Learning to see R -parity violating scalar top decays

Abstract

With this article, we introduce recent, improved machine learning methods from computer vision to the problem of event classification in particle physics. Supersymmetric scalar top decays to top quarks and weak-scale bino-like neutralinos, where the neutralinos decay via the U D D operator to three quarks, are difficult to search for and therefore weakly constrained. The jet substructure of the boosted decay products can be used to differentiate signal from background events. We apply the transformer-based computer vision models otet and axi to images built from jet constituents and compare the classification performance to a more classical convolutional neural network (CNN). We find that results from computer vision translate well to physics applications, and both transformer-based models perform better than the CNN. By replacing the CNN with axi, we find an improvement of S / B by a factor of almost 2 for some neutralino masses. We show that combining this classifier with additional features results in a strong separation of background and signal. We also find that replacing a CNN with a axi model in a simple mock analysis can push the 95% C.L. exclusion limit of stop masses by about 100 and 60 GeV for neutralino masses of 100 and 500 GeV.

Published by the American Physical Society 2024

Full text

Learning to see R-parity violating scalar top decays
Gerrit Bickendorf *and Manuel Drees †
Bethe Center for Theoretical Physics and Physikalisches Institut, University of Bonn, Bonn, Germany
(Received 16 June 2024; accepted 8 August 2024; published 4 September 2024)
With this article, we introduce recent, improved machine learning methods from computer vision to the
problem of event classification in particle physics. Supersymmetric scalar top decays to top quarks and
weak-scale bino-like neutralinos, where the neutralinos decay via the UDD operator to three quarks, are
difficult to search for and therefore weakly constrained. The jet substructure of the boosted decay products
can be used to differentiate signal from background events. We apply the transformer-based computer
vision models
C
o
A
t
N
et and
M
ax
V
i
T
to images built from jet constituents and compare the classification
performance to a more classical convolutional neural network (CNN). We find that results from computer
vision translate well to physics applications, and both transformer-based models perform better than the
CNN. By replacing the CNN with
M
ax
V
i
T
, we find an improvement of S= ffiffiffiffi
B
pby a factor of almost 2 for
some neutralino masses. We show that combining this classifier with additional features results in a strong
separation of background and signal. We also find that replacing a CNN with a
M
ax
V
i
T
model in a simple
mock analysis can push the 95% C.L. exclusion limit of stop masses by about 100 and 60 GeV for
neutralino masses of 100 and 500 GeV.
DOI: 10.1103/PhysRevD.110.056006
I. INTRODUCTION
The minimal supersymmetric extension of the standard
model (MSSM) is a promising candidate for physics
beyond the SM [1–6] that might solve the hierarchy
problem. Despite many experimental searches, most nota-
bly by the ATLAS and CMS collaborations at the CERN
LHC in recent years, no conclusive evidence of its
realization in nature has been found, pushing the parameter
space to ever higher masses (see Ref. [7] for an overview).
In models with conserved Rparity (RPC), many searches
leverage the large pmiss
Tdue to the stable lightest super-
symmetric particle (LSP) leaving the experiment unde-
tected [8]. Once the RPC assumption is dropped, these
strategies often become insensitive. In the context of the
R-parity-violating (RPV) MSSM, new terms are added to
the superpotential that break lepton- or baryon-number
conservation [9]. These additional terms imply that dras-
tically different search strategies are needed [10], especially
when prompt decays of supersymmetric particles become
drowned by the hadronic activity inside the detector.
At a hadron collider like the LHC, the production of
strongly interacting superparticles has the largest cross
section for a given mass. Among these, the stops—the
superpartners of the top quark—are often assumed to be the
lightest. On the one hand, for equal squark masses at some
very high (e.g., grand unified or Planckian) energy scale,
renormalization group effects reduce the masses of the
stops; mixing between the masses of the SUð2Þdoublet and
singlet stops will reduce the mass of the lightest eigenstate
even more [6]. On the other hand, simple naturalness
arguments [11–13] prefer not too heavy stop squarks, but
allow much heavier first- and second-generation squarks.
This motivates the analysis of scenarios where the mass of
the lighter stop squark lies well below those of the other
strongly interacting superparticles.
The same naturalness arguments also prefer rather
small supersymmetric contributions to the masses of the
Higgs bosons. In most (though not all [14]) versions of the
MSSM, this implies rather light Higgsinos, typically below
the stop. Since the mass splitting between the three
Higgsino-like mass eigenstates is small, they all behave
similarly if the LSP is Higgsino-like. In particular, in the
kind of RPV scenario we consider, all three states would
lead to very similar “fat jets”when produced in stop
decays; the recognition of such jets by exploiting recent
developments in computer vision is one of the central
points of our paper, which would apply equally to all three
Higgsino states. However, about half of all stop decays
would then produce a bottom, rather than a top, together
with a Higgsino, thereby complicating the analysis of the
*Contact author: bickendorf@th.physik.uni-bonn.de
†Contact author: drees@th.physik.uni-bonn.de
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
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PHYSICAL REVIEW D 110, 056006 (2024)
2470-0010=2024=110(5)=056006(15) 056006-1 Published by the American Physical Society

remainder of the final state. Moreover, Higgsinos, being
SUð2Þdoublets, have a sizable direct production rate. Their
nonobservation therefore leads to significant constraints on
parameter space, especially (but not only) if the bino has
mass comparable to or smaller than the Higgsinos [15–20].
In order to avoid such complications, we consider the
pair production of scalar top quarks, which decay to top
quarks plus two neutralinos with unit branching ratio. The
neutralinos in turn decay promptly via the UDD R-parity-
breaking term, which is fairly difficult to constrain [21].
Since each neutralino may form a (fat) jet, one can use
the substructure to differentiate it from background proc-
esses [22]. One may represent the jets as images made of
calorimeter cell hits that can be used by computer vision
techniques. Using a convolutional neural net (CNN) has
already been shown to work well on these images [23–34].
In recent years, computer vision techniques have
improved drastically with novel approaches such as the
vision transformer [35]. In standardized computer vision
tasks, these models have been shown to outperform CNN-
based models for large data sets. Fortunately, generating
large sets of simulated events is relatively cheap in particle
physics, which motivates the use of these new techniques.
Transformers have already been applied to classification in
particle physics scenarios [36–44], although these focus on
representing the jet as a set of particles, instead of as
an image.
In this article, we for the first time apply two modern
transformer-based computer vision techniques to find neu-
tralinos from scalar top quark decays and compare the results
to a classical CNN to see if the gain in performance translates
to detector images. Using gradient-boosted decision trees
(GBDTs), we combine the data from both neutralinos tagged
in this way and add further high-level features to construct
our final event classifier.
The remainder of this article is structured as follows. In
Sec. II we describe the specifics of the signal model we use.
In Sec. III we show how we generate the data sets to which
we apply the preprocessing outlined in Sec. IV. The novel
computer vision architectures we wish to adapt are
described in Sec. V. Section VI outlines the generation
of data sets from neutralino decay and background events,
which are used to train the neutralino taggers in Sec. VII.
The performance of these taggers is discussed in Sec. VIII.
