# Exploring Uncharted Soft Displaced Vertices in Open Data

Haipeng An${}^{\,(a,b,c)}$, Zhen Hu${}^{\,(a,b)}$, Zhen Liu${}^{\,(d)}$ ,
Daneng Yang${}^{\,(a)}$ (a)Department of Physics, Tsinghua University, Beijing
100084, China (b)Center for High Energy Physics, Tsinghua University, Beijing
100084, China (c)Center for High Energy Physics, Peking University, Beijing
100871, China (d)School of Physics and Astronomy, University of Minnesota,
Minneapolis, MN 55455, USA

###### Abstract

A cluster of soft displaced tracks corresponds to the dark matter co-
annihilation regime. The long-lived regime is, in particular, motivated by the
unexplored top partner physics. The background in this regime is extremely
challenging to model using a traditional simulation method. We demonstrate the
feasibility of handling the formidable background using the CMS Open Data. We
perform this analysis to search for compressed and long-lived top partners in
the 8 TeV CMS Open Data events with the integrated luminosity of 11.6 fb-1 and
obtain new limits. With 15-30 GeV mass splitting between the top partner and
the DM candidate, we exclude the top partner mass below 350 GeV, which is more
stringent than the ATLAS and CMS results using 8 TeV data with 20 fb-1
luminosity. Our study also shows that the CMS Open Data can be a powerful tool
to help physicists explore non-conventional new physics and even enable
deriving new limits on exotic signals from data directly.

Introduction—

The presence of long-lived particles can provide striking signals in many new
physics models Barbier:2004ez ; Giudice:1998bp ; Meade:2010ji ;
Arvanitaki:2012ps ; ArkaniHamed:2012gw ; Grober:2014aha ; Liu:2015bma ;
Chacko:2005pe ; Burdman:2006tz ; Kang:2008ea ; Craig:2015pha ; Davoli:2017swj
; Liu:2019ayx ; Hook:2019qoh . Requirement on the displaced decaying vertices
associated with these particles offers an efficient and novel way to reject
standard model (SM) background. In many well-motivated cases, the long-lived
particles decay into massive invisible particles, leaving only tracks of low
energies Grober:2014aha ; Baker:2015qna , which are hard to reconstruct. This
challenging signal can naturally arise from nearly degenerate two-level
systems of a neutral Lightest Stable Particle (LSP) and a next-to-lightest
stable particle (NLSP). In particular, even when the splitting is as large as
tens of GeV, the NLSP can still be long-lived, so long as the leading decay
channel is through heavy SM states, e.g., top quark. In this case, the decay
of the NLSP must go through a virtual top quark and a virtual $W$ boson. The
LSPs in these models are usually SM singlets, which lead to inefficient
annihilation in the early universe. The nearly degenerate NLSP, on the other
hand, opens up co-annihilation channels and restores the typical weakly
interacting thermal dark matter paradigm Griest:1990kh . Therefore, the
presence of both particles offers a way to control and match the observed
thermal dark matter abundance.

We consider two representative cases in this study. Case (A) is from
supersymmetry (SUSY) with bino ($\chi^{0}_{1}$) as the LSP, and the stop (top
squark $\tilde{t}_{1}$) as the NLSP, and the coupling of $\chi^{0}_{1}$ and
$\tilde{t}_{1}$ to the top quark can be written as

$\displaystyle{\cal
L}_{A}=\left(-y_{R}\tilde{t}_{1}\bar{t}_{R}\chi_{1}^{0}-y_{L}\tilde{t}_{1}\bar{t}_{L}\chi_{1}^{0}\right)+{\rm
h.c.}\ ,$ (1)

where $y_{R}=-(2\sqrt{2}/3)g_{1}\sin\theta_{t}$ and
$y_{L}=(\sqrt{2}/6)g_{1}\cos\theta_{t}$ with $g_{1}$ the hypercharge coupling
and the $\theta_{t}$ the mixing angle of the two top partners. In case (B),
the model we consider is a simplified model with the LSP a vector dark matter
state and the NLSP a colored Dirac spinor. This model can be traced to extra
dimension models with KK-parity or the little Higgs models with T-parity.
Therefore, we call the LSP $A_{H}$ and the NLSP $t_{H}$. The couplings of
$A_{H}$ and $t_{H}$ to the top quark can be written as

$\displaystyle{\cal
L}_{B}=\left(-g_{R}\bar{t}_{H}\gamma_{\mu}t_{R}-g_{L}\bar{t}_{H}\gamma_{\mu}t_{L}\right)A_{H}^{\mu}+{\rm
h.c.}\ ,$ (2)

where $g_{R}=g_{H}\sin\theta_{t}$, $g_{L}=g_{H}\cos\theta_{t}$. The typical
value of $g_{H}$ is around $g_{1}$. In both cases (A) and (B), to avoid
overproducing the dark matter in the early universe, $\Delta$ is constrained
to less than about 40 GeV deSimone:2014pda ; Ellis:2014ipa ; Ellis:2014ipa ;
Ibarra:2015nca ; Liew:2016hqo ; Mitridate:2017izz ; Keung:2017kot ;
Pierce:2017suq ; Keung:2017kot ; Ellis:2018jyl ; Garny:2018icg ;
Biondini:2018ovz .

In SUSY models, the top squark mass should be less than about 1 TeV to
effectively solve the fine-tuning problem Papucci:2011wy . However, when
$m_{\tilde{t}_{1}}>m_{\chi^{0}_{1}}+m_{t}$, traditional searches for the top
squark, which consider signals composed of energetic tracks of hadronic or
leptonic type, plus missing transverse energy (MET), have excluded the bulk
parameter region for this possibility Aad:2020sgw ; ATLAS:2020llc ;
ATLAS:2020dav ; Sirunyan:2019glc ; Sirunyan:2019xwh ; Sirunyan:2019ctn ,
making the complementary parameter region more interesting. There are fewer
studies for the extra dimension models or the little Higgs models. However,
with the KK-parity or the T-parity, the signal is similar, and if we recast
the SUSY result to these models, the constraints on the mass of the top
partner in the bulk region are also TeV scale. As a result, in these theories,
the compressed region, where the LSP and NLSP have small mass gaps, is
motivated to produce the electroweak scale naturally with the current LHC
data.

Figure 1:  Schematic flowchart of our analysis.

In this letter, we propose to search for the degenerate region of the two-
level system by looking for signals with displaced soft tracks and large MET
signatures. The complex experimental environment and appearance of nonstandard
background prevents a direct simulation-based search for this challenging
signal. We hence take a data-driven method. We design a hybrid analysis and
validation chain between experimental data, full simulation data, and fast
simulation data to carry out such an analysis. A schematic presentation of our
analysis method is shown in Fig. 1. We perform a vertex reconstruction based
on all the selected displaced tracks and search for signal-like vertices from
the products. By selecting displaced tracks and vertices, one can suppress
prompt decaying backgrounds. The backgrounds associated with intermediate B
mesons can still produce displaced vertices. However, the QCD-induced ones are
usually soft, being suppressed by the requirement of monojet and MET. The B
mesons from top decays are inherently energetic and vetoed by selecting only
soft tracks using a jet level upper-cut on the $H_{T}^{V}$, which is defined
as the scalar sum of the $p_{\mathrm{T}}$s of the tracks associated with the
reconstructed vertex.

With our analysis, we set limits on the signal in a conservative statistical
approach, based on the 11.6 fb-1 collision data collected by the CMS
experiment at 8 TeV. Compared to the other 8 TeV results, we demonstrate that
our method is significantly advantageous in the compressed region with
$15<\Delta<30$ GeV. The idea of our proposed new search can be further
extended to a general class of long-lived particle searches, exploring the
soft limit for the displaced vertex signatures, covering the blind spots of
current searches . We hope our study demonstrates and accelerates the new era
of open science for particle physics Larkoski:2017bvj ; Tripathee:2017ybi ;
Chen:2018drk ; Andrews:2019faz ; Komiske:2019jim ; Cesarotti:2019nax ;
Apyan:2019ybx ; Mccauley:2019uis ; Abdallah:2020pec ; Lassila-Perini:2021xzn .

