallness of the Yukawa coupling and is fairly
independent of the mass of $\chi$, we set $m_{\chi}=10$ GeV. These assumptions
of course mean a focus on a very specific scenario.

The exclusion results obtained with CheckMATE in the lifetime-mass plane are
shown in Fig. 5. The CMS displaced-lepton searches appear sensitive to the
considered scenario for LLP lifetimes up to $\sim 1$ ns and masses in the
range $100-500$ GeV. The Heavy Charged Particle search from ATLAS impacts the
high-lifetime area $c\tau>1$ m, in a comparable range of masses. The
disappearing track search does not show any sensitivity, due mainly to cross-
sections much weaker in the considered model than the targeted range of the
search: the scalar production cross section is indeed suppressed compared to
the fermionic one. Moreover, the original disappearing track final state
consists of multiple final states (all wino final state configurations), e.g.
the NLO cross section for scalar leptons of mass 125 GeV is about 0.045 pb
whereas the combined wino pair production cross section is about 1 pb.
Finally, the cross section is further suppressed since the disappearing track
search basically triggers on monojet events which requires a hard recoil of
the scalar pair against a hard jet. As a consequence, the current model cannot
be probed with the disappearing track search which is focused on mass
degenerate wino-like electroweakinos. Still, it is obvious that two LLP
searches aiming at quite different signatures can interplay and lead to
complementary exclusion bounds. Current prompt limits on pair production of
scalars that decay to electron or muon with missing energy are at 250 GeV with
full Run 2 data Aad:2019qnd .

###  3.2 Strongly-interacting LLP

Just as EW-charged LLPs manifest in the form of track-based signatures,
strongly charged LLPs result in jets originating from a secondary vertex. A
possible example is provided by the minimal supersymmetric standard model,
when small R-parity violating couplings Barbier:2004ez  open up decay channels
of coloured R-odd particles. We shall focus in particular on the LQD coupling
$\lambda^{\prime}_{223}$ where the subscripts refer to an interaction between
the second lepton doublet superfield, the second left-handed quark doublet
superfield and the right-handed bottom superfield. This results in various
possible decay channels like $\tilde{b}_{1}\rightarrow\mu c$ or
$\tilde{b}_{1}\rightarrow\nu s$, where $\tilde{b}_{1}$ is the lightest bottom
squark, which we assume to (nearly) coincide with the long-lived right-handed
sbottom.

To test this scenario, we employ a simplified supersymmetric model where all
the new-physics particles take a mass at the $10$ TeV scale, beyond the
discovery reach of the LHC, with the exception of the lightest bottom squark
of right-handed type $\tilde{b}_{1}$, whose mass is scanned over in the
electroweak-TeV range. The lifetime of $\tilde{b}_{1}$ is then entirely
determined by the size of the coupling $\lambda^{\prime}_{223}$ and can be
varied freely. We generate samples of $10^{6}$ events for
$pp\rightarrow\tilde{b}_{1}\tilde{b}^{*}_{1}$ using Pythia 8. The production
cross section is normalized to the NNLO${}_{\text{approx}}$ \+ NNLL values
provided by the LHC cross-section Working Group SQprod .

Several LLP searches are potentially sensitive to this scenario. The decay
channel muon + jet is an obvious target for the DV+muon analysis, while the
decay into neutrino + jet enters the scope of the DV+MET search. Finally, the
bottom squark could be detected as a heavy long-lived charged particle.
However, we do not compute limits from the corresponding search due to known
uncertainties in the hadronization of such long-lived strongly charged
scalars. In addition, the disappearing track search is insensitive here
because the displaced jets produced from the sbottom decays are hard, as a
general rule.

The limits obtained with CheckMATE are presented in Fig. 6 in the plane
corresponding to the lifetime and mass of the bottom squark. We observe that,
in this configuration where muon and MET productions are set equal, the
DV+muon search proves slightly more competitive than the DV+MET analysis
(which is consistent with the respective limits placed on the cross-sections
by these searches in their respective benchmark scenario). The most
constraining limits are placed on lifetimes $\sim 1$ ns and exclude sbottom
masses up to $\sim 1.6$ TeV. This is more competitive than limits from the
R-parity conserving scenario with $\tilde{b}\rightarrow b\tilde{\chi}^{0}$,
which places a limit of 1270 GeV for a massless $\tilde{\chi}^{0}$ Aad:2021jmg
.

Two types of prompt searches would also constrain the same model in a
different parameter space searches for promptly-decaying leptoquarks, or usual
sbottom limits (in the presence of some light LSP with the decay
$\tilde{b}\rightarrow c\tilde{\chi}^{+}\rightarrow
c\bar{\ell}\nu\tilde{\chi}^{0}$). There have not been direct searches for
sbottoms using this topology. Limits on squarks in the
$2\ell+2\mathrm{jets}+\mathrm{MET}$ or in $2\mathrm{jets}+\mathrm{MET}$ are
both expected to be much smaller than the standard topology
($\tilde{b}\rightarrow b\tilde{\chi}^{0}$). Limits on standard sbottom decay
with full run 2 data are currently between 600-1270 GeV Aad:2021jmg  depending
on the mass of the final invisible particle. Although there have been searches
for third generation leptoquarks, they focus on decays into tops Aad:2021rrh ;
Sirunyan:2020zbk  (requiring e.g. b-tagged jets) and therefore do not apply
directly to our model.

Figure 6: Limits in the mass-lifetime (left) and mass-$\lambda$
$(=\lambda^{{}^{\prime}}_{223})$ plane of the LLP sbottoms.

##  4 Comments and Outlook

In this paper, we present the implementation of a new class of analyses in the
CheckMATE package, dedicated to long-lived particle searches, which can be
used to reinterpret experimental limits on new physics models. We demonstrate
the interplay of these searches in detecting strongly or weakly charged LLPs
and obtain limits comparable to prompt limits in certain ranges of lifetime.
In addition, these searches provide a way to probe couplings that can even be
much smaller than those currently observable via low-energy intensity-frontier
experiments, e.g. by measuring meson decays.