In Sec. IX we combine information from both tagged
neutralinos into even more powerful classifiers. We also
demonstrate the power of combining the neutralino taggers
with other high-level features in Sec. X. The improved stop
mass reach is shown in Sec. XI, while Sec. XII contains a
brief summary and some conclusions.
II. SIGNAL MODEL
The MSSM contains two scalar top quarks, which are
mixtures of the SUð2Þdoublet ˜
tLand singlet ˜
tRweak gauge
eigenstates. We work with breaking parameters such that
the lighter mass eigenstate ˜
t1contains mainly the right-
handed top squark, which decays promptly into a top quark
and the bino-like neutralino ˜
χ0
1≡
˜
χ. We consider all other
scalar quarks to be decoupled. In order to avoid the
constraints from missing ET–based searches, we add
W
=
Rp¼1
2λ00
ijkUc
iDc
jDc
kð1Þ
to the superpotential, where UC
iand Dc
iare the up- and
down-type SUð2Þsinglet chiral quark superfields and i,j,k
are generation indices. Clearly, this term violates baryon
number conservation. In Eq. (1), antisymmetrization over
color (i.e., contraction with the totally antisymmetric tensor
in color space) is implied, and hence the coupling has to be
antisymmetric in the last two indices. Therefore, there
are in general nine independent coupling constants λ00
ijk.
When i¼3, this would allow the stop to decay directly to
two lighter quarks, which has already been extensively
studied [45–48]. A coupling with i≠3allows even a light
neutralino to decay into three quark jets via an off-shell
squark. The process we are interested in is shown in Fig. 1.
We also note that a mostly ˜
tReigenstate decaying into a
bino-like neutralino produces a predominantly right-
handed top quark. The same is true for a ˜
tLdecaying into
a neutral Higgsino. In contrast, a ˜
tLdecaying into a bino or
a˜
tRdecaying into a neutral Higgsino would produce a left-
handed top quark. Since we do not try to reconstruct the
polarization of the top (anti)quark in the final state, all four
reactions would have very similar signatures and could be
treated with the methods developed in this paper. However,
as already noted in the Introduction, a light neutral
Higgsino implies the existence of a nearly mass-degenerate
charged Higgsino (and a second neutral Higgsino), thereby
reducing the branching ratio for ˜
t→tþ˜
χdecays.
Moreover, by SUð2Þinvariance, a mostly ˜
tLstop eigenstate
would be close in mass to
˜
bL, leading to additional signals
from
˜
bLpair production. By focusing on a mostly ˜
tRlighter
stop and a bino-like LSP, we avoid these complications.
FIG. 1. Stop pair production with each stop decaying to a top
quark and a neutralino. The neutralinos decay via the RPV UDD
operator with nonzero λ00.
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III. DATA GENERATION AND PRESELECTION
For baseline selections, we follow roughly the CMS
search for this signal process [49]. We impose the following
preselection cuts:
(1) One muon with pT>30 GeV or electron with
pT>37 GeV and jηj<2.4.
(2) The lepton must be isolated within a cone radius
depending on the pTof the lepton like
R¼8
>
>
<
>
>
:
0.2;p
T<50 GeV;
10 GeV=pT;50 GeV <p
T<200 GeV;
0.05;p
T>200 GeV:
Together with the first cut, this isolation requirement
implies that in almost all events the lepton originates
from the semileptonic decay of one of the top
(anti)quarks in the final state. These two cuts satisfy
the requirements of the single lepton trigger.
Note that the events must contain exactly one
such isolated lepton; this largely removes Zþjets
backgrounds.
(3) We define “AK04 jets”via the anti-kT(AK) jet
clustering algorithm with distance parameter
R¼0.4, requiring pT>30 GeV and jηj<2.4for
each jet. We demand that the event contains at least
seven such AK04 jets, at least one of which is b
tagged. We note that our signal events contain at
least two b(anti)quarks from top decay. Moreover,
even if both tand ¯
tdecay semileptonically, signal
events contain eight energetic quarks even in the
absence of QCD radiation. They should therefore
pass this cut with high efficiency, except for very
light neutralinos where several of their decay prod-
ucts might end up in the same (quite narrow) AK04
jet. On the other hand, SM t¯
tevents with one top
decaying semileptonically contain only four hard
quarks. Hence, at least three additional jets would
have to be produced by QCD radiation, significantly
reducing the t¯
tbackground and reducing the
Wþjets background even more.
(4) HT>300 GeV, where HTis the scalar sum of the
transverse momenta of all AK04 jets. This cut is
mostly effective against W;Z þjets backgrounds.
(5) At least one combination of b-tagged jet and isolated
lepton must have an invariant mass between 50 and
250 GeV. Most events where the lepton and the b
quark originate from the decay of the same tquark
pass this cut, which helps to further reduce the
Wþjets background.
(6) At least one AK08 jet (defined with distance
parameter R¼0.8), with pT>100 GeV. We will
later try to tag these “fat jets”as coming from
neutralino decay. However, a boosted, hadronically
decaying top (anti)quark can also produce such a jet.
We will also consider even fatter jets. Since (nearly)
all particles inside an AK08 jet will end up inside
the same jet if R>0.8is used in the jet clustering,
while these fatter jets will contain additional
“nearby”particles, they will automatically also have
pT>100 GeV.
After these cuts, the remaining background is almost
exclusively due to top-quark pair production, as can be seen
in the original CMS publication [49]. In our simulation, we
therefore only consider this background process.
For the signal model, we set the masses of squarks
(except that of the stop), gluinos, and wino- and Higgsino-
like neutralinos to 5 TeV. We only set one RPV coupling
nonzero, λ00
223 ¼−λ00
232 ¼0.75; this leads to prompt neu-
tralino ˜
χ→csb decay even if the exchanged squark has a
mass of 5 TeV, τ˜
χ∼10−18 s·½m˜
χ=ð100 GeVÞ−5. We scan
over the stop mass from m˜
t¼700 GeV to m˜
t¼1200 GeV
in steps of 25 GeV. We also scan over the neutralino mass
from m˜
χ¼100 GeV to m˜
χ¼500 GeV in 10 GeV steps.
Background and signal events are simulated using
MadGraph5_aMC@NLO 3.2.0 [50]. The t¯
tbackground
is generated with between zero and three additional matrix
element partons, while the signal events contain up to two
additional partons. The NNPDF3.1 PDF set [51] is used.
We use
P
ythia 8.306 [52] for parton showering and hadro-
nization; background events are showered with the
CP5 tune, while signal events are showered with the
CP2 tune [53]. Events with different matrix element–level
final-state parton multiplicities are merged with the MLM
prescription [54], in order to avoid double counting
events where the parton shower produces additional jets.
Finally, detector effects are simulated with the CMS card of
D
elphes 3.5.0 [55,56].
IV. PREPROCESSING
The main novelty of this paper is the adoption of very
recent computer vision techniques to tag the hadronically
decaying neutralinos. To that end, we first have to translate
the simulated detector data to images.