NLSP decays—

From the Lagrangians for cases (A) and (B), one can derive the tree-level
effective Lagrangians that governs the decay of the NLPs.

$\displaystyle{\cal L}_{A,B}^{\rm tr}=C^{R}_{A,B}{\cal
O}^{R}_{A,B}+C^{L}_{A,B}{\cal O}^{L}_{A,B}\ ,$ (3)

where for the SUSY case (A) the effective operators are

$\displaystyle{\cal O}^{R}_{A}$ $\displaystyle=$
$\displaystyle(\bar{f}^{\prime}\gamma_{\mu}{\mathbb{P}}_{L}f)(\bar{b}\gamma^{\mu}{\mathbb{P}}_{L}\chi\tilde{t}_{1})\
,$ $\displaystyle{\cal O}^{L}_{A}$ $\displaystyle=$
$\displaystyle(\bar{f}^{\prime}\gamma_{\mu}{\mathbb{P}}_{L}f)(\bar{b}\gamma^{\mu}{\mathbb{P}}_{L}i\\!\not\\!\\!D(\chi\tilde{t}_{1}))\
.$ (4)

The corresponding Wilson coefficients are

$\displaystyle C^{R}_{A}=-\frac{y_{R}g_{2}^{2}}{2m_{W}^{2}m_{t}}\
,\;\;C^{L}_{A}=\frac{y_{L}g_{2}^{2}}{2m_{W}^{2}m_{t}^{2}}\ .$ (5)

For case (B) we have

$\displaystyle{\cal O}^{R}_{B}$ $\displaystyle=$
$\displaystyle(\bar{f}^{\prime}\gamma_{\mu}{\mathbb{P}}_{L}f)(\bar{b}\gamma^{\mu}{\mathbb{P}}_{L}\\!\not\\!\\!A_{H}t_{H})\
,$ $\displaystyle{\cal O}^{L}_{B}$ $\displaystyle=$
$\displaystyle(\bar{f}^{\prime}\gamma_{\mu}{\mathbb{P}}_{L}f)(\bar{b}\gamma^{\mu}{\mathbb{P}}_{L}\\!\not\\!\\!D(\\!\not\\!\\!A_{H}t_{H}))\
,$ (6)

with the Wilson coefficients

$\displaystyle C^{R}_{B}=-\frac{g_{R}g_{2}^{2}}{2m_{W}^{2}m_{t}}\
,\;\;C^{L}_{B}=\frac{g_{L}g_{2}^{2}}{2m_{W}^{2}m_{t}^{2}}\ .$ (7)

In both cases, the right-handed operators ${\cal O}^{R}_{A,B}$ are dimension
seven, whereas the left-handed operators are dimension eight. Therefore,
compared to the right handed operators, the contributions from left handed
operators are suppressed by $\Delta/m_{t}$, which is about $1/6$ in the
parameter region for this study. Especially in case (A),
$y_{R}/y_{L}=-4\tan\theta_{\tilde{t}}$. Therefore as long as
$\tan\theta_{\tilde{t}}>{\cal O}(0.1)$ the decay of $\tilde{t}_{1}$ is mostly
contributed by the right-handed operators. Indeed, as we can see from Fig. 2
that if $\tilde{t}_{1}$ is purely ${\tilde{t}}_{L}$, its lifetime is
approximately ${\cal O}(10^{3})$ times larger than the pure ${\tilde{t}}_{R}$
case. On the other hand, if the NLSP is mostly left-handed top partners, one
usually expects a bottom partner to contribute to the optimal channel for
collider search. The $c\tau$ values for case (B) are shown in Fig. 2. One can
see that for typical choices of $g_{R,L}$ the behavior of the $c\tau$ in case
(B) is almost identical to case (A). Thus, the same search strategy applies to
both cases. Therefore, in the following discussions, we focus on the SUSY case
(A) to illustrate our search strategy.

When $\tilde{t}_{1}$ decays, the typical energy scale in the internal virtual
particles is around $\Delta$. However, to get the effective operators, we
encounter energy scales of $m_{\tilde{t}_{1}}$, $m_{t}$, and $m_{W}$, which
are much larger than the minor splitting $\Delta$. Therefore, we include the
leading log corrections to the right-handed operators to calculate the decay
rate and the leptonic and hadronic branching ratios. To calculate the leading
part of the QCD corrections, we use the renormalization group method to crank
down the energy scale from $m_{\tilde{t}}$ to $\Delta$. We use the non-
relativistic quantum field theory technique, similar to the heavy quark
effective theory Manohar:2000dt . The detailed one-loop calculations are
presented in the supplemental material.

Figure 2:  Proper lifetime $c\tau$ of top partners as functions of
$\sin\theta_{t}$ for $500$ GeV top partner mass. The blue, green, and black
curves correspond to $\Delta=20,30$ and 40 GeV. The solid curves show results
for the SUSY model (A), with one-loop QCD RG corrections to the righthanded
operators included. The dashed curves are the tree-level results for model
(B).

Data processing—

We apply a single un-prescaled trigger HLT_PFMET150 for the 2012 MET primary
dataset CMSMET12 , which includes valid runs from CMS RunB, RunC and
corresponds to an integrated luminosity of $11.6$ $\mathrm{fb}^{\mathrm{-1}}$
CMSdata12 . The data contains high-level physics objects and additional event-
and sector-based information that allows for offline corrections and kinematic
refitting. The analysis is performed using the CMS SoftWare (CMSSW) version
5.3.32, provided by the Docker image released through the CMS Open Data portal
with build-in tools for offline analysis CMSWB .

We use reconstructed candidates in the Analysis Object Data (AOD). Primary
Vertices (PV) correspond to those lying on the beamline, including those from
pile-up events. Jets, together with the missing transverse momentum vector,
are reconstructed using the particle flow (PF) algorithm CMS-PAS-PFT-09-001 ;
CMS-PAS-PFT-10-001 ; Sirunyan_2017 . Besides jets and MET, our analysis is
mainly based on collections of reconstructed tracks. We use only the high
purity tracks with track $p_{\mathrm{T}}>1$ GeV and $|\eta|<2.4$ to ensure a
high quality Collaboration_2014 . For each track, the transverse impact
parameter $d_{xy}$ is computed from the beam-spot, which has been corrected in
each data-taking period. The track four-momentum is computed at the vertex
position, assuming the mass of a charged pion.

Displaced vertices are not provided in the AOD sample, and we reconstruct them
offline. We use the build-in Trimmed Kalman vertex filter and finder from the
CMSSW to fit and find vertices from displaced tracks Collaboration_2007 . We
select displaced tracks with $|d_{xy}|/\sigma_{d_{xy}}\geq 4$ and valid
vertices with $\chi^{2}>0$, $ndf>0$, $\chi^{2}/ndf<5$. Here $d_{xy}$ is the
fitted vertex displacement in the transverse plane, $\sigma_{d_{xy}}$ the
uncertainty of the fit, and $ndf$ the number of degrees of freedom. To avoid
contamination from interactions with the beam pipe or detector material, we
further require the reconstructed vertices to have a distance to the beamline
$d_{BV}$ from $0.1$ mm to $20$ mm Sirunyan:2018pwn . This very conservative
upper cut in transverse displacement is imposed mainly to avoid extrapolation
of detector and background estimation from data, given the lack of data and
full-simulation data in this regime. This conservative choice also simplifies
our study in that the signal tracks can be regarded as coming from a single
high purity category Collaboration_2014 .