For our two scalar models, we find that the electroweak model can be
constrained up to a mass of 480 GeV with the CMS dilepton search for lifetimes
under 10 cm. The charged track search is able to set bounds up to about 400
GeV for large lifetimes, larger than 10 m.

We find an important gap in the search for electroweakly charged LLPs decaying
to leptons in the intermediate lifetime range. Since the disappearing track
search employs a lepton veto, it is not possible currently to understand
whether models with decays into leptons can be observed (as the leptons emerge
with potentially very large impact parameters and not from the primary
vertex). It would be fruitful to both improve the experimental search criteria
as well as to provide efficiency for lepton identification based on $p_{T}$
and $d_{0}$ to remedy this situation.

For the strongly charged LLP, our limits are stronger than typical SUSY
searches for particles with the same quantum numbers because of the much
smaller backgrounds in these exotic searches. An important gap here is that
interaction of these particles with the detector material results in the
efficiencies being dependent on some more unknown parameters than just the
mass and lifetime. This is highlighted in Appendix C. Also, this same issue
prevents us from naively applying the charged track search to strongly
interacting LLPs where charge exchange with the detector material becomes
important.

Due to the absence of standardized detector objects in such searches, each re-
interpretation heavily depends on the parametrized efficiencies published by
the experiments. The validation of the five searches considered here has shown
that this method gives relatively good results, which should allow the recast
of the experimental limits for a very diverse range of models sharing an LLP
as a common feature. However, since the efficiencies do rely on identifying
the right truth-level particle via a user input and PDG code (possibly
assigned ad hoc by the user for new particles), we do advise user vigilance
when using these results.

Users can implement their own versions of an LLP analysis using the
AnalysisManager, which simplifies the setting up of detector parameters,
stores the expected and observed events reported by the experiment in the
correct format for future use in statistical calculations and provides a
skeleton C++ analysis code with access to all detector objects. Currently,
searches based on ionisation, like those for monopoles or multi-charged
objects, are not possible to implement because the relevant efficiencies are
not publicly available. However, if the efficiency tables based on mass,
charge and momentum were to be provided by the experiments in the future, it
would be possible to implement them as well, without further difficulties. The
latest version of CheckMATE can always be downloaded from
https://github.com/CheckMATE2. Documentation and validation notes can be found
at https://checkmate.hepforge.org/.

###### Acknowledgements.

We thank Giovanna Cottin and Andre Lessa for useful discussions. R. RdA
acknowledges partial funding/support from the Elusives ITN (Marie Skłodowska-
Curie grant agreement No 674896), the “SOM Sabor y origen de la Materia” (FPA
2017-85985-P). Z. S. W. is supported partly by the Ministry of Science and
Technology (MoST) of Taiwan with grant number MoST-109-2811-M-007-509, and
partly by the Ministry of Science, ICT & Future Planning of Korea, the Pohang
City Government, and the Gyeongsangbuk-do Provincial Government through the
Young Scientist Training Asia Pacific Economic Cooperation program of the Asia
Pacific Center for Theoretical Physics. Z. S. W. also thanks the Witwatersrand
University for hospitality where part of this work was initiated. KR is
supported by the National Science Centre, Poland under grants:
2016/23/G/ST2/04301, 2015/18/M/ST2/00518, 2018/31/B/ST2/02283,
2019/35/B/ST2/02008, and the Norwegian Financial Mechanism for years
2014-2021, grant DEC-2019/34/H/ST2/00707. F.D. acknowledges support of the
BMBF Verbundprojekt 05H2018 and the DFG grant SFB CRC-110 Symmetries and the
Emergence of Structure in QCD.

##  Appendix A Usage

In this section, we demonstrate the usage of CheckMATE with LLP searches on an
example, describing the initialization and running of a test. We assume that
CheckMATE has already been successfully installed.444For more detailed
information on CheckMATE operations, please refer to Refs. Drees:2013wra ;
Dercks:2016npn . We then focus on a simplified SUSY scenario where only the
gluino and the lightest neutralino are kinematically accessible, while the
rest of the R-odd spectrum is heavy and decoupled: we employ the same
benchmark point as in section 2.2, with $m_{\tilde{\chi}_{1}^{0}}=0.1$ TeV,
$m_{\tilde{g}}=1.4$ TeV, and $\tau_{\tilde{g}}=0.1$ ns. The gluinos are
produced at the LHC with $13$ TeV center-of-mass energy through strong
interactions: the corresponding events are generated by calling Pythia 8
internally. At this point, the CheckMATE user needs to provide input to the
program, either via a parameter card or via command line. We consider the
minimal input through text file (parameter card) below. For the command line
call, we refer the reader to Ref. Kim:2015wza .  checkmate_example.in
[Parameters]Name: ExampleRunAnalyses: atlas_1710_04901longlivedPIDs:
1000021invisiblePIDs: 1000022SLHAFile:
./example_run_cards/auxiliary/1710_04901_test_gluino_tau_0.1.slha[1710_04901_example_run]Pythia8Process:
p p > go goMaxEvents: 10000XSect: 0.0284 PB The text command file is
structured in blocks, which are introduced by expressions between brackets and
contain one or more `Key: Value` pairs.

`[Parameters]` provides general settings that are common to all processes in
the CheckMATE run. In the example above, the run is called ExampleRun, which
also determines the name of the result directory. The Analyses parameter
selects the list of analyses against which the generated events are tested: a
list of pre-defined identifiers involving LLP searches is provided in Table 4.
Only the displaced-vertex+MET search is kept in the file above. The PDG code
of the LLP (gluino) is defined in the longlivedPIDs entry, while that of
exotic stable electroweakly-interacting particles (the neutralino LSP in our
case) is fed via invisiblePIDs. In the current CheckMATE version, only a
single longlivedPIDs and invisiblePIDs are supported. Finally, the path to the
SLHA Skands:2003cj ; Allanach:2008qq  file defining the spectrum is provided
after the entry SLHAFile.

