††institutetext: 1 Centre for High Energy Physics, Indian Institute of
Science, Bengaluru 560012, India††institutetext: 2 Theoretical Physics
Division, Physical Research Laboratory, Shree Pannalal Patel Marg, Ahmedabad,
380009, Gujarat, India

# LLPNet: Graph Autoencoder for Triggering Light Long-Lived Particles at HL-
LHC

Biplob Bhattacherjee1, Partha Konar2, Vishal Singh Ngairangbam2, Prabhat
Solanki1<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In the search for exotic events involving displaced particles at HL-LHC, the
triggering at the level-1 (L1) system will pose a significant challenge. This
is particularly relevant in scenarios where low mass long-lived particles
(LLPs) are coupled to a Standard Model (SM)-like 125 GeV Higgs boson and they
decay into jets. The complexity arises from the low hadronic activity
resulting from LLP decay, and the existing triggers’ inability to efficiently
select displaced events. This study introduces a novel machine learning
approach to address this challenge, utilizing a lightweight autoencoder
architecture designed for low latency requirements at L1. Focusing on light
LLPs with decay lengths ranging from 1 to 100 cm, this approach employs “Edge
convolution” on L1 reconstructed tracks. The results show notable signal
acceptance at the permissible background rate, primarily originating from
minimum bias and QCD di-jet events. For LLPs of mass 10, 30, and 50 GeV at
decay length of 5 cm, the signal efficiencies are 33%, 70%, and 80%,
respectively. At a 50 cm decay length, these efficiencies are 20%, 39%, and
45% for the same respective masses.

###### Keywords:

Point clouds, Anomaly detection, Autoencoder, Triggers, Long-lived particles

##  1 Introduction

The Large Hadron Collider (LHC) is yet to discover any concrete signal of new
physics, with most measurements of the recently discovered Higgs boson
pointing towards its agreement with the Standard Model. With other
observations pointing toward the need for beyond-(the)-Standard-Model (BSM)
physics, it is important to be thorough while looking for possible deviations
from known physics at the LHC. One such signature traditionally overlooked is
displaced signature. Recently, there has been a growing interest in
investigating these signatures from both a phenomenological and experimental
perspective. On the phenomenological side, various studies have investigated
these signatures Ilten:2015hya ; Gago:2015vma ; Banerjee:2017hmw ;
KumarBarman:2018hla ; Bhattacherjee:2019fpt ; CidVidal:2019urm ;
Banerjee:2019ktv ; Jones-Perez:2019plk ; Bhattacherjee:2020nno ;
Bhattacherjee:2021qaa ; Adhikary:2022goh ; Gershtein:2020mwi ; Fuchs:2020cmm ;
Cheung:2020ndx ; Liu:2020vur ; Evans:2020aqs ; Linthorne:2021oiz ;
Alimena:2021mdu ; Bhattacherjee:2021rml ; Sakurai:2021ipp ; Du:2021cmt ;
Bhattacherjee:2023plj ; Bhattacherjee:2023kxw . Similarly, on the experimental
side, LHC experiment has also focused on investigating these signatures
CMS:2014hka ; CMS-PAS-EXO-16-022 ; CMS-PAS-EXO-16-036 ; CMS:2017kku ;
CMS:2018lab ; CMS:2018bvr ; CMS-PAS-FTR-18-002 ; CMS-PAS-EXO-19-013 ;
CMS:2019qjk ; CMS:2019zxa ; CMS:2020atg ; CMS:2020iwv ; CMS:2021juv ;
CMS:2021kdm ; CMS:2021yhb ; ATLAS:2015xit ; ATLAS:2015wsk ; ATLAS:2015itk ;
ATLAS:2017tny ; ATLAS:2018rjc ; ATLAS:2018lob ; ATLAS:2018niw ; ATLAS:2018tup
; ATLAS:2019qrr ; ATLAS:2019kpx ; ATLAS:2019fwx ; ATLAS:2019tkk ;
ATLAS:2019jcm ; ATL-PHYS-PUB-2019-002 ; ATLAS:2020xyo ; ATLAS-CONF-2021-015 ;
ATLAS-CONF-2021-032 ; ATLAS:2020wjh ; ATLAS:2021jig ; ATLAS:2023oti ;
LHCb:2016buh ; LHCb:2016inz ; LHCb:2016awg ; LHCb:2017xxn ; LHCb:2019vmc ;
LHCb:2020akw ; Feng:2017uoz ; Curtin:2018mvb ; Alpigiani:2020iam ;
Gligorov:2017nwh ; Aielli:2019ivi . While much of the investigation into
displaced signatures at the LHC has been conducted using detectors like CMS,
ATLAS, and LHCb, these studies have primarily relied on level-1 (L1) triggers
originally designed for prompt signatures. Nevertheless, there are exceptions,
such as specific studies conducted by ATLAS, like the one mentioned in
ATLAS:2019qrr , which employed a dedicated trigger known as CalRatio
ATLAS:2013bsk to identify events involving long-lived particles (LLPs)
decaying into jets, either within the hadronic calorimeter (HCAL) or near the
outer radius of the electromagnetic calorimeter (ECAL). In addition, some
phenomenological studies have proposed specialized triggers for displaced
jets, considering the unique characteristics of these jets, including their
trackless-ness nature and their substantial time delay within the ECAL
Bhattacherjee:2020nno ; Bhattacherjee:2021qaa . In this paper, we explore the
possibility of utilizing lightweight anomaly detection methods for point
clouds to design dedicated triggers to select LLP events at the L1 trigger
system in future runs of the LHC.