Simulations and efficiencies—

The signal pairly-produced top squark events are generated using MadGraph5
Alwall:2014hca  with up to two associated jets, with our benchmark model
parameters using the build-in MSSM model file deAquino:2011ub . The generated
parton-level events are interfaced to Pythia8 Sjostrand:2007gs ;
Sj_strand_2006  for parton shower and R-hadron decays, where an MLM matching
is performed and the jets are reconstructed using the FASTJET3 package
Cacciari:2011ma ; Cacciari:2005hq . The Pythia-level events are reweighted
with matrix element square to include kinetic information of the
${\tilde{t}}_{1}$ decay. We identify the intermediate top squarks, B mesons
and randomly displace the vertex positions for each simulated event, following
the decay local probability distribution based on the event-by-event lab-frame
lifetime and velocities.

The signal yield under event reconstruction and selection is computed based on
a mixture of requirements on the simulated kinematic variables, and on
applying dedicated track and vertex efficiencies. The track efficiency is used
to determine the number of displaced tracks under the selection
$|d_{xy}|/\sigma_{d_{xy}}\geq 4$ that are available to vertex reconstruction.
The vertex efficiency determines if a signal vertex of a given number of
displaced tracks can be reconstructed by the detector. We expect the inclusion
of the two efficiencies to capture most of the detector effect, making our
signal comparable to the data.

To obtain the $\sigma_{d_{xy}}$ distribution of the tracks from the stop and
B-meson decay, we consider a set of high fidelity tracks selected with
$|d_{xy}|<250~{}\rm\mu m$ and $|\eta|<2.4$ in the Open Data. For a simulated
track with $p_{\mathrm{T}}^{0}$ and $d_{xy}^{0}$, we select the Open Data
tracks in the corresponding $p_{T}$ bin and compute the efficiency as the
fraction of the number of tracks satisfying $\sigma_{d_{xy}}<|d_{xy}^{0}|/4$.
Besides the selection efficiency, we also include a constant 90%
reconstruction efficiency for each track according to the CMS track
performance Collaboration_2014 .

To estimate the displaced vertex identification and reconstruction efficiency
(hereafter vertex efficiency), we exploit the MC truth information from an 8
TeV full-detector simulated $t\bar{t}$ sample obtained from the CMS Open Data
portal CMSTTJets12 . In this $t\bar{t}$ sample, we select hadronically
decaying $B_{0}$ or $\bar{B}_{0}$ mesons by requiring the energies to be
within 10 to 30 GeV, and the vertex position to be larger than $0.5~{}$mm but
smaller than $18~{}$mm from the beam line. These $B_{0}/\bar{B}_{0}$ meson
candidates have properties mimicking our signals; hence can be used to
calculate the vertex efficiency. We further categorize these reconstructed
displaced candidates by the number of displaced tracks $N_{gen,tk}$, with each
track passing $d_{xy}>0.5$ mm, $p_{\mathrm{T}}>1$ GeV, $|\eta|<2.4$ and are
not coming from tertiary vertices. In each catalog, the vertex efficiency can
be computed by matching the reconstructed displaced vertices or their
associated tracks to the simulation truth-level ones. We use two methods to
perform this displaced vertices/tracks matching. The first method is based on
the fraction of tracks in a reconstructed vertex that can be matched to the
truth-level ones originating from the decaying vertex. The second method is
based on a requirement of the separation between the reconstructed and the
truth-level vertex. We found the matched candidates from the two methods have
$\gtrsim 60$% overlap in all considered catalogs. We use the results of the
first one for other parts of the analysis. We describe details of the
efficiency extraction in the supplemental material.

Figure 3:  Vertex $H_{T}^{V}$ distributions for templates of the CMS
simulation (bold bars) and of our signal sample with $\Delta=20~{}$GeV,
$m_{\tilde{t}_{1}}=360~{}$GeV. The agreement between the two simulations
verifies our modeling of signal efficiencies under selection and
reconstruction.

In our signal events, because the B-meson comes from long-lived R-hadron
decays, there are tertiary structures in the displaced vertices, depending on
the flying distance of the B-meson. To properly account for this effect, we
merge nearby pythia-level vertices that are below the reconstruction
resolution before applying the vertex efficiency. Details of the vertex
combination are contained in the supplemental material. When applying the
efficiencies, we note that vertices with $N_{{\sc pythia},tk}$ equals two or
three have a probability of passing our data selection $N_{vtx,tk}\geq$4\.
This is caused by contaminating tracks from other vertices.

After merging nearby vertices, we compare the vertex $H_{T}^{V}$ distribution
of the signal-like templates from the CMS simulation and a benchmark signal
sample with $\Delta=20~{}$GeV and $m_{\tilde{t}_{1}}=360~{}$GeV. This
comparison is performed in a one-vertex control region, defined by requiring
one displaced vertex with $N_{vtx,tk}\geq$4, MET larger than 150 GeV, and
transverse momentum of the ISR jet higher than 150 GeV. As shown in Fig. 3,
the selected templates do resemble the simulated signal samples in the
$N_{gen,tk}=2,3,4$ categories.

Results—

We select events with $p_{\mathrm{T}}^{j1}>300$ GeV,
$E^{\text{miss}}_{\mathrm{T}}>300$ GeV, and two displaced vertices with
$N_{vtx,tk}\geq$4 and $H^{V}_{\rm T}<40$ GeV. A selection flow is shown in
Table 1 for the data and a signal benchmark with $m_{\tilde{t}_{1}}=360~{}$GeV
and $\Delta=20~{}$GeV. In the table, we include the intermediate steps to
demonstrate the efficiency loss under the requirement of $N_{vtx,tk}$. With
higher statistics, one may require tighter $H^{V}_{\rm T}$ for improvement.
Based on the selected events, we set limits on the signal strength using the
Feldman-Cousins method PhysRevD.57.3873 . Our results are shown in Fig. 4. We
find a 68% (95%) confidence level (CL) limit corresponds to 1.28 (3.09) signal
events with the zero background post-selection.

Selection | Data | Signal BM
---|---|---
MET primary | 4.3$\times 10^{7}$ | -
$p_{\mathrm{T}}^{j1}>300$ GeV, $E^{\text{miss}}_{\mathrm{T}}>300$ GeV | 2.3$\times 10^{5}$ | 198
One displaced vertex ($N_{vtx,tk}\geq$2) | 4.7$\times 10^{4}$ | 83
One displaced vertex ($N_{vtx,tk}\geq$3) | 3.8$\times 10^{3}$ | 65
One displaced vertex ($N_{vtx,tk}\geq$4) | 3.0$\times 10^{2}$ | 39
One displaced vertex with $H^{V}_{\rm T}<40$ GeV | 245 | 39
Two displaced vertices with $H^{V}_{\rm T}<40$ GeV | 0 | 3.0
Table 1:  Selection flow table. Each selection criterion is applied on top of
the previous ones. A horizontal line is added to separate the selections
with/without the $H^{V}_{\rm T}$ requirement. The MC benchmark corresponds to
a top squark mass of 360 GeV and a mass gap of 20 GeV.