Analysis | CheckMATE identifier
---|---
ATLAS disappearing track EW | `atlas_1710_04901_ew`
ATLAS disappearing track QCD | `atlas_1710_04901_strong`
ATLAS heavy charged track (EW only) | `atlas_1902_01636`
ATLAS displaced vertex (DV with lepton veto) | `atlas_1710_04901`
ATLAS muon plus displaced vertex | `atlas_2003_11956`
CMS 8 TeV displaced leptons | `cms_1409_4789`
CMS 13 TeV displaced leptons | `cms_pas_exo_16_022`
Table 4: Table of implemented LLP analyses and their identifiers that can be
used in the CheckMATE run card.

The following [X] blocks list individual production processes (only one in our
example), with X the corresponding (and freely chosen) identifier.
Pythia8Process defines the considered production mode. MaxEvents sets the
number of generated events. XSect contains the corresponding cross-section
value at $13$ TeV center-of-mass energy, taken from the LHC cross-section
Working Group at NNLO${}_{\text{approx}}$+NNLL gluinoprod .

CheckMATE is executed with the following command:

$CMDIR/bin:  ./CheckMATE checkmate_example.in

The program responds with a summary of the inputted settings for the considered run and prompts confirmation from the user.  Terminal ____ _ _ __ __ _ _____ _____ ____ / ___| |__ ___ ___| | _| \/ | / \|_ _| ____|___ \| | | ’_ \ / _ \/ __| |/ / |\/| | / _ \ | | | _| __) || |___| | | | __/ (__| <| | | |/ ___ \| | | |___ / __/ \\____|_| |_|\\___|\\___|_|\\_\\_| |_/_/ \\_\\_| |_____|_____| The following settings are used:Analyses:atlas_1710_04901 ( )E_CM: 13.0Processes:Process Name: 1710_04901_example_runInput Cross section: 0.0284 PBAssociated event files and/or Monte-Carlo generation runs: Pythia8 Events \- internal identifier: ’1710_04901_example_run’ \- simplified SUSY process: p p > go go \- at most 10000 events are generated and analysedOutput Directory:$CMDIR/results/ExampleRunAdditional Settings: \- SLHA file path/1710_04901_test_gluino.slha will be used for event generation \- The following PIDs will be considered as invisible for the detector: 1000022 \- The following PIDs will be considered as long lived for the detector: 1000021Is this correct? (y/n) In this simple example, CheckMATE tests against the displaced-vertex+MET search from ATLAS.

Returning y launches the (Pythia+)CheckMATE run. During its course, updates of
the test status are printed on the screen. Finally, the program displays the
result summary:  Terminal Evaluating ResultsTest: Calculation of approximate
(fast) likelihood given in results folderResult: ExcludedResult for r:
150.498232611Analysis: atlas_1710_04901SR: SR1 In agreement with the ATLAS
result presented in Ref. Aaboud:2017iio , or the left plot in Fig. 2, the
benchmark point is thus excluded ($r\gg 1$) by the only analysis
(`atlas_1710_04901`) considered in the CheckMATE test, in the signal region
`SR1`.

### Note concerning the compatibility of LLP and prompt searches in CheckMATE

The explicit partitioning between prompt and LLP searches is in particular
motivated by the absence of in-built veto against LLP phenomena in the
implemented prompt searches, leading to inconsistent results when testing
events with particles having displaced production or decay points. Obviously
it would be possible to conservatively test LLP spectra against prompt
searches by calculating the fraction of LLPs decaying before hitting the
tracker in the simulated events, then multiplying the $r$-value obtained with
CheckMATE by this fraction. Such an approach was employed in a recasting tool
SModelS Ambrogi:2018ujg . Yet, such a test against prompt searches would still
require a separate CheckMATE run as compared to the test against LLP searches.

In fact, the fundamental reason for this separation is that the standard
efficiencies on reconstructed detector level objects (such as jets, leptons,
missing transverse energy) used in prompt searches do not necessarily apply to
the case of LLP searches. Individual LLP analyses e.g. employ truth-level
information (i.e. properties of the MC event generation prior to the fast
detector simulation), with recasting efficiencies provided by the experimental
collaborations explicitly applying on truth-level objects. We provide a new
directory in the top-level data directory called tables where any efficiency
tables can be stored. ROOT file format is preferred as it is easily available
from experimentalists but any format that can be read directly from the
analysis code in is allowed.

There are also situations where two different LLP searches should not be run
simultaneously. For example the disappearing track search has two different
efficiency tables published by the ATLAS disappearing track search for tracks
due to electro-weak particles and strong particles. Only one of these should
be on at a time. In general, we suggest that users build up by hand the list
of LLP analyses that they wish to run simultaneously. To facilitate this,
Table 4 lists the CheckMATE identifiers of all the implemented LLP analyses.

##  Appendix B Implementation details: Displaced Lepton

We use simulated Monte Carlo samples to evaluate the acceptance of the search
regions. The samples are generated for the process
$\textnormal{pp}\rightarrow\textnormal{\~{t}}_{1}\textnormal{\~{t}}_{1}^{*}$,
for a top squark mass of 500 (700) GeV, using MadGraph5 Alwall:2014hca  to
produce the LHE file for both the 8 TeV and 13 TeV runs. The top squark
further decays via an RPV vertex into electrons and muons with a branching
ratio of 0.5 each. Total production cross section is normalized to the stop-
pair production cross section at NLO.555The LHC SUSY cross section working
group publishes updated calculations of total cross sections at
https://twiki.cern.ch/twiki/bin/view/LHCPhysics/SUSYCrossSections To recover
the assumption of lepton universality, we multiply the overall cross section
with a factor of 2/3. Events are generated using Pythia 8, followed by Delphes
for simulation of the CMS detector acceptance and efficiencies respectively.
FastJet 3 Cacciari_2012  was used to construct jets, using the anti-kt
algorithm Cacciari_2008  with a distance parameter of 0.4. Ten thousand events
are generated for the described process, for each of the following proper
lifetimes of the top squark: 1 mm, 10 mm, 100 mm. Additionally, for the 13 TeV
analysis, a sample is generated for the stop proper lifetime 1 m.