The objects we are interested in are jets clustered with
the anti-kTjet algorithm as implemented by the
F
ast
J
et
package [57]. Choosing the optimal distance parameter
Rfor a given purpose can be somewhat nontrivial. A small
value of Rmeans that most particles inside a sufficiently
hard jet originated from the same parton, but some of the
energy of that parton might not be counted in this jet due to
final-state showering. On the other hand, a large Rlikely
leads to jets that capture all daughter particles while also
muddying the waters by including unrelated objects, e.g.,
from initial-state showering. One can use the fact that
the decay products of a resonance with a fixed mass m
and transverse momentum pTspread roughly like
ΔR¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Δϕ2þΔη2
p∝m=pTand the typical energy scale
of the process to arrive at a best guess for an optimal R
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parameter. This can be aided by the use of jet clustering
algorithms with variable R(e.g., [58,59]). In the case at
hand, this optimal value of Rwould depend on both the
stop and the neutralino mass. We therefore do not work
with a single fixed value of Rbut instead cluster each event
using several values of R, and ensemble the resulting jet
images to get a better per-event classification. Because we
consider rather large neutralino masses, m˜
χ≥100 GeV,
we consider AK08 (R¼0.8), AK10 (R¼1.0), and AK14
(R¼1.4) jets. This also allows us to keep the technique
general, i.e., to use the same algorithm over the entire
parameter space. Recall that the resulting fat jet has to
satisfy pT>100 GeV and jηj<2.4.
In order to get images out of the jets, we now consider
the calorimeter towers and tracks as jet constituents in the
ðη;ϕÞplane. As in the construction of top taggers [27],
we do not use the energy Eof the calorimeter towers
directly but rather opt for the transverse energy
ET¼E= cosh η. The relevant features are more readily
learned by the classifier if we normalize the coordinates.
First, we calculate the ETweighted center of the calo-
rimeter towers via
¯
η¼PiETiηi
PiETi
;ð2Þ
¯
ϕ¼PiETiϕi
PiETi
;ð3Þ
where the sums run over all constituents of a given fat jet.
We then shift the coordinates ηi→ηi−¯
ηand ϕi→ϕi−¯
ϕ
so that the image is centered on the origin. Next, we rotate
the coordinate system around the origin so that the
calorimeter tower with the highest ETpoints vertically
from the origin. We use the last degree of freedom to
make sure that the calorimeter tower with the second
highest ETlies in the right half of the coordinate by
flipping along the vertical axis if necessary.
Next, we pixelate the coordinates to a 0.04 ×0.04 grid.
The brightness/intensity of each pixel is given as the
measured ET. We use three channels, corresponding to
ETin the electromagnetic calorimeter (ECAL), hadron
calorimeter (HCAL), and pTof the tracks, analogous to
three color channels in classical images.1We divide each
pixel by the maximal value found in this image so that each
intensity is between 0 and 1. This makes learning more
efficient. It also removes information about the pTand
mass of the jet, which are powerful discriminators. We
partly remedy this by giving the classifier the mass of the
fat jet as another input; this will be explained in more detail
in a following section. We also note that we will later
introduce additional high-level features to our final clas-
sifier, which will reintroduce information about the overall
ETscale of the event.
In the last preprocessing step we crop the image to a
square centered around the origin with side lengths chosen
as 64 pixels for AK08 and AK10 jets and 128 pixels for
AK14 jets. This size is chosen to contain most of the
constituents, while also being a power of 2 which aids in
the application of the computer vision techniques. The
resulting images after all preprocessing steps, averaged
over the entire event sample, are shown in Fig. 2. These
average images look quite similar for signal and back-
ground, at least to the human eye; however, taking the
difference between the average images does reveal some
differences.
Moreover, there is more information available to the
computer vision techniques than can be displayed in the
figure. For instance, the number of nonzero pixel values is
useful for classification. On average the signal images
FIG. 2. Signal and background AK14 jet image averaged over the entire training data set. All three channels are aggregated by
summation. The rightmost plot shows the difference between the average signal and the average background jet image. Signal events are
more concentrated at the origin, while background jets are more spread out.
1In principle, the tracks have a much higher resolution
compared to the calorimeter towers and could thus be pixelated
into a finer grid. However, we do not expect these very fine details
to improve the discrimination between signal and background.
We therefore use the same grid spacing for all three channels.
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contain more nonzero pixels than the background images
do. Just cutting on this quantity allows, for one set of
parameters, to reach an accuracy of 74% when applied to a
sample containing an equal number of signal and back-
ground events. Of course, our final classifier should
perform much better than this.2
V. ARCHITECTURES
In this section, we describe how we process a single fat
jet, the goal being to distinguish jets due to the three-body
decay of neutralinos from SM background (“LSP tag-
ging”). At the core is one of three architectures adapted
from computer vision and described in more detail in the
following three subsections. In all three cases, the output of
this architecture is concatenated with the measured jet mass
and fed into the same multilayer perceptron classification
network. It is built from a dense layer with 256 neurons
followed by another dense layer with 128 neurons, which
connect to two output neurons. Between all three layers, the
ReLU activation function is used. The two neurons of the
last layer are passed into the softmax activation function,
such that the output can be interpreted as the predicted
probability of the image belonging to either the signal or
the background; since these probabilities should add to 1,
the values of the two output neurons are almost completely
correlated. In the following, we denote this as MLP Head.
All three architectures use convolution layers. Such a
layer convolutes cosets of learnable weights, the kernels,
with dimensions ci×kh×kwacross the height and width
of the image, where khand kware the height and width of
the weights and ciis the number of input channels. Recall
that we start with three input channels, akin to the three
colors commonly used in vision: ETmeasured in the
ECAL, ETmeasured in the HCAL, and pTof tracks.
The information of the three “colors”is therefore merged
already in the first convolution layer of a given architecture.
The results of each convolution are stacked such that the
output has cochannels. All architectures are built and
trained within the
P
y
T
orch [60] deep learning library.
A. CNN
The first architecture is a (comparatively) simple CNN.
We follow loosely an existing model used for top
tagging [23]. The first layers of the CNN are two blocks,
each containing a convolutional layer with 128 kernels of
size 4×4, stride 1, zero padding to keep the image
dimensions, the ReLU activation function, and average
pooling with kernel size 2 and stride 2. This halves the
spatial image dimensions. Next, we apply the same block
with only 64 kernels and without pooling. The last
convolution block contains again 64 kernels but this time
with the pooling operation. For AK08 and AK10 jet
images, this network produces outputs of shape 64 ×8×8;
here, 8×8refers to the size of one image after three
convolutions, and we use 64 different “filters”(i.e., sets of
weights in the convolution) for each image. In order to put
AK14 images of size 128 ×128 on the same footing, we
repeat the last block one more time. The output of shape
64 ×8×8is then flattened into 4096 features and fed into
the MLP head. In total, there are 1 350 850 learnable
parameters for AK08 and AK10 jets, while the model
for AK14 jets contains 1 416 450 learnable parameters.
The full architecture for AK08 and AK10 jets is shown
in Fig. 3. As already noted, this kind of architecture is
already being used for similar tasks; it serves as our
baseline, against which we compare the more advanced
architectures described in the following two subsections.