Figure 4:  Limits on the top squark mass and the mass splitting. CMS 8 TeV
limits from Ref. Khachatryan:2015pot ($\tilde{t}\to
bl\bar{l}\tilde{\chi}_{1}^{0}$ channel). On the left panel, the DM relic
density is taken from Ref. Keung:2017kot  with the bound state effects
included. On the right panel, the contour for the DM relic density is computed
using the MadDM program Backovic:2013dpa ; Arina:2020kko , considering
$\sin\theta_{\tilde{t}}=0.3$ and sbottom mass 1 TeV. The 8 TeV disappearing
track (DT) searches Aad:2013yna ; CMS:2014gxa  and heavy stable charged
particle (HSCP) searches ATLAS:2014fka ; Chatrchyan:2013oca  are re-
interpreted from the existing searches Chatrchyan:2013oca  using the framework
in Ref. Liu:2015bma . For a fair comparison, given the limited 8 TeV open data
energy and luminosity, we only include the current search results with 8 TeV
data. For a comparative understanding regarding the 13 TeV searches, please
see the discussion in the text and the supplemental material.

In the compressed region, the traditional search based upon kinematic
reconstructions, such as MT2 variables, in the top squark pair production
channels become insensitive, as most of the energy of the top squarks is
carried away by the neutralinos and observable tracks are very soft.
Nevertheless, relying on the initial state radiation (ISR), both CMS and ATLAS
conducted searches for missing energy plus jets Sirunyan:2018omt ;
Sirunyan:2017wif ; Aad:2020sgw . However, in these studies, the decay of
${\tilde{t}}_{1}$ is assumed to be prompt. Therefore, these results do not
fully apply to the ${\tilde{t}}_{1}$-bino co-annihilation model. We show in
blue-shaded regions in Fig. 4 constraints from these prompt searches, where we
cut away their coverage in parameter regions where the top squark proper
lifetime is beyond 0.2 mm. Note that the lifetime changes as eighth power of
the mass splitting, hence the prompt searches lose sensitivity rapidly as the
splitting decreases. Given the exact reconstruction for the prompt object in
the search is not publicly available, we choose 0.2 mm as the cut for prompt
searches.111Note that the prompt search at requires the leptons have an impact
parameter less than 0.1 mm, which is well-contained by our lifetime smaller
than 0.2 mm cut. In addition, the search uses a vetoing of b-tagged jets,
which is hard to model without knowing the full detail of the tagging
algorithm, especially for further displaced b-jets. In the regime with larger
splittings, e.g., mass splitting more than 30 GeV, our search sensitivity is
driven by the displacements from B-meson with prompt top squark decays,
approaching that of the CMS 8 TeV analysis. Note that our limit in the prompt
region is weaker than that of the CMS 8 TeV analysis. In the compressed regime
with $\Delta<30$, our method significantly improves the sensitivity and
excludes $m_{\tilde{t}}$ up to 350 (380) GeV for $\sin\theta_{t}=1$
($\sin\theta_{t}=0.3$).

Various long-lived particle search channels at the LHC cover complementary
regions. Based upon the framework established in Ref. Liu:2015bma , we
implement and recast the relevant search sensitivities for compressed stops.
In the parameter regions of benchmark models considered in this paper, the
heavy stable charged particle searches and disappearing track searches by both
ATLAS ATLAS:2014fka ; Aad:2013yna  and CMS Chatrchyan:2013oca ; CMS:2014gxa
are relevant and show complementary coverage. We show in Fig. 4 the
reinterpreted limits in our model from these 8 TeV long-lived particle
searches. We can see that the prompt and long-lived searches leave a critical
gap in the stop-bino mass plane, in particular, motivated by the relic
abundance considerations. The displaced vertex dijet search at 8 TeV
CMS:2014wda , which requires a pair of 30 GeV $p_{T}$ jets, is inefficient due
to the soft nature of our signal. Hence we did not recast it here. Our
proposed search, validated by the CMS Open Data, already shows stronger
constraints in the motivated regions, highlighting the importance of searches
for soft hadronic long-lived signatures.

Summary and outlook— In this work, we point out a well-motivated and yet
challenging regime of collider searches for new physics, namely, long-lived
soft and hadronic signatures. This region is particularly motivated by a broad
class of dark matter co-annihilation considerations. Yet, the softness and
hadronic nature make it a very challenging signal to look for at the LHC.
Simultaneously, due to the insufficient understanding and modeling of the
subtle background for them, it is preposterous for theorists to estimate how
viable the search would be at the LHC. Despite the difficulties, we directly
confront the challenge, beyond motivating the signatures, using the CMS Open
Data and simulation framework to understand the signal and background
behaviors.

To provide a concrete BSM model example, we consider the long-lived nature for
a SUSY-like signal and a composite-model-like signal from top partners as the
NLSP. As one of the key predictions in generically natural models, the top
partners can be long-lived in the compressed regime. Moreover, this regime
also provides a viable LSP DM candidate via the co-annihilation mechanism. We
are motivated to fully explore such possibilities by proposing a new long-
lived particle search that is poorly covered by conventional mono-jet searches
and long-lived particle searches.

We propose to search for displaced vertices within the MET-triggered samples.
We showed its excellent coverage in parameter space, complimenting current
prompt searches and long-lived signature searches. Our proposed search
achieves better coverage in the long-live regime and enjoys a great prospect
to improve with higher and higher luminosities from its low-background and
statistical-limited nature. This is in contrast to mono-jet search dominated
by systematic uncertainties. The idea of our proposed new search can be
further extended to a general class of long-lived particle searches, exploring
the soft limit for the displaced vertex signatures, covering the blind spots
of current searches . We hope our study demonstrates and accelerates the new
era of open science for particle physics.

Acknowledgment — We thank CERN, the CMS collaboration, and the CMS Data
Preservation and Open Access (DPOA) team for making the LHC data accessible.
We thank Alexander Ridgway for the work at the beginning stage of this
project. We thank Jesse Thaler for helpful comments and suggestions. HA is
supported by NSFC under Grant No. 11975134, the National Key Research and
Development Program of China under Grant No.2017YFA0402204 and the Tsinghua
University Initiative Scientific Research Program. ZH is supported by NSFC
under Grant No. 11975010 and No. 12061141002, and the Tsinghua University
Initiative Scientific Research Program. ZL acknowledges the Minnesota
Supercomputing Institute (MSI) at the University of Minnesota for providing
resources that contributed to the research results reported within this paper.

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Supplemental Materials

##  Appendix A Quantum corrections to the ${\tilde{t}}_{1}$ decay process

The effective Lagrangian determines the ${\tilde{t}}_{1}$ decay can be written
as

$\displaystyle{\cal L}^{\rm tr}=C_{R}^{(\ell)}{\cal
O}_{R}^{(\ell)}+C_{R}^{(h)}{\cal O}_{R}^{(h)}+C_{L}^{(\ell)}{\cal
O}_{L}^{(\ell)}+C_{L}^{(h)}{\cal O}_{L}^{(h)}\ ,$ (8)

where the superscript $\ell$ $(h)$ are for leptonic (hadronic) decay channels.
From perturbative calculation it is easy to show that

$\displaystyle{\cal
O}_{R}^{(h)}=(\bar{u}_{i}\gamma_{\mu}{\mathbb{P}}_{L}d_{j})V_{ij}(\bar{b}\gamma^{\mu}{\mathbb{P}}_{L}\chi\tilde{t}_{1}),$
(9) $\displaystyle{\cal
O}_{R}^{(\ell)}=(\bar{\nu}_{i}\gamma_{\mu}{\mathbb{P}}_{L}l_{j})(\bar{b}\gamma^{\mu}{\mathbb{P}}_{L}\chi\tilde{t}_{1})\,$
(10) $\displaystyle{\cal
O}_{L}^{(h)}=(\bar{u}_{i}\gamma_{\mu}{\mathbb{P}}_{L}d_{j})V_{ij}(\bar{b}\gamma^{\mu}{\mathbb{P}}_{L}i\\!\not\\!\\!D(\chi\tilde{t}_{1}))\,$
(11) $\displaystyle{\cal
O}_{L}^{(\ell)}=(\bar{\nu}_{i}\gamma_{\mu}{\mathbb{P}}_{L}l_{j})(\bar{b}\gamma^{\mu}{\mathbb{P}}_{L}i\\!\not\\!\\!D(\chi\tilde{t}_{1}))\
,$ (12)

where $V_{ij}$ is the CKM matrix, and $D$ is the covariant derivative for the
strong interaction.222The summation of color indices is also implicit. The
color structure of operators become important for loop corrections that we
include in this study. At tree level the Wilson coefficients are

$\displaystyle
C_{R}^{(h)}=C_{R}^{(\ell)}=-\frac{y_{R}g_{2}^{2}}{2m_{W}^{2}m_{t}}\
,\;\;C_{L}^{(h)}=C_{L}^{(\ell)}=\frac{y_{L}g_{2}^{2}}{2m_{W}^{2}m_{t}^{2}}\ .$
(13)

The right-handed operators ${\cal O}_{R}^{(h)}$ and ${\cal O}_{R}^{(\ell)}$
are dimension seven whereas the left-handed operators ${\cal O}_{L}^{(h)}$ and
${\cal O}_{L}^{(\ell)}$ are dimension eight.