Triggering events with displaced signatures will pose a major challenge at the
L1 trigger system at the high luminosity LHC (HL-LHC) Apollinari:2017lan where
instantaneous luminosity will reach $\mathrm{5\times 10^{34}cm^{-2}s^{-1}}$
with average number of pile-up (PU) interactions reaching up to 140. With $pp$
collision happening every 25 ns, event rate will be as high as 40 MHz which
need to be brought down by a factor of $\mathcal{O}(10^{3})$ at the L1 stage
of the trigger system using triggers dedicated for the displaced physics
searches. To deal with this challenge, the data acquisition process at L1 of
the trigger system will undergo a significant overhaul through the upgrade of
the inner and outer tracker and the implementation of Field Programmable Gate
Arrays (FPGA) CERN-LHCC-2020-004 . This upgrade will allow tracking
information to be available at L1. Consequently, the improved architecture for
data acquisition and processing at L1 will facilitate the utilization of
machine learning techniques, as well as higher-level object reconstruction,
within the trigger system. This advancement will prove invaluable in capturing
rare BSM events, particularly those involving displaced objects that would
otherwise have been missed. Furthermore, by implementing extended tracking at
L1, it will become possible to reconstruct displaced tracks up to a specific
transverse impact parameter, enabling the identification of events exhibiting
displaced signatures at L1 CERN-LHCC-2020-004 .

Triggering events containing displaced particles using standard PUPPI
Bertolini:2014bba ; CERN-LHCC-2020-004 jet triggers might not be efficient as
shown in Bhattacherjee:2021qaa . Specifically, triggering events that involve
displaced physics objects resulting from the decay of low-mass LLPs, such as
LLPs originating from the exotic decay of 125 GeV Higgs boson Curtin:2013fra ,
will pose a significant challenge at HL-LHC. In such scenarios, standard
$H_{T}$ triggers will also be inefficient to capture these events effectively
due to the limited hadronic activity in the calorimeters. Therefore, there is
an urgent requirement to develop dedicated triggers specifically designed to
efficiently select the events containing displaced objects while effectively
controlling the background rate.

Deep learning LeCun2015 ; Silver2016 ; devlin-etal-2019-bert methods offer
powerful ways to process the large and complicated data obtained at the LHC
PhysRevLett.65.1321 ; DENBY1988429 ; deOliveira:2015xxd ; Kasieczka:2019dbj ;
Guest:2018yhq , and anomaly detection techniques Heimel:2018mkt ;
Farina:2018fyg ; Blance:2019ibf ; Roy:2019jae ; Dillon:2022mkq ;
Anzalone:2023ugq based on deep neural networks offer a powerful way of
identifying displaced signatures at any point in the analysis chain at the
LHC. With the hardware upgrades to FPGAs at the L1 stage and enhancements to
the DAQ system at HL-LHC, the integration of unsupervised machine learning
algorithms into the L1 trigger system now seems achievable Duarte:2018ite ;
Summers:2020xiy ; Govorkova:2021utb ; Neu:2023sfh . In this paper, we
introduce LLPNet–a lightweight graph autoencoder Atkinson:2021nlt ;
Tsan:2021brw ; Atkinson:2022uzb ; Hao:2022zns that utilizes edge convolutions
wang2019dynamic and can detect anomalous signatures of LLPs against the
minimum bias and QCD di-jet backgrounds with the restricted tracker level
information available at the L1 trigger.

Rest of the paper is structured as follows- In Section 2, we outline the
signal benchmark points and background. Section 3 explains the network
architecture of the autoencoder and the training procedure employed in this
study. In Section 4, we study the performance of the autoencoder for various
signal benchmark points. Section 4.1 presents the rate calculation for the
various LLP scenarios and quantify results in terms of background rate and
signal efficiency. Finally, we conclude and summarise in Section 5.