B.
C
o
A
t
N
et
Since their inception in the context of natural language
processing [61], transformer models have been shown to
also be applicable for computer vision tasks. Because the
attention mechanism inside the transformer is computa-
tionally expensive it is often impractical to apply it directly
to the entire input image. In order to counteract this, the
vision transformer (ViT) model [35] splits the input image
into manageable patches. They are then flattened, linearly
projected, equipped with a position embedding, and fed
into the transformer encoder. Since both translation equiv-
ariance and locality are explicitly broken in this approach
one cannot expect to outperform CNNs that are known to
leverage both of these features and therefore generalize well
onto the unseen test set. In our tests, the vanilla ViT indeed
performs poorly, so we do not pursue it further.
However, it is possible to construct a model that
combines the global receptive field of view from self-
attention with the aforementioned advantages of CNNs.
FIG. 3. Architecture of the CNN for AK08 and AK10 jets.
2We note in passing that this multiplicity does contain some
information on the hardness of the event, since it correlates
positively with both the mass and the transverse momentum of
the fat jet. Hence, the normalization step described above does
not completely remove the information on these quantities.
However, these dependencies are only logarithmic and subject
to large event-by-event fluctuations. Explicitly adding the jet
mass as an input variable can therefore still be expected to aid in
the classification task.
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This is the aim of
C
o
A
t
N
et [62] (the name derives from the
combination of depth-wise convolution and self-attention).
Let us now briefly describe how
C
o
A
t
N
et is built; we refer
the interested reader to the original publication for a more
detailed description.
The model is constructed in five stages. The first stage
consists of three convolution layers which 3×3kernels,
where the first has a stride of 2. This halves the spatial
resolution of the input image. This is followed by two
stages of three MBConv blocks [63], which are computa-
tionally cheaper while maintaining most of the performance
of full convolutional layers. In both stages, the first layers
perform downsampling again with a stride size of 2. By
now the width and height of the input image are shrunk by a
factor of 23so global attention is feasible even for the large
AK14 jet. Thus, the last two stages consist of five and two
transformer blocks, respectively. In each transformer, 2D
relative attention is used, which adds a learnable weight that
depends only on the relative 2D position. These steps were
performed using the publicly available code.3Finally, we
average pool the outputs and feed the 768 features into the
classification head. This architecture contains the most
parameters. The version for AK08 and AK10 jets are built
of 17 356 666 learnable parameters, while the model for
AK14 jets contains 17 364 346 learnable parameters.
C. Multi-axis vision transformer
The third architecture we will use is the multi-axis vision
transformer (
M
ax
V
i
T
)[64]. This approach circumvents the
shortcomings of the vanilla ViT differently. Instead of
splitting the image in patches and only applying attention to
these flattened local patches, the goal is to apply it locally
and globally consecutively by decomposing the image with
two strategies into smaller bits. First, the image is separated
into equal-sized, nonoverlapping blocks along the spatial
dimensions. Self-attention is applied within each block
using only the local information. In order to leverage global
information, the next strategy groups every pixel that is
reached by a step of fixed size into the object to which self-
attention is applied. (For a nice visualization, we refer
to Fig. 3 of [64].) This approach essentially uses high-
resolution local information before using low-resolution
global information. Both attention mechanisms are applied
after a MBConv block forming the so-called
M
ax
V
i
T
block.
For our application, we chose the publicly available
implementation that is shipped with
P
y
T
orch,4with the only
change being the replacement of the MLP head that takes
the 256 output features. Concretely, the first stage consists
of two convolutional layers with 64 3×3kernels each,
where the first has stride 2, reducing the spatial dimensions
by half. This is followed by three stages with two
M
ax
V
i
T
blocks each. The convolution is strided with size 2 for the
first block of each stage. The partitions are of size 4×4
each. The
M
ax
V
i
T
stages have 64 128 and 256 channels,
respectively. Self-attention uses 32-dimensional heads.
This model contains 6 014 810 learnable parameters for
all jet sizes.
VI. DATA SET CREATION
In this section, we describe how the data set used to train
the LSP taggers is defined. In most events, there is more
than one fat jet that passes the selection criteria. Since we
investigate pair production, not all of the information useful
for event classification can be expected to be contained in
the hardest jet. It is therefore expected to be useful to
combine information from more than one jet in the analysis.
Wide jets produced from the ˜
χdecays are expected to be
hard because of the large stop mass. Therefore, the two jets
with the largest pTare expected to be signal enriched. The
preselection requirements imply that one top quark from
stop decay will generally decay semileptonically. However,
the second top quark might decay fully hadronically,
resulting in a third wide jet with large pT. Since the top
quarks and the LSPs have very similar pTdistributions, the
third largest-pTjet may also well be from an LSP.
In order to design taggers that perform well on all three
leading jets, and hence for a wide range of pT, we therefore
include samples of all three leading fat jets in our training
data set in the ratios that the respective number of jets are
present in the full events. To this end, we add up to three fat
jets present in an event as images to the data set. Of course,
an event may also contain only one or two such jets; in fact,
this is generally the case for background events. We
generate as many events as required to reach the desired
size of the training set for each jet size.
VII. TRAINING THE LSP TAGGERS
We start by verifying that the different Pythia tunes that
we adopt do not significantly influence our results. To this
end, we train the CNN model described in Sec. VA to
differentiate not between signal and background samples,
but between background events generated with the CP2
tune and background samples generated with the CP5 tune.
We combine 1 000 000 jet images generated with each tune
into a data set and split it equally between training and test
sets for each jet size. The initial learning rate ηLis chosen
as 5×10−4. This value worked best in tests of the LSP
taggers. At the end of each epoch, the learning rate is
lowered by a factor of 0.7 and the entire training data set is
shuffled. The batch size is 64. We minimize the averaged
binary cross-entropy loss
l¼−1
NX
N
i
yiln xiþð1−yiÞlnð1−xiÞ;ð4Þ
3https://github.com/chinhsuanwu/coatnet-pytorch.
4https://github.com/pytorch/vision/blob/main/torchvision/
models/maxvit.py.
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where the index iruns through all N¼64 images in the
batch, yiis the true label, and xiis the predicted label.
Adam [65] is chosen as the optimizer. All taggers are
trained for a total of 15 epochs.
The minimum validation losses for AK08, AK10, and
AK14 jets are found to be 0.6917, 0.6918, and 0.6920,
respectively. When the classifier is tasked to assign the
label 0 to the first class (e.g., CP2 tune) and the label 1 to
the second class (CP5 tune) and the classifier is perfectly
confused (i.e., unable to distinguish between the classes), it
will assign labels close to 0.5 regardless of the true class.
The binary cross-entropy per image is then lnð2Þ¼0.6931.
Evidently, our observed losses are only very slightly below
the value expected for a classifier that learns nothing. We
therefore conclude that the difference in
P
ythia tunes can be
neglected in the following.