#### Event Selection

Throughout this section, the 13 TeV criteria (wherever they differ) are shown
within a bracket following the 8 TeV requirements. At the preselection stage,
events with exactly one electron and one muon with opposite charge are singled
out. Further conditions on the leptons to pass the detector trigger
requirements are as follows:

  * •

$p_{T}^{\mu}>25(40)\textnormal{ GeV}$

  * •

$p_{T}^{e}>25(42)\textnormal{ GeV}$

  * •

$|\eta^{l}|<2.5(2.4)$

Additional isolation requirements are placed on the leptons by requiring that
the sum of $p_{T}$ of all tracks that fall within some $\Delta R$ of the
lepton track are smaller than a fraction $f$ of the lepton-$p_{T}$. Here are
the requirements in terms of $(\Delta R,f)$ for the different cases.

  * •

Muons: (0.5, 0.15)

  * •

Electrons with $1.56(1.57)<|\eta^{e}|<2.5(2.4)$: (0.3, 0.065)

  * •

Electrons with $|\eta^{e}|<1.44$: (0.3, 0.035)

Electron candidates are rejected if they have $1.44<|\eta|<1.56$ due to
reduced reconstruction performance of electrons in the overlap region between
the barrel and endcap detectors. For the 8 TeV search, there must be no jets
within a cone of $\Delta R=0.5$ around either lepton and the electron and muon
must be separated by $\Delta R>0.5$. There is no jet-isolation requirement for
the 13 TeV search.

Events passing the preselection criteria are further classified according to
the impact parameters of the leptons. The impact parameter $|d_{0}|$ is
defined as the distance of closest approach of the helical trajectory of the
lepton in the transverse plane to the beam axis.

$|d_{0}^{\ell}|=\frac{|p_{x}^{\ell}L_{y}^{\ell}-p_{y}^{\ell}L_{x}^{\ell}|}{p_{T}^{\ell}}$

where $p_{x}^{\ell}$ and $p_{y}^{\ell}$ are the radial components of the
lepton’s momentum while $L_{x}^{\ell}$ and $L_{y}^{\ell}$ are the radial
components of the decay vertex (position) of the top squark from which the
lepton originates. By requiring the leptons to have a large enough impact
parameter, one reduces the likelihood of the leptons of having originated from
standard model processes. Both leptons are therefore required to have a
minimum impact parameter $d_{0}^{\ell}>0.1(0.2)$ mm. Three signal regions are
defined based on these impact parameters:

  * •

SR1: $d_{0}>0.2$ mm for both leptons and $d_{0}<0.5$ mm for at least one;

  * •

SR2: $d_{0}>0.5$ mm for both leptons and $d_{0}<1$ mm for at least one;

  * •

SR3: $d_{0}>1$ mm for both leptons.

Furthermore, events in which either lepton has an impact parameter
$d_{0}>20(100)$ mm is rejected, in order to ensure that the leptons originate
within the pixel layer.

#### Comparison

To compare the results of this analysis with the values determined by the CMS
collaboration CMS-PAS-EXO-16-022 ; Khachatryan_2015 , we define the percentage
difference as

$\textnormal{diff}=\frac{|n_{\textnormal{CheckMATE}}-n_{\textnormal{CMS}}|}{n_{\textnormal{CMS}}}$
(2)

where $n$ is the number of events that pass all selections in each signal
region.

The results as well as the comparison with the CMS analysis are shown in
Tables 5 and 6.

$c\tau$ | Search Region 1 | Search Region 2 | Search Region 3
---|---|---|---
| CM | CMS | diff | CM | CMS | diff | CM | CMS | diff
1 mm | 27.2 | 30.1 $\pm$ 5.35 | 9.63% | 6.86 | 6.54 $\pm$ 1.21 | 4.89% | 1.26 | 1.34 $\pm$ 0.28 | 5.97%
10 mm | 37.6 | 35.3$\pm$ 6.25 | 6.52% | 32.0 | 30.3 $\pm$ 5.35 | 5.61% | 50.0 | 51.3 $\pm$ 9.06 | 2.53%
100 mm | 4.78 | 4.73 $\pm$ 0.88 | 1.06% | 6.13 | 5.57 $\pm$ 1.03 | 10.0% | 26.5 | 26.3$\pm$ 4.65 | 0.760%
Table 5: Comparison of the number of expected events in the three search regions for the Displaced Lepton search with the CMS detector at 8 TeV with CheckMATE (CM), for the process $pp\rightarrow\textnormal{\~{t}}_{1}\textnormal{\~{t}}_{1}^{*}$, with $\textnormal{M}_{\textnormal{\~{t}}_{1}}=500$ GeV and stop quark lifetimes of 1, 10 and 100 mm. The CMS results are presented here with statistical and systematic uncertainties combined in quadrature. $c\tau$ | Search Region 1 | Search Region 2 | Search Region 3
---|---|---|---
| CM | CMS | diff | CM | CMS | diff | CM | CMS | diff
1 mm | 3.85 | 3.80 $\pm$ 0.2 | 1.24% | 0.99 | 0.94 $\pm$ 0.06 | 4.8% | 0.19 | 0.16 $\pm$ 0.02 | 20%
10 mm | 5.18 | 5.20 $\pm$ 0.4 | 0.38% | 4.6 | 4.1 $\pm$ 0.3 | 11% | 7.6 | 7.0 $\pm$ 0.3 | 8.0%
100 mm | 0.86 | 0.80 $\pm$ 0.1 | 7.0% | 1.04 | 1.0 $\pm$ 0.1 | 3.97% | 5.4 | 5.8 $\pm$ 0.2 | 7.5%
1000 mm | 0.03 | 0.009 $\pm$ 0.005 |  $\sim$260% | 0.05 | 0.03 $\pm$ 0.01 | 59% | 0.45 | 0.27 $\pm$ 0.03 | 67%
Table 6: Comparison of the number of expected events in the three search
regions for the Displaced Lepton search with the CMS detector at 13 TeV with
CheckMATE (CM), for the process
$pp\rightarrow\textnormal{\~{t}}_{1}\textnormal{\~{t}}_{1}^{*}$, with
$\textnormal{M}_{\textnormal{\~{t}}_{1}}=700$ GeV and stop quark lifetimes of
1, 10, 100 and 1000 mm. The CMS results are presented here with statistical
and systematic uncertainties combined in quadrature.