##  2 Signal and background

In this work, we explore the specific BSM scenario of LLPs coupling to the 125
GeV SM-like Higgs boson. The quest for LLPs within this model is substantially
motivated given that the Higgs portal serves as one of the prominent
renormalizable channels through which new gauge-singlet particles can interact
with the SM Curtin:2013fra ; Matsumoto:2018acr ; Arcadi:2021mag ;
Bhattacherjee:2021rml . Exotic signatures of the Higgs boson decaying into
LLPs are prevalent in various BSM theories, the details of which are
elaborated in Alimena:2019zri and references therein. We consider decay of LLP
into a pair of $b$-quarks with 100% branching fraction. The whole process can
be represented as-

$\mathbf{pp\rightarrow h(125GeV)\rightarrow XX,X\rightarrow jj}$

Where SM-like 125 GeV Higgs boson ($h$) is primarily produced through gluon-
gluon fusion. It then decays to two LLPs ($X$), with each LLP subsequently
decaying to two jets ($jj$). In this particular scenario, the mass of the LLPs
is relatively small, with values below half the mass of the Higgs boson,
making it challenging to trigger such events efficiently using existing
triggering strategies. BSM searches looking for LLPs in such scenarios have to
rely on the associated production modes of Higgs boson like vector-boson
fusion (VBF) and Higgs-strahlung (VH) to trigger events at L1 using physics
objects like jets and leptons produced in association with Higgs boson which
comes at the cost of reduced cross-section. Our study focuses on LLPs with
mass, $m_{X}=$ 10, 30 and 50 GeV with mean proper decay lengths ($c\tau$)
spanning from 1 cm to 100 cm. Due to the small boosts of LLPs in this specific
scenario, displaced jets originating from the decay of LLPs are soft and
exhibit low hadronic activity within the detectors. Consequently, triggering
on these LLPs becomes more challenging. For producing signal events and
handling showering and hadronization, we use Pythia8 Sjostrand:2014zea . We
generate the signal samples using the CTEQ6L1 PDF (Parton Distribution
Function) DanielStump_2003 , along with the CUETP8S1-CTEQ6L1 CMS tune
CMS:2015wcf .

The primary sources of background in the present study are “minimum bias”
(Minbias) events and QCD di-jet events. Minbias events refer to inelastic
proton-proton ($pp$) collisions where momentum transfer is very low. Such
events will be produced at the nominal HL-LHC luminosity with huge cross-
section of approximately 80 millibarns (mb). These events contribute
significantly to the overall background due to their huge cross-section. QCD
di-jet events also contribute significantly to the background in the current
study. We divide the QCD background events in various ranges of parton level
$p_{T}$ ($p_{T}^{gen}$) to generate sufficiently significant background events
in different $p_{T}^{gen}$ bins along with minbias events. QCD events are
generated in bins of $p_{T}^{gen}\in$ {30, 50}, {50, 75}, {75, 100}, {100,
125}, {125, 150}, {150, 175}, {175, 200} and $>200$ GeV. A large background
dataset in the $p_{T}^{gen}$ bin, {30, 50} GeV, for training the autoencoder
is produced using MadGraph5_aMC@NLO Alwall:2014hca ; Frederix:2018nkq while
showering is done using Pythia8. Simple detector simulation is carried out
with Delphes-3.5.0 deFavereau:2013fsa for both the signal and background
datasets. To mimic the high PU environment of the HL-LHC, we incorporate the
effect of PU by merging 140 PU events111The average number of soft
interactions is taken to be 140 following a Poisson distribution. with each
primary collision in Delphes. We made slight modifications to the PileUpMerger
code in Delphes to properly account for the offset in the z-direction and
timing values for the first stable particle which in current study comes from
the decay of the LLPs. This adjustment ensures accurate positioning of the
decay product of the LLPs and properly factor in the effects of PU in our
analysis.

##  3 Network architecture and training

The convolution-like operation referred to as “EdgeConv” (Edge convolution)
was introduced for point clouds, as detailed in wang2019dynamic . In this
approach, the point cloud is interpreted as a graph, with individual points
symbolizing nodes, and edges being formulated by linking each point to its
closest neighboring points. The EdgeConv operation accepts a graph as input,
where each node signifies a data point, and the edges describe the connections
or relationships among those data points. Through the application of
convolutional operations to the features of adjacent nodes in the graph, it
generates updated features for every node. In essence, EdgeConv exploits the
connectivity data between points within the graph structure of the point
cloud, thereby allowing the operation to extract local information and conduct
convolutions on the point cloud data.

Mathematically, the EdgeConv operation for the $l^{th}$ message passing
operation can be represented as follows:

$\mathbf{h}_{i}^{(l+1)}=\Delta_{j\in\mathcal{N}(i)}\Theta^{(l)}\cdot(\mathbf{h}_{i}^{(l)},\mathbf{h}_{j}^{(l)}-\mathbf{h}_{i}^{(l)},\mathbf{e}_{ij}^{(l)})$

where:

  * •

$\Delta$ is a permutation invariant aggregation operation, e.g., max, sum, or
mean.

  * •

$\mathbf{h}_{i}^{(l)}$ and $\mathbf{e}_{ij}^{(l)}$ are the input node and edge
features (if present) at the $l$-th message passing operation, respectively.