We now turn to the actual training of the taggers to select
LSP-like fat jets. Signal samples are generated for ˜
χmasses
between 100 and 500 GeV in 10 GeV steps, and for stop
masses between 700 and 1200 GeV in 25 GeV steps. For
each combination of stop and neutralino mass, we take
4750 sample images from ˜
t1
˜
t
1signal events. In order to
generate an almost pure signal sample for training, we only
include images of jets that are within ΔR<0.5of a parton
level ˜
χ. Since we want the LSP tagger to work for all
combinations of m˜
t1and m˜χ, we combine all 41 ×21 ×
4750 ¼4 089 750 images into a single training set. We take
the same number of background images, 4 089 750, from
t¯
tþjets events.
Finally, we split the 8 179 500 images into 5 725 650
images for training and 2 453 850 images for validation.
After training the model state at the epoch with the lowest
validation loss is selected to define the tagger.
VIII. RESULTS FOR NEUTRALINO TAGGERS
In order to compare the performance of our classifiers,
we neglect any systematic uncertainties and define the
signal significance Zas
Z¼S
ffiffiffiffi
B
p¼ϵS
ffiffiffiffiffi
ϵB
p·σS
ffiffiffiffiffi
σB
pffiffiffiffiffiffiffiffi
Lint
p;ð5Þ
where Sand Bare the number of signal and background
samples passing a cut (e.g., on the value of an output
neuron of the MLP), ϵS=B is the selection efficiency of this
cut, σS=B is the fiducial production cross section, and Lint is
the integrated luminosity of the data set considered. Instead
of comparing signal significances directly, we compare the
significance improvement ϵS=ffiffiffiffiffi
ϵB
p, which captures the gain
due to the sophisticated event classifiers and is independent
of the assumed luminosity.
Figure 4shows the performance of all neutralino taggers
on the entire test data set, i.e., with all signal masses present
and with the three leading jets mixed, as mentioned in
Sec. VII. As a working point for the following analysis,
we choose the cut on the MLP output neuron such that
ϵS¼0.3. Even lower values of ϵScan still increase ϵS=ϵB,
but the significance improvement is already close to the
maximum at the chosen point. Moreover, for smaller ϵS
the background efficiency ϵBbecomes so small that the
statistical uncertainty on the accepted background becomes
sizable, in spite of the large number of generated back-
ground events.
Both
C
o
A
t
N
et and
M
ax
V
i
T
show superior performance
in classical image classification tasks compared to CNN-
based models, as was reported in the respective original
publications. We expect this to carry over to jet classi-
fication. Indeed, this is the case here and both models
outperform the classical CNN by up to a factor of 2 for
AK14 jets. The most performant classifiers are the trans-
former-based models trained on the large-radius jets. These
large jets still contain the entire narrow jets from small LSP
masses, while the small jets might miss important features
for larger neutralino masses. We also observe that
M
ax
V
i
T
performs slightly better than
C
o
A
t
N
et, as is the case in the
original
M
ax
V
i
T
publication. Evidently, improvements in
modern computer vision translate well to the classification
of jet images. Even the worst transformer-based model (i.e.,
AK08
C
o
A
t
N
et) matches the best CNN. Interestingly, despite
the transformer models showing a clear hierarchy (the
larger a jet is, the better), this is not the case for the CNN,
which performs best for AK08 jets.
So far we have considered classification of singlet jets. In
the next section, we will show how this can be used for
event classification.
IX. BOOSTED CLASSIFIERS
As previously mentioned, our signal model always
produces two neutralinos that subsequently decay into
three quarks (plus possible gluons from final-state
FIG. 4. Significance improvement curves for all three neutra-
lino taggers for all single jet samples in the test data set. The
shaded regions show one bootstrapped standard deviation.
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radiation). It is therefore instructive to combine multiple jet
images into our predictions. To this end, we apply one of
our LSP taggers described above to the three leading fat jets
in an event; from now on, we drop the merging requirement
since it is not meaningful anymore. The three resulting
MLP outputs are used as inputs for a GBDT classifier. If an
event contains less than three fat jets with pT>100 GeV
we assign the label −1for the missing jets. The GBDT is
implemented using the
XGB
oost [66] package. We use 120
trees with a learning rate of 0.1 with other hyperparameters
left unchanged from the default values. In order to train the
GBDT and calculate its results, we use 3000 and 2500
events, respectively, for each combination of stop and LSP
masses. This corresponds to a total of 2 583 000 signal
events for training and 2 152 500 signal events for calcu-
lating results. We again generate an equal number of
t¯
tþjets background events.
Figure 5shows the significance improvement after a cut
on the signal probability given by the GBDT. The differ-
ence in performance between the two transformer-based
models has shrunk significantly for all jet sizes, especially
for AK10 and AK08 jets. Comparing this with Fig. 4, the
gain by combining the three jets is not very large. One has
to keep in mind that the merging requirement is now
dropped. If we calculate the significance improvement for
only the jet with the highest pTwithout requiring it to be
close to a (truth-level) LSP,
M
ax
V
i
T
reaches 6.79 0.05 at
ϵS¼0.3. Comparing this with 9.92 0.12 for the same
base model after combining the LSP tagger output for the
three hardest jets shows an improvement of almost 50%,
equivalent to doubling the integrated luminosity in Eq. (5).
The CNN now also works best with AK14 jets, even though
the AK08 version is still better than the AK10 version,
contrary to the hierarchy of the other models.
Overall, the level of improvement between the results of
Fig. 5, which use information from up to three jets per
event, and Fig. 4for single jets, might seem somewhat
disappointing. After all, in the absence of QCD radiation
a˜
t1
˜
t
1signal event contains two signal jets plus one fat
background jet from the hadronically decaying top quark,
whereas a generic t¯
tevent with one top quark decaying
semileptonically contains only a single background fat jet.
In such a situation simply requiring at least one fat jet to be
tagged as signal would increase the signal efficiency (for
ϵS≫ϵB) from ϵSto 1−ð1−ϵSÞ2, while the background
efficiency remains unchanged. However, recall that we
require each event to contain at least seven AK04 jets. This
greatly reduces the t¯
tbackground since at least three
additional partons need to be emitted for the event to pass
this cut; on the other hand, it also means that background
events frequently contain several fat jets, in which case
a simple single tag requirement would not increase the
significance. In any case, as noted above, there is a
significant improvement in performance when information
of the three leading fat jets is combined using a GBDT;
of course, the GBDT output is not equivalent to simply
demanding a fixed number of jets in a given event being
tagged as LSP-like.
In Fig. 6we show how the performance of the GBDT
depends on the LSP mass. For small masses, the two
transformer-based models perform comparably for all three
jet sizes. Here the decay products are usually contained
even in the AK08 jet so all three jet sizes contain the
necessary information for our task. Evidently, the trans-
former networks are able to filter out the noise from
particles not related to LSP decay that are present in the
AK10 and AK14 jets, while the simpler CNN cannot;
hence, the GBDT using the CNN applied to AK10 or AK14
jets performs relatively poorly for a small LSP mass. On the
other hand, for an LSP mass above 200 GeV the GBDT
performs significantly worse when used on the smaller jets,
FIG. 5. Significance improvement curves for all GBDT clas-
sifiers built to combine the LSP tagger outputs for the three
highest-pTjets. The shaded regions are one bootstrapped
standard deviation.