The analysis was conducted over top squark masses ranging from 300 (400) GeV
to 850 (900) GeV, and top squark lifetimes $c\tau$ from 0.1 mm to 1 m. A
simultaneous counting experiment was performed on the three bins of the three
signal regions, and the resulting exclusion contours compared to those
provided by CMS CMS-PAS-EXO-16-022 ; Khachatryan_2015 . Figure 1 of section
2.1 shows these exclusion contours for the 8 TeV and 13 TeV analyses. It is
observed that the CheckMATE 8 TeV exclusion contours agree well with CMS,
except for $c\tau$ between about 7 mm to 70 mm, where the CheckMATE limits are
stronger.

For the $13$ TeV analysis, the absence of available efficiencies complicate
the recast. The use of the $8$ TeV values results in satisfactory agreement at
small lifetimes up to about $5$ mm. However, as the $13$ TeV search extends
the allowed $d_{0}$ up to $100$ mm, it is necessary for the recast to assess
the efficiency in the $d_{0}$-bin $20-100$ mm. The lepton $d_{0}$ efficiency
is linearly extrapolated at 22 mm using the values in the range [12 mm, 20 mm]
for both electrons and muons. Keeping the electron $d_{0}$ efficiency in the
added bin constant, we vary the muon $d_{0}$ efficiency in the added bin and
minimize the $\chi^{2}$ fit over all resulting signal regions for the
benchmark stop lifetimes 100 mm and 1000 mm. The minimum $\chi^{2}$ value
occurs for a muon $d_{0}$ efficiency of 0.01 in the added bin. Using this
value, we similarly minimize the $\chi^{2}$ fit for electron $d_{0}$
efficiency values in the added bin. This minimum occurs for an electron
$d_{0}$ efficiency of 0.06. The resulting exclusion plot still exhibits
stronger limits than those of the CMS publication, but shows a significant
improvement over the exclusion plot obtained with only the 8 TeV lepton
reconstruction efficiencies.

##  Appendix C Implementation details: Displaced Vertex and MET

#### Monte Carlo Simulation Samples

As a template for this search, the ATLAS collaboration considered the (strong)
production of a pair of long-lived gluinos, then decaying into light quarks
and a stable neutralino: $pp\to\tilde{g}\tilde{g}$, $\tilde{g}\to
q\bar{q}\tilde{\chi}_{1}^{0}$. The gluino decay branching ratios are set to
$50\%$ for both the $d\bar{d}\tilde{\chi}_{1}^{0}$ and
$u\bar{u}\tilde{\chi}_{1}^{0}$ channels. Two values of the gluino mass are
studied, $m_{\tilde{g}}=1.4$ TeV and $2$ TeV, with varying lifetimes between
$\tau_{\tilde{g}}=0.003$ ns and $50$ ns. The neutralino mass is either set to
$m_{\tilde{\chi}_{1}^{0}}=100$ GeV or kept at $m_{\tilde{g}}-100$ GeV. We
generate events internally to CheckMATE, via the Pythia 8 interface, also
carrying out the hadronization. Jets are reconstructed with FastJet 3
Cacciari_2012 , using the anti-kt algorithm Cacciari_2008 , with a distance
parameter of $0.4$. The cross-section is varied by rescaling the number of
events by the ratio (imposed cross-section)/(SUSY cross section), so that the
limits are independent of the actual SUSY production cross-sections (or their
LO calculation). Samples of $10^{6}$ events are considered for the relevant
mass, lifetime and cross-section values.

While we principally consider unmatched events, we will also discuss the
impact of jet radiation on individual scenarios. For this, we employ MadGraph5
2.7.0  Alwall:2014hca  and add up to two jets using the MLM method with xqcut
parameter of $100$ GeV.

#### Event selection

The event selection follows the recast instructions provided by the ATLAS
collaboration DVrecast . It includes a preselection defining the event- and
vertex-level acceptances for truth-level particles, and then the application
of parameterized efficiencies. This recast strategy has been successfully
applied in the past (see Contribution $22$ in Brooijmans:2018xbu  and the
publicly available codes Cottincode ; Lessacode ).

#### Preselection

The missing energy $E_{\text{T,true}}^{\text{miss}}$ is defined at truth-level
as the vector sum of the momenta of the stable invisible weakly-interacting
particles ($\tilde{\chi}_{1}^{0}$ and neutrinos in the considered benchmark
scenario). It is requested to satisfy the cut:
$E_{\text{T,true}}^{\text{miss}}>200$ GeV.

Jet requirements are applied on $75\%$ of the events. These demand either one
truth jet with $p_{\text{T}}>70$ GeV or at least two jets with
$p_{\text{T}}>25$ GeV, satisfying in both cases a trackless requirement: the
scalar sum of the charged particle $p_{\text{T}}$ in the jet should not exceed
$5$ GeV for particles with small impact parameter ($d_{0}$). We interpret the
small impact-parameter condition as $d_{0}<2$ mm.

DVs are reconstructed from stable charged particles. The event should contain
at least one DV in the fiducial volume: the transverse distance from the
interaction point should lie between $4~{}\text{mm}$ and $300~{}\text{mm}$, as
well as $|z|<300~{}\text{mm}$. The DVs should contain at least $5$ selected
decay products, i.e. stable and charged particles with $p_{\text{T}}>1$ GeV
and an approximate transverse impact parameter $d_{0}>2$ mm. The condition on
the $p_{\text{T}}$ of the selected decay product assumes a charge $|q|=1$,
which we regard as implicitly fulfilled.

#### Efficiencies

Efficiencies of two origins are considered. The first one corrects
$E_{\text{T,true}}^{\text{miss}}$. The second one applies on the
reconstruction of the DV. Both are provided by the experimental collaboration
DVdata .