  * •

$\mathbf{h}_{i}^{(l+1)}$ represents the updated feature of node $i$ at the
$(l+1)$-th layer.

  * •

$\mathcal{N}(i)$ denotes the set of neighboring nodes of node $i$.

  * •

$\Theta^{(l)}$ denotes the learnable function at the $l$-th layer.

The EdgeConv operation captures the local relationships and information within
the graph structure by considering the differences between the features of
neighboring nodes. It enables the network to learn meaningful representations
from the graph-structured data.

In this study, we focus on tracks reconstructed at L1 with a transverse
momentum $p_{T}>2$ GeV and pseudorapidity $|\eta|<$ 2\. Each of these tracks
is treated as an individual node, while each event consisting of these tracks
corresponds to the graph. Without any a priori given graph structure, we
investigate two approaches to construct graphs–radius graphs and k-nearest
neighbour ($k$-NN) graphs using the Euclidean distance in the spatial vertex
in $x$, $y$, and $z$ coordinates. Radius graphs are constructed by connecting
pairs of points within a specified radius ($r_{G}$) of each other–an edge is
drawn between two nodes if their spatial separation is less than or equal to
$r_{G}$. The parameter $r_{G}$ can be seen as a threshold that determines how
close points must be to each other to be considered neighbors, while $k$-NN
graphs are formed by connecting each point to its $k$ nearest neighbors,
regardless of the actual distances between those points. Here, $k$ is a fixed
integer representing the number of nearest neighbors to be considered. In
radius graphs, we can control the granularity of the relationships by varying
the radius. Smaller radii will result in sparser graphs, while larger radii
will capture broader relationships. Similarly, the number of neighbors $k$
controls the sparsity (or density) of a $k$-NN graph. The relative number of
edges for these two approaches will vary depending on the spatial distribution
of the nodes. Additionally, we note that in a $k$-NN graph, a fixed number of
nearest neighbors are chosen for each track, even if multiple neighbors have
the same distance measure close to the track (many tracks can originate from
the same point when their separation is lower than the resolution of the
tracker). In contrast, radius graphs consider all these tracks regardless of
their multiplicity, which offers a more accurate representation of the event
structure.

In analyzing events at L1, where computational efficiency is a crucial factor,
the sparsity of the graph is an important factor in determining the graph
construction algorithm. To this end, we look into the distribution of edge
multiplicity for radius graphs and a relative $k$-NN construction. The
histogram in Figure 1 shows the difference in the number of edges between
these two methods for various values of $r_{G}$ and $k=10$ in minbias events.

Figure 1:  Number of edges in a graph for minbias events, constructed by
considering neighbors within a specific radius ($r_{G}$) or by selecting a
fixed number of neighbors using the $k$-nearest neighbors method.

As we can see, the median number of edges for radius graphs and graphs
constructed using the $k$-NN method differ considerably, with the former being
significantly sparser. Therefore, we prefer to choose radius graphs over
$k$-NN graphs in our study as they capture event structure better and have
less computational overhead.

Each node is associated with ten node features. These features include the
track’s transverse momentum ($p_{T}$), momentum ($p$), transverse impact
parameter ($d_{0}$), vertex position in cartesian coordinates (x, y, and z),
outer position in the tracker in $\eta-\phi$ plane at the radial distance of
1.16 m, the distance of the closest approach to the beam line in the
z-direction ($z_{0}$) and radial distance from the beamline ($D_{T}$). We
additionally construct three edge attributes for every edge: the spatial
separation along the x, y, and z axes between the connected nodes, the
momentum and transverse momentum ratio between two linked nodes.

The network architecture of the autoencoder employed in the present study is
depicted in Figure 2. Autoencoders are generic neural network models where the
input features undergo compression to a latent representation via an encoder.
This latent representation again gets mapped to the same dimensions as the
input features via a decoder, and the whole encoder-decoder setup is trained
to reconstruct the input features faithfully via a loss function like mean
squared error. In anomaly detection methods, the autoencoder is trained to
reconstruct the most probable samples in the data (background processes at the
LHC). In contrast, any sample (possible signals) with inherently different
features from the training data would lead to a poor reconstruction, leading
to larger values of the loss function.

Figure 2: The figure shows a schematic diagram of LLPNet, with latent node
dimensions $n_{lat}$.