FIG. 6. Significance improvements depending on the LSP mass
for all GBDT classifiers built to combine the LSP tagger outputs
for the three highest-pTjets. The cut on the GBDT output has
been set such that ϵS¼0.3for each given LSP mass. The shaded
regions are one bootstrapped standard deviation.
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which no longer contain all particles originating from
LSP decay.
We also note that using the CNN applied to AK14 jets
performs far worse than the other models for small LSP
mass, but matches the performance of the
C
o
A
t
N
et-based
model for m˜χbetween 450 and 500 GeV. This curve also
shows the strongest LSP mass dependence. We will revisit
this point later in this section.
Finally, while the
M
ax
V
i
T
architecture with AK10 and
AK14 jets again shows the best overall performance, the
resulting ϵS=ffiffiffiffiffi
ϵB
pshows a shallow minimum at m˜χ≃mt.
For a given pT, fat jets originating from LSP and top decay
will then have similar overall features, and the additional
information about the jet mass will not help at all; more-
over, recall that in our scenario the LSP decay products
contain exactly one bquark, just like nearly all jets from top
decay. Nevertheless, the model performs quite well even in
this difficult mass region. Presumably, it exploits the
fact that top decays into three quarks proceed via two
two-body decays with a color-singlet on-shell Wboson in
the intermediate state, whereas the LSP decays via the
exchange of a (far) off-shell squark.
At this point, we still have nine predictions for each event
(the output of three architectures applied to AK08, AK10,
and AK14 jets). Of course, these nine numbers are highly
correlated. Nevertheless, a further improvement of the
performance might be possible by either combining results
from different jet definitions within a given architecture or
vice versa. Comparing these results might also allow us to
infer in which aspect a single model has room for improve-
ments that might be gained by another architecture.
We start by combining LSP tagger outputs for different
jet sizes. We show the results in Table Iand compare the
performance to that of the best single jet definition, which is
achieved for AK14 jets, as we saw in Fig. 5. Evidently, the
improvement is barely statistically significant for the two
transformer-based models. These models extract most of
the useful information from the images of the large AK14
jets, even when there is a lot of clutter present. The
improvement is larger for the CNN-based classifier, which,
however, still performs somewhat worse than the other
models. It seems to benefit from the multiple jet definitions
intended to extract high-level features such as the mass in
classical applications. In particular, the combination allows
to compensate the degraded performance when using the
large jets for an LSP mass below 250 GeV by information
from the AK08 jets, which is more useful in this parameter
region, as we saw in Fig. 6.
Next, we combine the outputs of different LSP taggers
into a single GBDT, for fixed jet definition. The results
are shown in Table II. This time the combination leads to a
slight but significant improvement over the best single
model (the one based on
M
ax
V
i
T
). This shows that, even
though the transformer-based models perform almost
equally well for the AK14 jets while the CNN is noticeably
weaker, each model misses complementary information
that the GBDT can combine into a stronger classifier.
In Fig. 7we show how the performance of various
strategies to combine LSP taggers varies with the neutralino
TABLE I. Significance improvement, ϵS=ffiffiffiffiffi
ϵB
p, for ϵS¼0.3
when only using AK14 jets (second column) and when combin-
ing the LSP tagger outputs on AK08, AK10, and AK14 jets using
a larger GBDT (third column). The uncertainties are bootstrapped
standard deviations.
Model AK14 Combined jets
C
o
A
t
N
et 9.32 0.10 9.63 0.10
M
ax
V
i
T
9.91 0.11 10.09 0.12
CNN 7.62 0.06 9.16 0.09
TABLE II. Significance improvement with ϵS¼0.3when
feeding the outputs of all three LSP taggers simultaneously to
the GBDT, keeping the jet definition fixed. For comparison, the
third column shows the significance improvement for the
M
ax
V
i
T
-
based model, which performs best for all three jet sizes. The
uncertainties are bootstrapped standard deviations.
Jet Combined Best single model
AK08 7.53 0.06 7.34 0.06 (
M
ax
V
i
T
)
AK10 8.94 0.09 8.15 0.07 (
M
ax
V
i
T
)
AK14 10.51 0.11 9.91 0.11 (
M
ax
V
i
T
)
FIG. 7. Significance improvement as a function of the LSP
mass for GBDT classifiers built by combining the output of
different LSP taggers, with ϵS¼0.3in each case. The shaded
regions are one bootstrapped standard deviation. The curves
labeled
C
o
A
t
N
et (blue), CNN (orange), and
M
ax
V
i
T
(red) show the
performance of GBDTs built from combining the jet sizes for the
given model, as in the third column of Table I. The green curve is
for the GBDT that uses the outputs of all LSP taggers, but only
for the AK14 jets, as in the third row of Table II. The purple line
results from combining both transformer-based LSP taggers for
all jet sizes, while the brown line is for a GBDT that combines all
LSP taggers and all jet sizes.
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mass. Combining all transformer-based predictions into a
single GBDT does not show any significant improvement
over the performance of the
M
ax
V
i
T
-based tagger. This
indicates that these models use the same features of the jet
images and do not find complementary information. The
combination of all CNN predictions is comparable to the
weaker transformer-based model,
C
o
A
t
N
et, for an LSP mass
above 200 GeV, while
M
ax
V
i
T
is still more sensitive for all
LSP masses.
Because our LSP taggers generally perform best on
AK14 jets, we also show the combination of all three
architectures using only AK14 jets, as in the last row of
Table II. Comparing to Fig. 6, we see that for an LSP mass
below ∼160 GeV this combination does not further
improve on the
M
ax
V
i
T
-based model applied to AK14 jets.
Between ∼160 GeV and ∼300 GeV, the performance
closely follows that of the two combined transformer
models shown in purple. Since we already showed that
one does not gain much by combining the
C
o
A
t
N
et and
M
ax
V
i
T
models, this shows that the CNN does not yield
useful information in this region of parameter space either.
However, as we saw in Fig. 6, the CNN-based model
applied to AK14 jets improves more with increasing LSP
mass than the transformer-based models do, even matching
C
o
A
t
N
et at 500 GeV. The combination profits from this fact
and outperforms above 300 GeV the GBDTs using only
input from the transformer-based LSP taggers. This shows
that the CNN learns something about the sample that the
other models miss.
Finally, we show the result of a GBDT that is trained on
the LSP tagger outputs of all three models and all three jet
sizes, and thus has 27 inputs in total for each event.
Compared to the AK14-only case, this does benefit
from the inclusion of smaller jets, in particular at smaller
LSP masses where the AK08 and AK10 jets already
capture most LSP decay products. For larger LSP masses,
the performance is only slightly better than that of the
AK14-only case.
These various comparisons show that for the given signal
process, the largest improvement in significance ϵS=ffiffiffiffiffi
ϵB
pis
achieved by the transformer-based models applied to AK14
jets. Both models capture details of the jet images that the
CNN misses. Nevertheless, also feeding the output of the
CNN-based LSP tagger into a larger GBDT leads to a
further slight improvement of the performance. This
indicates that one might be able to find new architectures
that perform even better than
M
ax
V
i
T
.