The results presented in Fig. 2 seem to indicate a satisfactory performance of
the recast strategy in scenarios involving a sizable mass-splitting between
the LLP and the LSP. In addition, we were able to largely recover the cutflow
information DVrecast  (as far as a comparison between truth- and detector-
level events is possible).

#### Observed inconsistencies in provided efficiency parametrisation

We then test the scenario with compressed SUSY spectra, i.e. a $100$ GeV mass-
splitting between gluino and the neutralino: see Fig. 7. In this case, the
limiting cross-sections obtained with the recast information differ from the
experimental observed limits by a sizeable deviation: on average, a factor $2$
to $3$.

Figure 7: Scenario with compressed spectrum – validation plots. Same
conventions as in Fig. 2. We only present the results obtained with the
simplified CheckMATE criterion.

This issue appears to be generic with this recast strategy, as was confirmed
by the authors of Cottincode ; Lessacode . The impact of R-hadronization on
the definition of the missing transverse energy was put forward as a possible
factor. We also note that the perspectives for implementing a successful
recast look bleak in a region where cross-section limits show a strong
variation with the kinematics, such as the compressed regime of the DV+MET
search (see Fig. 10a of Aaboud:2017iio ), due to the difficulty of matching
simulation objects with their collider counterparts. In any case, the poor
performance of the recast in this compressed configuration hints at the
possible failure of the approach with parametrized efficiencies to
simultaneously address all relevant spectra. As a possible way to mitigate
incomplete efficiency parametrisations, the LHC Reinterpretation Forum white
paper LHCReinterpretationForum:2020xtr  and the LLP white paper
Alimena:2019zri  recommend using multiple topologies or multiple mass
benchmarks for unweighting.

#### Matching events

Finally, we discuss the impact of jet radiation (matched events) for a gluino
lifetime of $1$ ns. The results are collected in Table 7, allowing for up to
two radiated jets. We observe that the radiation of jets affects the limiting
cross-section by only a few percent in the scenario with large gluino-
neutralino mass-splitting, and by $\sim 10\%$ in the compressed case. Matching
events thus leads to a minor effect in the considered search, justifying our
focusing on unmatched events in Figs. 2 and 7. We note that the added jets
affect the limiting cross sections in opposite directions for the two
scenarios, relaxing the bound in the uncompressed case, while tightening it in
the other case. This can be qualitatively understood as, in the compressed
case, the radiation of additional jets provides possibly more energetic
candidates for the jet cuts, while the jets originating in gluino decays have
reduced energy due to the narrow phase space. On the contrary, in the scenario
with large mass-splitting, the jets originating in gluino decays are already
energetic, so that radiating further jets decreases their ability to withstand
the jet cuts.

Uncompressed scenario: $m_{\tilde{\chi}_{1}^{0}}=100$ GeV.
---
$m_{\tilde{g}}=1.4$ TeV |  $m_{\tilde{g}}=2$ TeV
unmatched | matched | unmatched | matched
$\sigma^{\text{rec}}_{\text{lim}}=0.272$ | $\sigma^{\text{rec}}_{\text{lim}}=0.281$ | $\sigma^{\text{rec}}_{\text{lim}}=0.220$ | $\sigma^{\text{rec}}_{\text{lim}}=0.228$
Compressed scenario: $m_{\tilde{\chi}_{1}^{0}}=m_{\tilde{g}}-100$ GeV.
$m_{\tilde{g}}=1.4$ TeV |  $m_{\tilde{g}}=2$ TeV
unmatched | matched | unmatched | matched
$\sigma^{\text{rec}}_{\text{lim}}=2.07$ | $\sigma^{\text{rec}}_{\text{lim}}=1.81$ | $\sigma^{\text{rec}}_{\text{lim}}=1.96$ | $\sigma^{\text{rec}}_{\text{lim}}=1.76$
Table 7: Impact of matched events on the DV search for a gluino lifetime of
$1$ ns. Matched events allow for up to two radiated jets.
$\sigma^{\text{rec}}_{\text{lim}}$ represents the upper bound on the cross-
section in fb, obtained from the recast procedure.

##  Appendix D Implementation details: Displaced vertex and a muon

### Monte Carlo Samples

As a benchmark process for the search involving displaced vertices and
displaced muons Aad:2020srt  the stop pair production was considered, followed
by the RPV decay: $pp\to\tilde{t}_{1}\tilde{t}_{1}$, $\tilde{t}_{1}\to\mu\,q$.
The Monte Carlo samples for cutflows and exclusion limits were generated using
MadGraph 2.7.2 Alwall:2014hca  with up to two additional partons matched using
CKKW-L procedure Lonnblad:2011xx . Showering was performed by Pythia 8.244
Sj_strand_2015 . To minimise the impact of statistical uncertainty the sample
for each grid-point or cutflow scenario were 10–20 times larger than the
nominal number of events. The samples were normalized to the approximate next-
to-next-to-leading order in the strong coupling constant with a soft gluon
resummation (approximate NNLO+NNLL) Beenakker:2016lwe ; Beenakker:1997ut ;
Beenakker:2010nq ; Beenakker:2016gmf .

### Event selection and signal regions

The event selection defines two mutually exclusive trigger-based signal
regions: $E_{\mathrm{T}}^{\text{miss}}$ Trigger SR and Muon Trigger SR. The
first SR should be selected by the MET trigger and requires
$E_{\mathrm{T}}^{\text{miss}}>180$ GeV and a muon with $p_{T}>25$ GeV, while
the second one should be selected by the muon trigger with muon $p_{T}>60$ GeV
and $E_{\mathrm{T}}^{\text{miss}}<180$ GeV. For both signal regions the muon
vertex should be at least 2 mm away from the primary vertex. The selection
also requires a displaced vertex at least 4 mm away from the primary vertex,
with at least 3 associated tracks and its visible invariant mass calculated
from the four-momenta of the associated tracks $m_{\text{DV}}>20$ GeV
(assuming each track originates from a charged pion).