In the autoencoder architecture used in this work, both encoder and decoder
have two EdgeConv blocks where each EdgeConv block takes the node features and
edge features (if present) and undergoes a message-passing operation, as
illustrated in Figure 2 (left) with both linear layers having an equal number
of perceptrons to the specified updated node feature dimension. With the input
node features and three edge features, the first EdgeConv operation takes a
23-dimensional input and produces 32-dimensional updated node features. The
second EdgeConv layer accepts a 64-dimensional input (32 * 2) and leads to a
$n_{lat}$-dimensional latent space where $n_{lat}$ is a hyperparameter which
we optimize as explained in the following sections. On the decoder side, the
initial EdgeConv layer receives a 2*$n_{lat}$-dimensional input from the
latent space and maps it to a 32-dimensional output. The second decoder layer,
taking a 64-dimensional input (32 * 2), returns the reconstructed node
features for each event. We use “max” aggregation operation ($\Delta$) for
updating the nodes in all message-passing operations. With its relatively
fewer learnable parameters ($\approx 6000$ at latent space dimension,
$n_{lat}=6$), this network architecture demonstrates suitability for
deployment within the L1 stage of the trigger system

We train the autoencoder using QCD di-jet events with parton level $p_{T}$
ranging from 30 to 50 GeV, which includes 140 PU events. Rather than solely
relying on minbias events for training, we incorporate the QCD di-jet events
as the hard interaction with the PU for training the network. This approach
enables us to capture simultaneously the characteristics of the hard QCD
interaction and minbias events present in each event in terms of PU. On the
other hand, an admixture of minbias samples with QCD di-jet events would lead
to inefficient learning of the hard background features since the relative
proportion (based on the cross-section) of QCD di-jet events would be much
lower. Moreover, the harder $p_{T}$ bins of QCD di-jet events have relatively
much lower cross-sections than the first bin but have similar features. For
training the autoencoder, we utilize 600k events for training and 300k for
validation, with 50k events taken for inference. For optimization, we employ
the ADAM optimizer with a learning rate of 0.001 aided by a reduce-on-plateau
condition with a 0.5 decay factor and patience of three epochs. To calculate
the anomaly score for each event, we use the mean squared error (MSE) between
the reconstructed event and the original event.

##  4 Results

We evaluate the autoencoder using two key metrics: the Area Under the Curve
(AUC) for the Receiver Operator Characteristics (ROC) and the signal
efficiency when the background rejection rate is 99.9% (denoted as
$\epsilon_{s}$). The choice of a 99.9% background rejection rate in the second
metric takes into consideration the order of magnitude of background rejection
needed at L1, which is on the order of $10^{3}$. The values for AUC and
$\epsilon_{s}$ are calculated from a dataset with eighteen benchmark points
for LLPs and the minbias background. Each of these benchmarks, including the
minbias sample, contains 140 PU events. The LLP benchmarks are characterized
by masses, $m_{X}=$ 10, 30, and 50 GeV and $c\tau$ that span from 1 cm to 100
cm. Our analysis includes a comprehensive scan of hyperparameters,
specifically the graph radius, $r_{G}$, and dimension of the latent space,
$n_{lat}$. Figures 3, 4, and 5 illustrate how $\epsilon_{s}$ changes with
$c\tau$ across different $r_{G}$ and $n_{lat}$ for LLP masses of 10, 30, and
50 GeV respectively. We vary the $r_{G}$ from 0.5 to 1 mm and the $n_{lat}$
from 2 to 12. We also highlight the configurations that maximize
$\epsilon_{s}$. Corresponding plots with AUC values are provided in the
Appendix A.

Figure 3: Variation of signal efficiency at 99.9% minbias background rejection
($\epsilon_{s}$) with graph radius ($r_{G}$) and latent space dimension
($n_{lat}$) for various $c\tau$ values of LLP with mass, $m_{X}=$ 10 GeV.
Figure 4: Variation of signal efficiency at 99.9% minbias background rejection
($\epsilon_{s}$) with graph radius ($r_{G}$) and latent space dimension
($n_{lat}$) for various $c\tau$ values of LLP with mass, $m_{X}=$ 30 GeV.
Figure 5: Variation of signal efficiency at 99.9% minbias background rejection
($\epsilon_{s}$) with graph radius ($r_{G}$) and latent space dimension
($n_{lat}$) for various $c\tau$ values of LLP with mass, $m_{X}=$50 GeV.

As seen in Figures 3, 4, and 5, there’s no single common point in latent space
and graph radius that yields the maximum signal efficiency $\epsilon_{s}$ for
every LLP benchmark point. Therefore, we must identify optimal points that
provide the best possible signal efficiency for LLPs. This includes both very
low mass LLPs ($m_{X}=$ 10 GeV) and relatively heavy mass LLPs ($m_{X}=$ 50
GeV), as well as LLPs with short decay lengths ($c\tau=$ 1 cm) and those with
large decay lengths ($c\tau=$ 100 cm).

To identify the optimal combination of $n_{lat}$ and $r_{G}$ that maximize the
$\epsilon_{s}$ for our anomaly detector, we employed the Gradient Boosting
Regressor (GBR). Our dataset was structured such that combinations of
$n_{lat}$ and $r_{G}$ served as features, and their corresponding
$\epsilon_{s}$ acted as the target. The GBR, with parameters set at
‘n_estimators=1500’, ‘learning_rate=0.01’, ‘max_depth=6’, and
‘loss=squared_error’, was trained on this dataset. Post training, the model
predicted the $\epsilon_{s}$ for all potential combinations, and the one
yielding the highest predicted $\epsilon_{s}$ was considered the optimal
hyperparameter set. The optimization was performed separately for different
masses. Subsequently, we adopted a similar approach for different decay
lengths.