Neglecting the masses of the light quarks and leptons, as well as the bottom
quark mass, and taking the limit that $\Delta\ll m_{t},m_{\chi}$, in the case
of $\cos\theta_{\tilde{t}}=0$, the decay rate of $\tilde{t}_{1}$ can be
calculated analytically as

$\displaystyle\Gamma=\frac{g_{2}^{4}g_{1}^{2}\Delta^{8}}{20160\pi^{5}m_{W}^{4}m_{t}^{2}m_{\tilde{t}_{1}}}.$
(14)

However, with nonzero $\cos\theta_{\tilde{t}}$ and $m_{b}\neq 0$, we need to
calculate the stop decay rate numerically.

In the process of the $\tilde{t}_{1}$ decay, the typical energy scale in the
internal virtual particles is around $\Delta$. However, to get the effective
operators, we encounter energy scales of $m_{\tilde{t}_{1}}$, $m_{t}$ and
$m_{W}$, which are much larger than the small splitting $\Delta$. Therefore,
to accurately calculate the decay rate and the leptonic and hadronic branching
ratios, we need to include the leading QCD corrections to the right-handed
operators. The ratio of one-loop QCD corrected $c\tau$ value of purely right
handed $\tilde{t}_{1}$ to the leading order value is shown in Fig. 5. One can
see that the one-loop QCD effects induce a 20% correction.

Figure 5:  The ratio of the one-loop $c\tau$ value to the leading order result
as a function of $\Delta$ for $m_{\tilde{t}_{1}}=500$ GeV.

To collect the leading log QCD corrections, we use the renormalization group
method to crank down the energy scale from $m_{{\tilde{t}}_{1}}$ to $\Delta$.
We use the non-relativistic quantum field theory technique, similar to the
heavy quark effective theory Manohar:2000dt , to simplify the calculation.

The QCD correction mixes the ${\cal O}_{R}^{(h)}$ operator with

$\displaystyle{\cal O}_{R}^{\prime(h)}$ $\displaystyle=$
$\displaystyle(\bar{b}\gamma_{\mu}{\mathbb{P}}_{L}D_{j})V_{ij}(\bar{U}_{i}\gamma^{\mu}{\mathbb{P}}_{L}\chi\tilde{t}_{1})\
.$ (15)

where the combination inside each bracket is a color singlet. Then RG
evolution of their Wilson coefficients follows

$\displaystyle\frac{d}{d\log\mu}\left(\begin{array}[]{c}C_{R}^{(h)}\\\
C_{R}^{\prime(h)}\end{array}\right)=\frac{\alpha_{S}}{4\pi}\left(\begin{array}[]{cc}-5{}&~{}3\\\
3{}&~{}-5\\\ \end{array}\right)\left(\begin{array}[]{c}C_{R}^{(h)}\\\
C_{R}^{\prime(h)}\end{array}\right)\ .$ (22)

Whereas for the leptonic operator we have

$\displaystyle\frac{d}{d\log\mu}C_{R}^{(\ell)}=-\frac{\alpha_{S}}{\pi}C_{R}^{(\ell)}\
.$ (23)

Eqs. (22) and (23) are used to calculate the leading order RG effect from the
$m_{W}$ to $\Delta$. We can also include the RGE effect for the energy scale
between $m_{W}$ and $m_{t}$ although this effect is not large. The running
from $m_{{\tilde{t}}_{1}}$ to $m_{t}$ can also be described by Eq. (23) since
the color structures of the $\tilde{t}_{1}\bar{t}_{R}\chi^{0}_{1}$ is the same
as ${\cal O}_{R}^{(\ell)}$.

In the following, we present the main steps of the derivation of the RG
equations (22) and (23). In the rest frame of the decaying ${\tilde{t}}_{1}$,
$\chi_{1}^{0}$ is non-relativistic. In loop diagrams the ${\tilde{t}}_{1}$
internal line is also almost on-shell since $\Delta\ll m_{{\tilde{t}}_{1}}$.
Therefore, following Ref. Manohar:2000dt  we redefine the ${\tilde{t}}_{1}$
field as

$\displaystyle{\tilde{t}}_{1}=e^{-im_{{\tilde{t}}_{1}}v\cdot x}{\tilde{T}}\ ,$
(24)

where $v$ is the four-velocity of ${\tilde{t}}_{1}$. Then the kinetic term and
the mass term of ${\tilde{t}}_{1}$ can be written as

$\displaystyle|\partial^{\mu}{\tilde{t}}_{1}|^{2}-m_{{\tilde{t}}_{1}}^{2}|{\tilde{t}}_{1}|^{2}=|\partial_{\mu}{\tilde{T}}|^{2}+im_{{\tilde{t}}_{1}}v\cdot({\tilde{T}}\partial{\tilde{T}}^{*}-{\tilde{T}}^{*}\partial{\tilde{T}})\
.$ (25)

Since we have already factorized the static energy of ${\tilde{t}}_{1}$ by the
redefinition (24), on the righthand side of Eq. (25) the size of
$\partial{\tilde{T}}$ is of the order of magnitude of $\Delta{\tilde{T}}$.
Therefore, we have

$\displaystyle-
im_{{\tilde{t}}_{1}}v\cdot({\tilde{T}}\partial{\tilde{T}}^{*}-{\tilde{T}}^{*}\partial{\tilde{T}})\gg|\partial_{\mu}{\tilde{T}}|^{2}\
.$ (26)

Therefore, we can neglect the $|\partial_{\mu}{\tilde{T}}|^{2}$ term. With
strong interaction, the Lagrangian for ${\tilde{T}}$ is

$\displaystyle{\cal
L_{\tilde{T}}}=-im_{{\tilde{t}}_{1}}v\cdot({\tilde{T}}D{\tilde{T}}^{*}-{\tilde{T}}^{*}D{\tilde{T}})\
,$ (27)

where $D_{\mu}$ is the covariant derivative for the strong interaction. With
the convention that $D_{\mu}=\partial_{\mu}+igT^{a}A_{\mu}$, where $T^{a}$ is
the SU(3) generator, the strong interaction vertex between gluon and
${\tilde{T}}$ is

$\displaystyle-2igm_{{\tilde{t}}_{1}}T^{a}_{ij}\ .$ (28)

The propagator of ${\tilde{T}}$ is

$\displaystyle\frac{i}{2m_{{\tilde{t}}_{1}}v\cdot q+i\epsilon}\ ,$ (29)

where $q$ is defined as $q\equiv p-mv$ and $p$ is the four momentum of
${\tilde{t}}_{1}$. It can also be seen as the off-shellness of the propagator.