Displaced vertices are reconstructed internally by CheckMATE using a DVfinder
class. Firstly, charged stable particles with production vertex separated from
the primary vertex are searched for. These are then merged into vertices based
on the relative position, if the distance is smaller than 1 mm. The position
of the combined vertex is calculated as a weighted (with invariant momentum)
average of the positions of input vertices. The procedure continues until no
further vertices can be combined and the remaining single track vertices are
dropped.

### Validation

For the reconstruction of events at the CheckMATE level no specific truth
particles efficiencies were used (these are available on HEPdata). The ATLAS
analysis rejects vertices from interaction with dense detector material. The
veto is imposed based on a three dimensional map of the detector and rejects
$42\%$ of the events. Because the map was not public when the CheckMATE
analysis was implemented, the veto is applied as a flat rejection probability
on each reconstructed displaced vertex. Due to inhomogeneous distribution of
vetoed points this can potentially result in a bias for certain lifetimes
compared to the experimental analysis.

The validation procedure was performed both in terms of the exclusion contour
for the benchmark model, see Figure 3, and of the cutflows for three different
parameter points. As already noted in Section 2.3, generally a good agreement
is observed for the exclusion contour, except a range of lifetimes
$0.01$–$0.1$ ns where the recast exclusion is significantly weaker, though
still within the 2-sigma band. In order to get a better insight, in Table 8 we
also compare cutflows published by the ATLAS collaboration for
$m_{\tilde{t}}=1.4$ TeV and three lifetimes.

For the longest lifetime, 1 ns, the number of reconstructed events within
CheckMATE is by $40\%$ higher than that of ATLAS. This corresponds to the
slightly stronger recast exclusion observed in Figure 3 for lifetimes
$\sim$0.2–6 ns. For the intermediate lifetime, 0.1 ns, the number of CheckMATE
reconstructed events is half of the ATLAS number. This corresponds to a weaker
exclusion observed in Figure 3 for lifetimes $<0.2$ ns. We note however that
this falls inside the expected 2-$\sigma$ exclusion limit calculated by the
collaboration. Finally, for the shortest lifetime, 0.01 ns, we observe almost
a 3-fold difference. However, this region is close to the experimental
sensitivity limit, as can be seen in Figure 3, where the exclusion line
becomes almost vertical. For this reason a poor agreement between the full
simulation and recasting is not surprising. On the other hand, it also has a
minimal impact on the exclusion contour.

|  $\tau=0.01$ ns |  $\tau=0.1$ ns |  $\tau=1$ ns
---|---|---|---
Selection |  ATLAS |  CM |  ATLAS |  CM |  ATLAS |  CM
All |  64.2 |  64.2 |  64.2 |  64.2 |  64.2 |  64.2
$E_{T}^{\mathrm{miss}}$ trigger |  63.0 |  63.2 |  63.0 |  63.3 |  62.7 |  63.2
$E_{T}^{\mathrm{miss}}>180$ GeV |  60.9 |  61.2 |  61.0 |  61.3 |  60.6 |  61.0
$\geq 1\mu$; $p_{T}>25$ GeV, $|\eta|<2.5$ |  57.8 |  60.8 |  58.7 |  61.0 |  52.5 |  60.6
$2<|d_{0}(\mu)|<300$ mm |  11.3 |  12.8 |  49.1 |  52.5 |  49.5 |  59.2
$|z_{0}(\mu)|<500$ mm |  11.3 |  12.8 |  49.1 |  52.5 |  49.3 |  58.1
Fake/HF/cosmic veto |  9.1 |  9.9 |  40.0 |  42.4 |  39.4 |  48.1
At least one DV |  8.5 |  4.4 |  37.6 |  29.8 |  32.6 |  39.6
DV fiducial volume |  8.4 |  3.7 |  37.1 |  29.1 |  31.2 |  32.7
Material veto |  5.3 |  2.2 |  31.0 |  16.9 |  22.2 |  19.0
$n_{\text{tracks}}^{\text{DV}}\geq 3$ |  3.8 |  1.8 |  26.0 |  15.5 |  13.7 |  17.3
$m_{\text{DV}}>20$ GeV |  3.4 |  1.2 |  22.7 |  11.9 |  10.3 |  14.0
Table 8: Cutflow for the displaced vertex + muon search in the missing
transverse energy SR.

##  Appendix E Implementation details: Heavy Charged Particle Track

#### Monte Carlo Simulation Samples

In order to validate our implementation of this search in CheckMATE we
simulated the production of pairs of charginos and staus with MG5_aMC@NLO
(version 2.6.6) Alwall:2014hca  including two additional partons at leading
order in combination with Pythia 8 Sj_strand_2015 . Matching was performed
using the MLM scheme Mangano:2006rw . The resulting events in HepMC format
Dobbs:2001ck  were processed by CheckMATE which uses Delphes for the
simulation of the ATLAS detector.

The reference benchmarks used for the validation are provided in SLHA format
by the ATLAS collaboration in the HEPData repository Maguire:2017ypu . The
pair production of charginos is based on a mAMSB Randall:1998uk  scenario
while the staus production on a GMSB Giudice:1998bp  scenario. For reproducing
the ATLAS results we have chosen a set of chargino and stau masses and
simulated $2\times 10^{4}$ events for each of the benchmarks considered.

#### Preselection

The following event preselection is applied:

  * •

Full-detector candidates have to be electrically single-charged particles and
pass the full ATLAS detector before decaying (R $>$ 12.0 m, $|z|>$ 23.0 m).

  * •

Events with $E_{\text{T}}^{\text{miss}}>$ 300 GeV are accepted, otherwise a
trigger efficiency is applied.

  * •

Single-muon trigger objects have to reach the ATLAS muon spectrometer (stable
within R $>$ 12.0 m, $|z|>23.0$ m), have a $\beta>0.2$, a pseudorapidity
$|\eta|<2.5$ and a minimum transverse momentum of 26 GeV. In addition a muon
trigger efficiency which is a function of $\beta$ and $|\eta|$ of the particle
is applied.

Maps of trigger efficiencies are provided by the ATLAS collaboration in the
HEPData repository hepdata .