Table 1 presents the optimal $r_{G}$ and $n_{lat}$ for three distinct LLP
masses, $m_{X}=$10, 30, and 50 GeV.

$m_{X}$ (GeV) |  $r_{G}$ (mm) | $n_{lat}$
---|---|---
10 | 0.9 | 6
30 | 1 | 10
50 | 1 | 10
Table 1: Optimal parameters (graph radius, $r_{G}$, and latent space
dimension, $n_{lat}$) for different LLP masses ($m_{X}$).

Similarly, Table 2 enumerates the same parameters but for various decay
lengths ranging from 1 cm to 100 cm.

$c\tau$ (cm) |  $r_{G}$ (mm) | $n_{lat}$
---|---|---
1 | 1 | 10
5 | 0.6 | 8
10 | 0.9 | 6
30 | 0.9 | 6
50 | 0.9 | 6
100 | 0.9 | 6
Table 2: Optimal parameters (graph radius, $r_{G}$, and latent space
dimension, $n_{lat}$) for different LLP decay lengths ($c\tau$).

These tables present the results of a hyperparameter scan, highlighting the
configurations that yielded the best performance for each LLP scenario. From
these tables, it becomes clear that the optimal performance for each LLP
benchmark can be reached using one of the following three combinations in
graph radius and latent space dimensions: [0.6, 8], [0.9, 6] and [1, 10].
Thus, we have identified three working points (WPs) corresponding to these
values, referred to as WP-1, WP-2, and WP-3. In Figure 6 (top), we explicitly
show the ROC curve for WP-1 with $r_{G}=0.6$ mm and $n_{lat}=$ 8 for above
mentioned LLP scenarios. In Figure 6 (bottom), the anomaly scores for each of
these LLP benchmark points, along with the minbias background, are presented
explicitly. Similar plots for WP-2 and WP-3 are included in Appendix B.

Figure 6:  ROC curves (top), along with anomaly scores (bottom), for minbias
and specific LLP benchmark points with $m_{X}=$ 10 GeV (left), 30 GeV
(middle), and 50 GeV (right) for WP-1. The decay length of LLP ranges from 1
cm to 100 cm.

###  4.1 Background rate calculation

The QCD samples across different $p_{T}^{gen}$ bins, as explained in Section
2, are combined with minbias events to calculate the event rate using
“stitching” method as explained in Ehataht:2021rkh and validated in one of our
previous studies Bhattacherjee:2021qaa . When calculating the background rate
for triggers by combining different $p_{T}^{gen}$ bins can lead to unavoidable
overlap in phase space. The stitching method helps to overcome this issue by
applying selected weights to the simulated events. It also lets us gradually
increase the number of simulated events in specific areas of phase space
($p_{T}^{gen}$) if the existing MC samples don’t cover those areas well. The
calculation of the event weight in terms of rate is performed using the
following formula-

$w^{I}=\frac{F}{N_{incl}+\sum_{j}N_{j}\times\frac{n_{j}}{(N_{PU}+1)\times
p_{j}}}$ (4.1)

Here, $F$ is the $pp$ collision frequency (around 28 MHz222Although proton
beams cross each other every 25 ns yet only 70% of these intersections result
in $pp$ collisions.). $N_{incl}$ refers to the total number of events with
$N_{PU}$+1 minbias events, where $N_{PU}$ is the average PU events in a
collision. $N_{j}$ is the event count for the $j^{th}$ $p_{T}^{gen}$ bin, and
$n_{j}$ is the number of inelastic $pp$ interactions in that bin. $p_{j}$ is
the probability of a single inelastic collision in the $j^{th}$ $p_{T}^{gen}$
bin, determined by comparing the cross-section for that bin to the cross-
section without any $p_{T}^{gen}$ conditions.

The technical design report for the L1 trigger at the CMS for the HL-LHC
outlines the design of two dedicated triggers tailored towards triggering jets
originating from LLPs with short and large decay lengths. The combined
allowable rate bandwidth for these triggers is approximately 40 KHz so we need
to keep the background rate below 40 KHz. For the current study, we constrain
the background rate at 30 KHz.

In Figure 7, 8, and 9 the variation of trigger rate is plotted against signal
efficiency for three LLP benchmark points for WP-1, WP-2 and WP-3
respectively, with LLP masses ($m_{X}$) of 10, 30, and 50 GeV. Within these
points, the decay length is considered across a range from 1 cm to 100 cm.