In this work, we only calculate the QCD correction for the case that
${\tilde{t}}_{1}$ is purely right-handed. Then the gauge-Yukawa coupling of
${\tilde{T}}$, $\chi_{1}^{0}$ and the top quark can be written as

$\displaystyle y_{R}e^{-im_{{\tilde{t}}_{1}}v\cdot
x}{\tilde{T}}\bar{t}{\mathbb{P}}_{L}\chi_{1}^{0}\ .$ (30)

The factor $e^{-im_{{\tilde{t}}_{1}}v\cdot x}$ ensures the energy-momentum
conservation. With these Feynman rules we can calculate the loop diagrams.

###  A.1 Self energy of ${\tilde{T}}$

The Feynman diagram for the ${\tilde{T}}$ self energy is shown in Fig. 6. We
use the dimensional regularization ($d=4-2\epsilon$) to evaluate the loop
diagrams. Then the self-energy of ${\tilde{T}}$ in the non-relativistic
formalism defined above can be evaluated as

$\displaystyle i\Sigma(p)$ $\displaystyle=$
$\displaystyle(-2igm_{{\tilde{t}}_{1}}v_{\mu})(-2igm_{{\tilde{t}}_{1}}v_{\mu})\mu^{2\epsilon}T^{a}_{ik}T^{a}_{kj}\int\frac{d^{d}q}{(2\pi)^{d}}\frac{i}{2m_{{\tilde{t}}_{1}}~{}v\cdot(p+q)}\frac{-i}{q^{2}}$
(31) $\displaystyle=$
$\displaystyle-\frac{8}{3}\delta_{ij}g^{2}\mu^{2\epsilon}m_{{\tilde{t}}_{1}}\int\frac{d^{d}q}{(2\pi)^{d}}\frac{1}{v\cdot(p+q)}\frac{1}{q^{2}-m_{{\tilde{t}}_{1}}^{2}}\
.$

Following the steps in sec. 3.1 in Manohar:2000dt  we have

$\displaystyle
i\Sigma(p)\rightarrow-\frac{ig^{2}}{3\pi^{2}}m_{{\tilde{t}}_{1}}v\cdot
p\frac{1}{\epsilon}+{\rm finite}\ .$ (32)

Comparing to the tree-level Lagrangian in (27), we have at one-loop

$\displaystyle Z_{\tilde{T}}=1+\frac{g^{2}}{6\pi^{2}\epsilon}\ .$ (33)

Figure 6:  Left: Self energy diagram of ${\tilde{T}}$ in non-relativistic
theory. Right: One loop correction to the
${\tilde{t}}\bar{t}{\mathbb{P}}_{L}\chi^{0}_{1}$ operator.

###  A.2 Anomalous dimension of
${\tilde{t}}\bar{t}{\mathbb{P}}_{L}\chi^{0}_{1}$

The Feynman diagram for the QCD correction to the ${\cal
O_{y}}={\tilde{t}}\bar{t}{\mathbb{P}}_{L}\chi^{0}_{1}$ operator is shown in
the right panel of Fig. 6. It can be evaluated as

$\displaystyle\mu^{2\epsilon}(-ig)(-2igm_{{\tilde{t}}_{1}}v^{\mu})t^{a}_{ik}t^{a}_{kj}\int\frac{d^{d}q}{(2\pi)^{d}}\bar{u}_{t}(k_{1})\gamma_{\mu}\frac{i(\not\\!k_{1}+\not\\!q)}{(k_{1}+q)^{2}}{\mathbb{P}}_{L}v_{\chi}(k_{2})\frac{i}{2m_{{\tilde{t}}_{1}}v\cdot
q}\frac{-i}{q^{2}}\ .$ (34)

Follow the same trick in Ch. 3 of Manohar:2000dt  the divergent part of the
above loop correction is

$\displaystyle\frac{g^{2}}{12\pi^{2}\epsilon}\delta^{ij}\bar{u}_{t}(k_{1}){\mathbb{P}}_{L}v_{\chi}(k_{2})\
.$ (35)

Then we have

$\displaystyle\frac{Z_{\tilde{T}}^{1/2}Z_{t}^{1/2}}{Z_{y}}-1=-\frac{g^{2}}{12\pi^{2}\epsilon}\
,$ (36)

where

$\displaystyle Z_{t}=1-\frac{g^{2}}{12\pi^{2}\epsilon}\ ,$ (37)

is the renormalization factor of the relativistic quark field. Then we have

$\displaystyle Z_{y}=1+\frac{g^{2}}{8\pi^{2}\epsilon}\ .$ (38)

The anomalous dimension of the operator is

$\displaystyle\gamma_{y}=\frac{\mu}{Z_{y}}\frac{dZ_{y}}{d\mu}=-\frac{g^{2}}{4\pi^{2}}\
.$ (39)

Then the gauge-Yukawa coupling $y_{R}$ can be seen as the Wilson coefficient
of the operator ${\cal O}_{y}$ and as a result its evolution can be written as

$\displaystyle\frac{dy_{R}}{d\log\mu}=-\frac{g^{2}}{4\pi^{2}}y_{R}\ ,$ (40)

which is the same as in Eq. (23).

###  A.3 The operator
$W_{\mu}\bar{b}\gamma^{\mu}{\mathbb{P}}_{L}\chi_{1}^{0}{\tilde{T}}$

When the energy scale $\mu$ is below $m_{t}$, the top quark is integrated out
and we get an effective operator ${\cal O}_{W}\equiv
W_{\mu}\bar{b}\gamma^{\mu}{\mathbb{P}}_{L}\chi_{1}^{0}{\tilde{T}}$. Compared
to ${\cal O}_{y}$, the difference is that in ${\cal O}_{W}$, there is a gamma
matrix. However, a detailed calculation can show that this does not change the
result of the one-loop anomalous dimension. The reason is that Eq. (34) we can
see that in the numerator of the top propagator, only the $\not\\!q$ part
contributes to the divergent part. Therefore, the coefficient of the divergent
part only depends on the coefficient of $\not\\!q$. Therefore, the evolution
of its Wilson coefficient $C_{W}$ is the same as in Eq. (40),

$\displaystyle\frac{dC_{W}}{d\log\mu}=-\frac{g^{2}}{4\pi^{2}}C_{W}\ .$ (41)

Figure 7:  Left: Self energy diagram of ${\tilde{T}}$ in non-relativistic
theory. Right: One loop correction to the
${\tilde{t}}\bar{t}{\mathbb{P}}_{L}\chi^{0}_{1}$ operator.

###  A.4 Quantum corrections to ${\cal O}_{R}^{(\ell)}$ and ${\cal
O}_{R}^{(h)}$

The color structure of ${\cal O}_{R}^{(\ell)}$ is of no difference from ${\cal
O}_{W}$. Therefore its evolution is the same as ${\cal O}_{W}$ and ${\cal
O}_{y}$, and this is how we get Eq. (23).