#### Search Regions

The searches for pair-produced long-lived staus and charginos are based on a
loose and on a tight candidate selection. The main difference between them is
that the loose selection requires two candidates while the tight one only one.
In both cases the candidates are required to have $p_{T}>70$ GeV and while for
the loose selection $p>100$ GeV is imposed, the tight one requires $p>200$
GeV. In addition a tighter pseudorapidity $|\eta|$ $<$ 1.65 for the tight
selection is required. Finally trigger efficiencies for loose and tight
candidates in function of $\beta$ and $|\eta|$ of the candidates are applied.

The signal regions are based on cuts on the loose and tight selections plus
applying inclusive cuts on the time of flight (ToF) measurements of the
particle mass $m_{\mathrm{ToF}}$. The reconstructed $m_{\mathrm{ToF}}$ is
given as a function of the truth mass for the candidates and is sampled from a
Gaussian using the mean and width from the respective bin in the truth mass.
For candidates passing the loose cuts two masses have to be sampled for a
given truth mass and the lower of the two is used for the final counting. For
tight candidates only one mass is sampled using the respective mean and
resolution. The tables parametrizing the reconstructed $m_{\mathrm{ToF}}$ are
provided in the HEPData repository hepdata .

Following the prescription above, the signal regions are defined according to
the loose and tight cuts plus a condition on the minimum reconstructed
$m_{\mathrm{ToF}}$ as summarized in Table 9. The results of the validation are
presented in Fig. 4 and establish the reliability of this recast search in
CheckMATE.

$m^{\mathrm{min}}_{\mathrm{ToF}}$ [GeV]
---
tight | 175 | 375 | 600 | 825
loose | 150 | 350 | 575 | 800
Table 9: Definition of the signal regions implemented in CheckMATE.

##  Appendix F Implementation details: Disappearing Track

#### Monte Carlo Simulation Samples

We generated the SUSY signal samples in the framework of minimal AMSB model
Randall:1999ee  assuming $\tan\beta=5$, with a positive higgsino mixing mass
parameter, while the scalar masses are decoupled by setting $m_{0}=5$ TeV. The
chosen benchmark contains a chargino with a 0.2 ns lifetime and a mass of 400
GeV. For the strong production channel, we select a gluino mass of 1600 GeV, a
chargino mass of 500 GeV and a proper lifetime of 0.2 ns. The SUSY mass
spectrum files are generated with ISASUSY 7.80 Baer:1993ae . The gluino decays
via off-shell squarks into the following decay channels,

$\tilde{g}\rightarrow q\bar{q}\tilde{\chi}^{0}_{1},\quad
q\bar{q}^{\prime}\tilde{\chi}^{-}_{1},\quad
q\bar{q}^{\prime}\tilde{\chi}^{+}_{1}.$ (3)

assuming only first and second generation partons intervene. The wino–like
chargino LLP decays into a charged pion and a neutralino LSP.

The MC events for the signal are generated together with two additional
radiated partons in the hard process with MadGraph5 2.6.1Alwall:2014hca . The
parton level events were showered and hadronized with Pythia 8.230
Sj_strand_2015 . We used the NNPDF 2.3LO parton distribution function
Ball:2012cx . Renormalisation and factorisation scales were kept at the
default scale of MadGraph5. The combination of the parton shower and the
matrix elements partons was performed in the CKKW-L merging scheme
Lonnblad:2001iq  and the merging scale was set to a quarter of the wino mass
for the electroweak production channel or a quarter of the gluino mass for the
strong production channel. The cross sections for electroweak production are
calculated at NLO using Prospino2 Beenakker:1999xh  and the normalisation for
the strong production is determined at NLO+NLL accuracy with NLLfast
Beenakker:1996ch ; Kulesza:2008jb ; Kulesza:2009kq ; Beenakker:2009ha ;
Beenakker:2011fu .

#### Preselection

We apply the object removal procedure given in Ref. Aad:2019vnb . Electron and
muon candidates must satisfy $p_{T}>10$ GeV and $|\eta|<2.47$ and $2.7$,
respectively. The isolation criteria for final state leptons are given by the
requirement that the scalar sum of the transverse momentum of tracks inside
variable-size cone around leptons is less than 15$\%$ of lepton transverse
momentum while the cone radius is defined as min($\Delta R=10$ GeV$/p_{T}$,
$\Delta R=0.2\,(0.3)$) for electrons (muons). Jets are reconstructed with
anti–kT algorithm and $\Delta R=0.4$. Jet candidates must pass $p_{T}>20$ GeV
and $|\eta|<2.4$. The missing transverse momentum is given as the negative
vector sum of all reconstructed detector level objects, i.e. electrons, muons,
photons and jets.

#### Search Regions

For the electroweak and strong signal region, a set of common kinematic
preselection cuts are demanded and for both signal regions we require a lepton
veto, i.e. no isolated electrons and muons are allowed in the event. Two
signal regions targeting the electroweak production channel (EW SR) and strong
production channel via gluinos (strong SR) are considered. We apply parton
level cuts on the tracklet (i.e. the long lived chargino). We demand isolated
charginos with $p_{T}>100$ GeV. Moreover, we require geometric acceptance cuts
$0.1<|\eta|<1.9$. For the EW SR, the events must have at least one jet with
$p_{T}>140$ GeV and $E_{\text{T}}^{\text{miss}}>140$ GeV (90
GeV$<E_{\text{T}}^{\text{miss}}<$140 GeV) in the high– (low–) missing
transverse momentum region. Multijet background suppression is achieved by a
$\Delta\Phi$ cut. Here, the difference in azimuthal angle between the missing
transverse momentum and each of the up to four highest–$p_{T}$ jets with
$p_{T}>50$ GeV is required to be larger than 1.0.

The strong SR demands that events have a jet with $p_{T}>100$ GeV, at least
two additional jets with $p_{T}>50$ GeV and $E_{\text{T}}^{\text{miss}}>150$
GeV (100 GeV $<E_{\text{T}}^{\text{miss}}<$ 150 GeV) in the high–(low–)
missing transverse momentum. The $\Delta\Phi$ between the missing transverse
momentum and each of the up to four leading jets with $p_{T}>50$ GeV is
required to be larger than 0.4.

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