Figure 7:  Background rate versus signal efficiency for LLPs with mass,
$m_{X}=$ 10 GeV, 30 GeV and 50 GeV for decay lengths, $c\tau$, varying from 1
to 100 cm for WP-1. Annotated values correspond to signal efficiency at
background rate of 30 KHz. Figure 8:  Background rate versus signal efficiency
for LLPs with mass, $m_{X}=$ 10 GeV, 30 GeV and 50 GeV for decay lengths,
$c\tau$, varying from 1 to 100 cm for WP-2. Annotated values correspond to
signal efficiency at background rate of 30 KHz. Figure 9:  Background rate
versus signal efficiency for LLPs with mass, $m_{X}=$ 10 GeV, 30 GeV and 50
GeV for decay lengths, $c\tau$, varying from 1 to 100 cm for WP-3. Annotated
values correspond to signal efficiency at background rate of 30 KHz.

In these plots, the signal efficiency for various benchmark points is
annotated at a background rate of 30 KHz. In Table 3, we also enumerate the
signal efficiency for various signal benchmark points for LLPs with $m_{X}=$
10, 30 and 50 GeV and $c\tau$ varying from 1 cm to 100 cm for three WPs: WP-1,
WP-2 and WP-3 after constraining the background rate at 30 KHz.

|  $c\tau$ (cm)
---|---
$m_{X}$ (GeV) | 1 | 5 | 10 | 30 | 50 | 100
WP-1
10 | $0.40$ | $0.33$ | $0.24$ | $0.12$ | $0.08$ | $0.05$
30 | $0.63$ | $0.70$ | $0.60$ | $0.37$ | $0.27$ | $0.17$
50 | $0.64$ | $0.80$ | $0.76$ | $0.56$ | $0.44$ | $0.29$
WP-2
10 | $0.39$ | $0.52$ | $0.47$ | $0.28$ | $0.20$ | $0.12$
30 | $0.35$ | $0.60$ | $0.59$ | $0.48$ | $0.39$ | $0.27$
50 | $0.30$ | $0.62$ | $0.64$ | $0.53$ | $0.45$ | $0.34$
WP-3
10 | $0.34$ | $0.25$ | $0.18$ | $0.09$ | $0.06$ | $0.04$
30 | $0.64$ | $0.70$ | $0.60$ | $0.37$ | $0.27$ | $0.17$
50 | $0.66$ | $0.80$ | $0.76$ | $0.57$ | $0.45$ | $0.29$
Table 3: Signal efficiency values for different LLP signal benchmark points
across three working points (WP-1, WP-2, WP-3) at background rate of 30 KHz.

Table 3 illustrates that good signal efficiency is achievable for LLPs with
both higher and lower decay lengths. As a result, a single anomaly-based
trigger might be sufficient to trigger the LLPs, regardless of whether the
decay length is short or large. For example, for LLPs with a decay length of 5
cm, signal efficiencies of approximately 52%, 60%, and 62% can be achieved for
LLPs with $m_{X}=$ 10 GeV, 30 GeV, and 50 GeV with WP-2, respectively. At a
decay length of 100 cm, the signal efficiency retains respectable values at
around 12%, 27%, and 34% for LLPs with $m_{X}=$ 10 GeV, 30 GeV, and 50 GeV
respectively.

For context, when considering a decay length of 5 cm for $m_{X}$= 30 GeV
resulting from the decay of a 125 GeV Higgs boson, the CMS experiment
estimates a signal efficiency of roughly 8%. This is evaluated at a background
rate of 30 KHz in a 200 PU scenario, utilizing the displaced tracker $H_{T}$
trigger. This trigger is designed for LLPs with shorter decay lengths at HL-
LHC CERN-LHCC-2020-004 . Meanwhile, for a $c\tau$ value of 100 cm and $m_{X}$=
50 GeV, the CMS experiment projects a signal efficiency close to 20% at a
background rate of 25 KHz in a 200 PU environment using the displaced Calo-Jet
trigger, primarily designed for LLPs with a significant lifetime at HL-LHC
CERN-LHCC-2020-004 .

##  5 Summary and conclusion

Triggering exotic events within the level-1 (L1) trigger system continues to
be a substantial challenge for many searches involving displaced particles.
High pile-up (PU) environment at HL-LHC will further add to this challenge
making it exceedingly difficult to trigger such events efficiently at L1 while
keeping the background rate under control. In scenarios where long-lived
particles (LLPs) are coupled to the Standard Model (SM)-like 125 GeV Higgs
boson, the triggering of such events is primarily dependent on the associated
production modes of the Higgs boson where these searches predominantly depend
on triggering events with objects produced in association with the Higgs
boson. However, this approach comes with a reduced production cross-section,
making the search for such particles exceedingly challenging. In the specific
scenario where the Higgs boson is SM-like with a mass of 125 GeV, a further
challenge arises. The hadronic activity resulting from the decay of LLPs,
which originate from the decay of the Higgs boson, is very small. This low
level of hadronic activity further complicates the triggering process, making
detection more difficult using standard $H_{T}$ or jet triggers. Therefore,
there is an immediate and pressing requirement to develop L1 triggers capable
of independently capturing LLP events, without relying on existing triggers.