For ${\cal O}_{R}^{(h)}$ the one-loop diagrams for quantum corrections are
shown in Fig. 8. There are six diagrams contributing to the one-loop QCD
correction to ${\cal O}_{R}$ as shown in Fig. 8. One can see that the result
of diagram (a) is the same as the one-loop result in Fig. 7, and therefore its
divergent part has the same coefficient as in (35), and can be written as

$\displaystyle\frac{g^{2}}{12\pi^{2}\epsilon}\delta^{ij}\delta^{kl}\bar{u}_{b}\gamma^{\mu}{\mathbb{P}}_{L}\chi{\tilde{T}}\bar{u}_{u}\gamma_{\mu}{\mathbb{P}}_{L}v_{d}\
.$ (42)

Explicit calculation of diagrams (b) and (c) show that their divergent parts
are canceled with each other. The loops in diagrams (d) and (e) are only
composed of light quarks and therefore can be done with standard calculation.
In these two diagrams the exchange of the gluon mixes the color indices. The
results are, for diagram (d)

$\displaystyle\frac{g^{2}}{16\pi^{2}\epsilon}t^{a}_{ij}t^{a}_{kl}\bar{u}_{b}\gamma^{\mu}{\mathbb{P}}_{L}\chi{\tilde{T}}\bar{u}_{u}\gamma_{\mu}{\mathbb{P}}_{L}v_{d}\
,$ (43)

and for diagram (e)

$\displaystyle-\frac{4g^{2}}{16\pi^{2}\epsilon}t^{a}_{ij}t^{a}_{kl}\bar{u}_{b}\gamma^{\mu}{\mathbb{P}}_{L}\chi{\tilde{T}}\bar{u}_{u}\gamma_{\mu}{\mathbb{P}}_{L}v_{d}\
.$ (44)

With the formula
$t^{a}_{ij}t^{a}_{kl}=\frac{1}{2}\delta_{il}\delta_{jk}-\frac{1}{6}\delta_{ij}\delta_{kl}$,
we can see that the operator ${\cal O}^{\prime(h)}_{R}$ appears from the
quantum correction. For diagram (f), from the well-known result of Ward
identity, we know that its contribution to the correction of ${\cal
O}^{(h)}_{R}$ is canceled by the contributions from the wave function
renormalization. Collecting the above results, we get the renormalization
factors for $O_{R}^{(h)}$ and $O_{R}^{\prime(h)}$

$\displaystyle
Z_{R}^{(h)}=\left(\begin{array}[]{cc}1+\frac{5g^{2}}{32\pi^{2}\epsilon}{}&~{}-\frac{3g^{2}}{32\pi^{2}\epsilon}\\\
-\frac{3g^{2}}{32\pi^{2}\epsilon}{}&~{}1+\frac{5g^{2}}{32\pi^{2}\epsilon}\\\
\end{array}\right)\ .$ (47)

Then we can get the anomalous dimensions following the standard procedure that

$\displaystyle\gamma_{R}^{(h)}=\left(\begin{array}[]{cc}\frac{5g^{2}}{16\pi^{2}}{}&~{}-\frac{3g^{2}}{16\pi^{2}}\\\
-\frac{3g^{2}}{16\pi^{2}}{}&~{}\frac{5g^{2}}{16\pi^{2}}\\\ \end{array}\right)\
.$ (50)

This leads to the evolution of the Wilson coefficients in Eq. (22).

Figure 8:  Left: Self energy diagram of ${\tilde{T}}$ in non-relativistic
theory. Right: One loop correction to the
${\tilde{t}}\bar{t}{\mathbb{P}}_{L}\chi^{0}_{1}$ operator.

##  Appendix B Details of Open Data Treatment

###  B.1 Selection efficiency of track impact parameter significance

Each displaced track needs to satisfy an impact parameter significance
selection $|d_{xy}|/\sigma_{d_{xy}}>4$. To estimate the efficiency of this
selection, we need to know the $\sigma_{d_{xy}}$ distribution, which is a pure
detector effect and cannot be obtained without full detector simulation. For a
simple estimate, we use the $\sigma_{d_{xy}}$ distribution of the high purity
prompt tracks, selected with $|d_{xy}|<250~{}\rm\mu m$ and $|\eta|<2.4$ of the
CMS Open Data, to approximate the true distribution. We use prompt tracks
because they are mostly truly identified, similar to our tracks from top
squark and B meson decays. As the error of a distance between two points is
dependent only on that of the two endpoints, we expect the $\sigma_{d_{xy}}$
distribution to have insignificant $d_{xy}$ dependence for physically
originated tracks.

Based on the $p_{\mathrm{T}}-\sigma_{d_{xy}}$ histogram of the selected high
purity prompt tracks, we calculate the selection efficiency in
$p_{\mathrm{T}}$ bins from $0-40$ GeV, with a bin-width of 1 GeV. For a track
with $p_{\mathrm{T}}^{0}$ and $d_{xy}^{0}$, the efficiency is computed as the
ratio of the number of tracks satisfying $\sigma_{d_{xy}}<|d_{xy}^{0}|/4$ and
the total number of tracks in that $p_{\mathrm{T}}$ bin, i.e.,
$N(\sigma_{d_{xy}}<|d_{xy}^{0}|/4~{}|~{}p_{\mathrm{T}}^{0})/N(\sigma_{d_{xy}}>0~{}|~{}p_{\mathrm{T}}^{0})$.
We show in Fig. 9 the two dimensional $p_{\mathrm{T}}-\sigma_{d_{xy}}$
histogram used for this computation.

Figure 9:  Two dimensional $p_{\mathrm{T}}-\sigma_{d_{xy}}$ histogram used for
the computation of the track impact parameter significance selection
efficiency.

###  B.2 Displaced vertex efficiency

Displaced vertices in simulated events should be reconstructed using the same
algorithm as for data events. Without the capability of doing full detector
simulation for the signal samples, we extract the vertex efficiency using a
fully simulated 8 TeV $t\bar{t}$ sample obtained from the CMS Open Data
portal. Our method is viable because both generator- and detector-level
information are available in the fully simulated AOD sample. At the generator
level, we select hadronically decaying $B_{0}$ or $\bar{B}_{0}$ mesons whose
vertex position is at least $0.5$ mm, but smaller than $18$ mm from the beam-
line, and have energies within 10 to 30 GeV. Here, we make a harder upper cut
on the signal displacement to be on the conservative side. We further
categorize these candidates by their number of MC charged tracks $N_{gen,tk}$,
where the tracks satisfy $d_{xy}>0.5$ mm, $p_{\mathrm{T}}>1$ GeV and
$|\eta|<2.4$ and are not coming from tertiary vertices. In this way, we
prepared several categories of signal-like displaced vertices.

To calculate the vertex efficiency in each category, we reconstruct displaced
vertices in the same way as we have done for data and check if a signal-like
vertex candidate is reconstructed as a displaced one. For the latter purpose,
one needs to introduce some matching criteria. We consider two methods to
ensure the reliability of our computation. In the first method, hereafter
track fraction (TF) method, we consider matching criteria based on the
fraction of tracks associated with the reconstructed displaced vertex from
B-meson decays. First, we consider the following criteria for matching a
track:

  * •

$\Delta R(p_{1},p_{2})<0.1$, where $\Delta
R(p_{1},p_{2})\equiv\sqrt{(\eta_{1}-\eta_{2})^{2}+(\phi_{1}-\phi_{2})^{2}}$ is
the $\eta-\phi$ separation between the candidates;

  * •

absolute value of the $p_{\mathrm{T}}$ difference between the two tracks is
smaller than 10%;

  * •

distance in the transverse plane is smaller than $0.5$ mm.

We have also checked that the matching quality is not sensitive to these
parameters. Based on the matched tracks, we consider the following criteria
for matching a displaced vertex:

  * •

the fraction of matched tracks among all iterated reconstructed tracks is
higher than 0.4;

  * •

the reconstructed vertex position is at least $0.1$ mm but smaller than $20$
mm from the beam-line;

  * •

invariant mass for the sum of iterated reconstructed tracks is smaller than
6.5.

The requirements of the number of iterated tracks and the vertex position to
be at least $0.1$ mm from the beam-line eliminate misassociations to primary
vertices.

The second method, hereafter the vertex distance (VD) method, we do not match
tracks but simply require that the distance between simulated vertex and the
reconstructed vertex to be smaller than its one sigma uncertainty, which is
computed as:

$\displaystyle\sigma(|\bm{d}_{VV}|)=\dfrac{1}{|\bm{d}_{VV}|}\sqrt{\sum_{i,j}(\bm{d}_{VV})_{i}c_{ij}(\bm{d}_{VV})_{j}}.$
(51)

where $\bm{d}_{VV}$ is the vector from the position of the simulated vertex to
that of the reconstructed one, and $c_{ij}$ is the covariant matrix of the
vertex error where $i,j$ runes over $x,y,z$.