X. ADDING HIGH-LEVEL FEATURES
The cuts discussed in Sec. III are only preselections.
They ensure that the event passes the single lepton trigger
and contains at least one fat jet to which the LSP tagger
can be applied. They also reduce the background, but
even after including information from the LSP tagger
these cuts are not likely to yield the optimal distinction
between signal and background. A full event has addi-
tional features that allow to define additional, potentially
useful cuts, even if they may show some correlation with
the output of the LSP tagger.
In particular, so far the only dimensionful quantities we
used in the construction of our classifier are the masses of
the hardest three fat jets, which we use as input of the LSP
tagger. Therefore, we now introduce as additional input
variables for the final GBDT the sum of the masses of all
AK14 jets [67],
MJ¼X
AK14
m; ð6Þ
and the total missing transverse momentum pmiss
T.In
addition, we use the total number Njof all AK04 jets as
well as the scalar sum HTof their transverse momenta.
Moreover, information about the angular separation of
the jets might be helpful. Inspired by Ref. [49], we capture
this information via the Fox-Wolfram moments [68] Hl,
defined by
Hl¼X
i;j¼1
pTipTj
ðPkpTkÞ2Plðcos ΩijÞ;ð7Þ
where i,j,krun over all AK04 jets in the event, pTi is the
pTof the ith jet, Plis the Legendre polynomial, and
cos Ωij ¼cos θicos θjþsin θisin θjcosðϕi−ϕjÞð8Þ
is the cosine of the opening angle between the jets iand j.
We combine these features into two sets. The first is the
small set DS1¼½pmiss
T;H
T;M
J;N
j, which includes the
most commonly used features for new physics searches in
hadronic final states. In addition, we also consider a slightly
larger set DS2, which also includes the second to sixth
Fox-Wolfram moments. We combine these features with
the output of the LSP tagger based on
M
ax
V
i
T
applied to
AK14 jets (i.e., the most performant single model) and
derive predictions with a similar GBDT as before.
Results are shown in Fig. 8.WeseethatevenGBDT
classifiers that only use the kinematic information of sets
DS1 or DS2 are quite capable of separating signal from
background, especially for larger LSP masses; this recon-
firms the usefulness of these variables for new physics
searches at the LHC. In fact, for an LSP mass above
300 GeV these classifiers even outperform the GBDT that
only uses information from the
M
ax
V
i
T
-based LSP tagger. On
the other hand, except for m˜χ¼100 GeV, adding kinematic
information to the output of the LSP tagger clearly improves
the performance of the event classifier; the Fox-Wolfram
moments prove useful for an LSP mass above 250 GeVor so.
Conversely, adding information from the LSP tagger to
the purely kinematic variables raises the significance
improvement by an amount that is nearly independent of
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the LSP mass. We expect the gain of performance to be
even larger when compared to a classical selection based
purely on kinematical cuts.
XI. APPLICATION AT 137 fb−1
We are now ready to discuss how the different classifiers
fare in terms of the reach in stop mass for exclusion or
discovery. Here we set the integrated luminosity to
Lint ¼137 fb−1, as in the original CMS publication [49].
For simplicity, we ignore the systematic uncertainty on the
signal as well as the uncertainty from the finite size of our
Monte Carlo samples. The former is much less important
than the systematic error on the background estimate, and
the latter should be much smaller than the statistical
uncertainty due to the finite integrated luminosity. The t¯
t
background is normalized to the next-to-leading-order
(NLO) production cross section [50]. The simulated stop
pair samples are normalized to NLO + next-to-leading-
logarithm accuracy [69]. This corresponds to 273 084
background events and a stop mass–dependent number
of signal events. We calculate exclusion limits from the
expected exclusion significance [70]:
Zexcl ¼2S−2BlnBþSþx
2B−2B2
Δ2
B
lnB−Sþx
2B
−ðBþS−xÞBþΔ2
B
Δ2
B1=2
;ð9Þ
where Band Sare the expected number of background and
signal events, ΔBis the absolute systematic uncertainty on
the background, and
x¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðSþBÞ2−4SBΔ2
B
BþΔ2
B
s:ð10Þ
We chose ΔB¼0.06B, as described below. Zexcl is the
expected number of standard deviations with which the
predicted signal Scan be excluded if the background-only
hypothesis, described by the background B, is correct; note
that Zexcl →S= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
BþΔ2
B
pif B≫S. This quantity is com-
puted for every combination of stop and LSP masses
introduced in Sec. III, using four different event classifiers.
The results are shown in Fig. 9. We again define signal-
like events through a cut on the GBDT output correspond-
ing to ϵS¼0.3. The top-left frame is for a GBDT that
uses only kinematic information about the AK04 jets, as in
the green curve of Fig. 8. Associating the contour along
Zexcl ¼1.645 with the 95% confidence level exclusion
bounds of this “traditional”analysis, we find an expected
exclusion reach in m˜
t1of about 740 GeV for an LSP mass
of 100 GeV. This is rather close to the expected reach
of about 710 GeV for the same LSP mass achieved in
the CMS search,5which is based on a neural network
“trained to recognize differences in the spatial distribution
of jets and decay kinematic distributions”[49].
Unfortunately, they do not show results for other LSP
masses. This agreement is not accidental; we chose the
systematic background uncertainty, ΔB¼0.06B, accord-
ingly. Presumably, even closer agreement would have been
possible for somewhat larger ΔB. However, it would then be
significantly larger than the actual systematic error on the
background estimate found by CMS, which is below 5%.
We note that for Δ2
B≫Bthe significance scales ∝1=B,
rather than ∝1=ffiffiffiffi
B
p,ifΔBis a fixed percentage of B.
A larger ΔBtherefore increases the relative improvement in
reach achieved by including information from one of our
LSP taggers; recall that this leads to a significant improve-
ment of ϵS=ffiffiffiffiffi
ϵB
p, and hence to an even bigger improvement
in ϵS=ϵB.
The other three frames show results for GBDTs that
also use the outputs of an LSP tagger as input variable; we
apply this tagger to the three leading AK14 jets. We see
that the simpler CNN-based tagger (top right) increases
the reach in stop mass only by less than 10 GeV. Recall
from Fig. 6that the CNN tagger applied on AK14 jets
does not perform well for a small LSP mass. For a larger
LSP mass, and hence larger angular spread of the LSP
decay products, the kinematic information on the AK04
FIG. 8. Significance improvements as functions of the LSP
mass for various GBDT classifiers. In all cases, the cut on the
GBDT output has been set such that ϵS¼0.3for each given LSP
mass. The upper two curves show results from classifiers that
combine the output of the
M
ax
V
i
T
-based LSP tagger applied to
AK14 jets with additional kinematical features. The feature set
DS1 contains ½pmiss
T;H
T;M
J;N
j, while DS2 contains, in addi-
tion, the second to sixth Fox-Wolfram moments. For comparison,
the red curve results when using only LSP tagger information, as
in Fig. 6, while the lower blue and green curves are for GBDTs
that only use kinematical information. The shaded regions are one
bootstrapped standard deviation.
5We note in passing that the actual CMS limit on the stop mass
is only 670 GeV for this LSP mass, due to a small (not
statistically significant) excess of events.
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jets, many of which are components of AK14 jets, already
seems to capture much of the physics found by the CNN.
Recall that the kinematic GBDT includes information on
the angular separation of these jets via the Fox-Wolfram
moments of Eq. (7).
In contrast, using the transformer-based LSP taggers does
improve the reach considerably. As before,
M
ax
V
i
T
(bottom
right) performs slightly better than
C
o
A
t
N
et (bottom left); the
reach in stop mass increases by 100 GeV for m˜
χ¼100 GeV
andbyabout60GeVform˜
χ¼500 GeV. This again
indicates that the kinematic information on the AK04 jets
allows some effective LSP tagging for large LSP masses.
For stop masses in the interesting range, the ˜
t1
˜
t
1
production cross section of [69] can be roughly para-
metrized as
σðpp →
˜
t1
˜
t
1Þ≃0.08 pb · m˜
t1
700 GeV−7.8
:ð11Þ
Increasing the reach from 740 to 840 GeV (for
m˜
χ¼100 GeV) thus corresponds to reducing the bound
on the stop pair-production cross section by a factor of ∼2.7.
Note that the limit-setting procedure is quite nonlinear
because the background falls by nearly 2 orders of
FIG. 9. Exclusion significance Zexcl defined in Eq. (9) for an integrated luminosity of 137 fb−1. In all cases, the cut on the GBDT
output has been chosen such that the signal efficiency ϵS¼0.3and ΔB¼0.06B. The top-left frame is for a GBDT using kinematical
information only, corresponding to the green curve in Fig. 8. The other three frames are for GBDTs that also use the output of LSP
taggers applied to AK14 jets, based on the CNN (top right),
C
o
A
t
N
et (bottom left), and
M
ax
V
i
T
(bottom right). Solid, dashed, and dotted
lines denote contour lines corresponding to a signal significance of 1, 1.281, and 1.645, respectively. These are smoothed by a Gaussian
filter with standard deviations 10 and 25 GeV on the neutralino-mass and stop-mass axis, respectively, applied to the logarithm of the
signal significances.
GERRIT BICKENDORF and MANUEL DREES PHYS. REV. D 110, 056006 (2024)
056006-12

magnitude when m˜
t1is increased from 700 to 1200 GeV,
while keeping ϵS¼0.3fixed.
XII. CONCLUSION
The large hadronic activity in pp collisions makes the
search for physics beyond the SM in purely hadronic
processes at the LHC especially challenging. This problem
can be mitigated by the use of sophisticated analysis
methods. In particular, jet substructure has proved a power-
ful discriminator between various production processes.
In this article, we studied the feasibility of applying
modern computer vision techniques in detecting RPV stop
decays. As a benchmark, we used ˜
t1pair production, where
each stop decays to a top and a neutralino LSP, which
subsequently decays via the UDD operator to three quarks.
For not too small mass splitting between the stop and the
LSP, the decay products of the latter tend to reside in a single
fat (e.g., AK14) jet. One can build images from the
constituents of such jets by using the angle ϕand pseudor-
apidity ηas spatial positions and deposited energy into the
detector as pixel intensity. One can then use computer vision
techniques on this representation to build classifiers (“LSP
taggers”) that aid in amplifying the signal process.
In recent years, transformer-based architectures have
been shown to trump the performance of more classical
convolutional neural network–based structures in standard
classification tasks. We studied how well these novel
architectures work on jet images by training LSP taggers
based on
M
ax
V
i
T
,
C
o
A
t
N
et, and a CNN architecture. The
training was done on single jet images. The combined
training time of all nine models was less than 1 month on a
consumer-grade RTX 3060 Ti GPU. Therefore, the results
can be obtained without specialized computing resources.
We then combined the output of the LSP tagger applied to
the three jets with the highest pTusing a gradient-boosted
decision tree into a more robust classification score. We
found that the CNN-based tagger improves the statistical
significance of the signal by a factor between 5 and 10
for fixed signal efficiency ϵS¼0.3, with the exact factor
depending on the neutralino mass and the definition of the
fat jets. In contrast, the transformer-based models lead to an
improvement factor between 8 and 11, outperforming the
CNN over the entire parameter space.
In addition, we showed a method to quantify the
systematic uncertainty implied by the different Pythia tunes
between background and signal events. For this, we trained
a CNN to tell background events generated with the CP2
tune apart from background events with the CP5 tune. We
found no significant difference between both tunes and
concluded that the systematic uncertainty can be neglected.
Although the modern models are highly complex and
therefore hard to interpret, one might speculate that the
reason for the large improvement lies in the properties of
the jet images. Due to the large neutralino masses, the
decay products are spread far apart. An efficient classifier
needs to leverage the correlation of all decay products,
which is therefore a global feature. This is seen by the
transformer-based models earlier due to the global recep-
tive field of view.
We also combined the predictions of all architectures for
each jet size separately and found a modest improvement,
hinting that even the transformer-based models do not use
the entire information present in the images; hence, an
investigation of further improvements of the architecture
might be worthwhile.
Since the kinematic preselection cuts are not optimized
for sensitivity, we also used high-level features such as Fox-
Wolfram-moments, pmiss
T,HT,MJ, and Njas inputs to a
GBDT, in combination with the output of one of our LSP
taggers. This leads to a total gain of sensitivity by a factor
of 20 for 500 GeV LSPs, on top of the effect due to the
acceptance cuts.
Finally, we estimated the reach in stop and LSP mass
that could be expected from the full Run-2 data set. We
chose the systematic uncertainty on the background
such that a GBDT that only uses kinematic information
on AK04 jets leads to a reach (for an LSP mass of
100 GeV) similar to that found by CMS [49].Usingin
addition the output of the relatively simple CNN-based
LSP tagger then leads to almost no further improvement
of the reach. By instead using the
M
ax
V
i
T
-based tagger,
one can improve the reach by 100 GeV (60 GeV) for
neutralino masses of 100 GeV (500 GeV), under the
assumption that the relative size of the systematic
uncertainty remains the same. This corresponds to a
reduction of the bound on the stop pair-production cross
sectionbyuptoafactorof2.7.
We conclude that LSP taggers built on modern trans-
former-based neural networks hold great promise in
searches for supersymmetry with a neutralino LSP where
Rparity is broken by the UDD operator. This result can
presumably be generalized to models with different LSPs,
e.g., a gluino decaying via the same operator or a slepton
decaying into a lepton and three jets via the exchange of a
virtual neutralino.
In fact, it seems likely that these advanced techniques
can also be used to build improved taggers for boosted,
hadronically decaying top quarks or weak gauge or Higgs
bosons. We did not attempt to construct such taggers
ourselves since this field is already quite mature.
Convincing progress would therefore have to be based
on fully realistic detector-level simulations, for which we
lack the computational resources. Moreover, a careful
treatment of systematic uncertainties would be required,
which ideally uses real data. However, we see no reason
why the improvement relative to CNN-based taggers that
we saw in our relatively simple simulations should not
carry over to fully realistic ones.
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056006-13

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