We summarize results from the two methods in Table 2. We adopt results using
the TF method for the analysis in this study and consider the VD method proof
of concept. We find the matched candidates have $\gtrsim 60$% overlap for all
the catalogs. In Fig. 3, we check property of the signal-like categories by
comparing their $H^{V}_{T}$ distribution to that of a signal benchmark with
$\Delta=20~{}$GeV and $m_{\tilde{t}_{1}}=360~{}$GeV, in the one-vertex control
region.

As shown in Table 2, the vertex efficiency is not monotonically increasing
with $N_{gen,tk}$. The reason for this is due to contamination of additional
tracks from the collider environment. To demonstrate that, we put extra
matching conditions $\Delta R<0.1$, $|\Delta
p_{\mathrm{T}}/p_{\mathrm{T}}|<0.05$ on the set of reconstructed tracks and
reconstruct displaced vertices based on these purified tracks. As expected,
the efficiency increased significantly, reaching 85.1% in the TF method for
the $N_{gen,tk}=4$ catalog.

Catalog | $N_{gen,tk}=2$ | $N_{gen,tk}=3$ | $N_{gen,tk}=4$ | $N_{gen,tk}=5$ | $N_{gen,tk}\geq 6$
---|---|---|---|---|---
Efficiency from TF (%) | $23.8\pm 0.4$ | $36.6\pm 1.0$ | $46.1\pm 2.9$ | $45.3\pm 6.2$ | $32.4\pm 10.8$
Efficiency from VD (%) | $17.5\pm 0.3$ | $25.7\pm 1.0$ | $32.6\pm 2.4$ | $32.6\pm 5.0$ | $40.5\pm 12.4$
Overlapping fraction (%) | 59.7 | 62.0 | 64.3 | 70.5 | 83.3
Vertex error ($\rm\mu m$) | 173 | 170 | 164 | 175 | 155
Vertex error RMS ($\rm\mu m$) | 110 | 110 | 103 | 119 | 94.5
Probability of passing $N_{vtx,tk}\geq$2 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0
Probability of passing $N_{vtx,tk}\geq$3 | 0.61 | 0.78 | 0.83 | 0.82 | 0.83
Probability of passing $N_{vtx,tk}\geq$4 | 0.23 | 0.39 | 0.54 | 0.64 | 0.58
Table 2:  Efficiencies and errors of the reconstructed displaced vertices. The
overlapping fraction is computed as the ratio of the number of vertices
matched in both methods to that matched only in the TF method.

###  B.3 Merging MC vertices

In the experiment, two displaced vertices can not be resolved if they are too
close. To take into account this effect, we merge nearby simulated vertices in
each event before applying the efficiencies in Table 2. We merge nearby
displaced vertices through the following steps:

  * •

For all simulated tracks associated with displaced vertices, apply the track
efficiencies. Count the number of selected tracks associated with each vertex.

  * •

Assign errors to these vertices. Based on the number of selected tracks, the
error is generated from a Gaussian random distribution considering error and
RMS values in Table 2. As we have chosen the impact parameter selection
efficiency to mimic that of the impact parameter significance selection one,
we expect $N_{{\sc pythia},tk}\approx N_{gen,tk}$ and do not distinguish them.

  * •

Merge nearby vertices into one if their distance is smaller than the sum of
their errors in quadrature. The new position is calculated as an average of
candidate vertices weighted by their number of selected tracks. The error of
the merged vertex is taken as those of the candidates summed in quadrature.

  * •

Based on the number of $N_{{\sc pythia},tk}$ of the merged vertices, apply the
vertex identification and reconstruction efficiency in Table 2.

###  B.4 Supplementary figures

In Fig. 10, we show the importance of beam-spot correction. Performing this
correction, we found the distribution to be uniform in the $\phi$ direction.

Figure 10:  Transverse track impact parameter distribution, computed with
(right) and without (left) beam-spot correction. The figure is made using
tracks from 10k data events.

In Fig.11, we compare the signal and data features in the one-vertex control
region. The one-vertex control region is defined by requiring one displaced
vertex with $N_{vtx,tk}\geq$4, MET larger than 150 GeV, and transverse
momentum of the ISR jet higher than 150 GeV.

Figure 11:  Comparisons of signal and background kinematic features. The
signal sample is generated considering a top squark mass of 360 GeV and a mass
gap of 20 GeV. The background sample is taken from data in the one-vertex
control region, where we expect the signal contribution to be negligible.
Their differences reveal features that may be exploited to enhance the signal
significance. In this analysis, we cannot exploit all features due to limited
statistics.

##  Appendix C Current limits with 13 TeV Data

In the text, for the purpose of a fair comparison, we show the obtained limit
contour with the 8 TeV ones. To understand the current status of the
constraints, we show in Fig. 12 a comparison of our limit with the 13 TeV
ones. The 13 TeV CMS limit is taken from Ref. Sirunyan:2018omt ($\tilde{t}\to
bff^{\prime}\tilde{\chi}_{1}^{0}$ channel, MVA approach) which exploited a
data set corresponding an integrated luminosity of 35.9 $fb^{-1}$. As in Fig.
4, we also include limits from DT and HSCP experiments. These limits are
recasted from 13 TeV experiments Sirunyan:2018ldc ; Khachatryan:2016sfv ;
Aaboud:2019trc , using the framework in Ref. Liu:2015bma ; Kraml:2013mwa ;
Ambrogi:2018ujg . From Fig. 12, we see the 13 TeV experiments exclude the
majority of parameter space that corresponds to the prompt or the stable heavy
charged particle. Similar to our reinterpretation of the prompt search at 8
TeV, we choose 0.2 mm as the cut for prompt searches. Note that the lifetime
changes as the eighth power of the mass splitting; hence the prompt search
lose sensitivity rapidly as we decrease the splitting. Admittedly, if the
exact prompt search parameters are known, one can perform a more faithful
reinterpretation of the 13 TeV prompt analysis. In this case, we would
anticipate a smooth transition, rather than the abrupt kink in the sensitivity
curve near our $c\tau$ cut in the current figure, the prompt search probably
will start loosing efficiency at $c\tau$ of 0.5 mm, as in the case of Higgs
exotic decay experimental reinterpretation ATLAS:2018pvw . The prompt analysis
loses signal efficiency by more than one order of magnitude from 0.5 mm to 2
mm. Our analysis approach provides a promising test of the parameter region in
the middle, which corresponds to the existence of the long-lived decaying
particle.

Figure 12:  Same as Fig.4, but with existing exclusion limits replaced by the
13 TeV ones. The 13 TeV CMS limits come from Ref. Sirunyan:2018omt
($\tilde{t}\to bff^{\prime}\tilde{\chi}_{1}^{0}$ channel, MVA approach). The
limits from disappearing track (DT) Sirunyan:2018ldc  and heavy stable charged
particle (HSCP) Khachatryan:2016sfv ; Aaboud:2019trc  searches are also
replaced by the ones re-interpreted from 13 TeV analyses, using the framework
in Ref. Liu:2015bma ; Kraml:2013mwa ; Ambrogi:2018ujg . The orange-dashed
curves show the contours above which the models would lead to over-abundant
DM. The corresponding relic densities, for the $\sin\theta_{\tilde{t}}=1$
case, is taken from Ref. Keung:2017kot  with the bound state effects included;
for the $\sin\theta_{\tilde{t}}=0.3$ case, is computed using the MadDM program
Backovic:2013dpa ; Arina:2020kko , considering a sbottom mass of 1 TeV.