With the introduction of advanced FPGAs and forthcoming upgrades to the
triggering and DAQ system at HL-LHC, the implementation of machine learning,
particularly graph-based machine learning triggers, becomes viable due to
relaxed latency constraints at L1 and reliable and efficient deployment of
such networks on FPGAs. In this work, we study a novel method to trigger
displaced events using a machine learning approach where we utilize message
passing autoencoder employing “Edge convolution” on the tracks reconstructed
at L1 where these collection of tracks act as point cloud.

In consideration of the low latency requirement at L1, the network
architecture of the autoencoder is deliberately designed to be simple,
featuring a comparatively small number of learnable parameters. For the LLP
benchmark points, we focus on the LLPs with mass of 10, 30, and 50 GeV and
decay lengths extending from 1 cm to 100 cm. Each LLP is assumed to decay into
two displaced jets, with a branching ratio of 100%, resulting in a final state
marked by displaced jet signatures. The kinematic features of these tracks,
along with their displacement in the transverse plane, are utilized to
construct the node and edge features for event-based graphs. Instead of the
$k$-nearest neighbours method, we adopt the radius method to form these
graphs. This approach better captures the event structure and offers a reduced
computational load, a crucial aspect for L1 triggers. A hyperparameter scan is
conducted over the graph radius and latent space dimensions. Subsequently,
based on regression analysis with the “Gradient Boosting Regressor” performed
for each mass and decay length independently, three distinct working points
(WPs) are established. However, it is of significance to acknowledge that the
optimal radius of the graphs employed in the experiments may vary depending
upon the attainable spatial track resolution at HL-LHC, taking into account
factors such as the track $p_{T}$ and position within the tracker along with
the mean number of PU events.

Results are presented in terms of background rate versus signal efficiency.
The background rate computation employs the “Stitching method”, taking into
consideration both minbias and QCD di-jet backgrounds taking into account the
potential overlaps in phase space. QCD background is segmented into multiple
$p_{T}^{gen}$ bins to ensure precise rate calculations. We observe notable
signal efficiency for signal benchmark points at the permissible background
rate of 30 KHz. Although, signal efficiency decreases with increase in decay
length as expected, it retains a respectable value for higher decay lengths.
For example, we observe signal efficiency of 33%, 70% and 80% for LLPs of mass
10, 30 and 50 GeV with WP-1. At decay length of 50 cm, signal efficiency of
20%, 39% and 45% is observed for 10, 30 and 50 GeV LLP with WP-2.

In conclusion, the graph-based autoencoder highlighted in this study suggests
a potential for enhancing signal acceptance for light LLPs at the L1 trigger
level for HL-LHC. However, it is imperative to underscore that additional
comprehensive studies are warranted to evaluate the overall performance and
suitability of such autoencoders, especially in relation to FPGA
implementations while carefully considering the inherent L1 latency
constraints.

## Acknowledgement

B.B. acknowledges the support provided by the MATRICS Grant (MTR/2022/000264)
from the Science and Engineering Research Board (SERB), Government of India.
P.S. thanks Rhitaja Sengupta for valuable discussions. Additionally, P.S.
expresses gratitude to the Indian Institute of Science (IISc), Bengaluru, and
the Physical Research Laboratory (PRL), Ahmedabad, for facilitating P.S.’s
visit to PRL for scientific collaboration. This work utilized the Param
Vikram-1000 High-Performance Computing Cluster and TDP resources at the PRL
and computing facility at the Centre for High Energy Physics (CHEP), IISc.

## Appendix

##  Appendix A AUC values in radius vs. latent plane

Figure 10, 11 and 12 represents the area under curve (AUC) values of ROC for
combinations of graph radius ($r_{G}$) and latent space dimension ($n_{lat}$)
for LLPs of mass, $m_{X}=$ 10, 30 and 50 GeV for decay length, $c\tau$,
varying between 1 cm and 100 cm. Combination of $r_{G}$ and $n_{lat}$ with
maximum AUC is highlighted.

Figure 10: Variation of area under curve (AUC) with graph radius ($r_{G}$) and
latent space dimension ($n_{lat}$) for various $c\tau$ values of LLP with
mass, $m_{X}=$ 10 GeV. Figure 11: Variation of area under curve (AUC) with
graph radius ($r_{G}$) and latent space dimension ($n_{lat}$) for various
$c\tau$ values of LLP with mass, $m_{X}=$ 30 GeV. Figure 12: Variation of area
under curve (AUC) with graph radius ($r_{G}$) and latent space dimension
($n_{lat}$) for various $c\tau$ values of LLP with mass, $m_{X}=$ 50 GeV.

##  Appendix B ROC curves for WP-2 and WP-3

Figures 13 and 14 (top) depict the ROC curve for WP-2 with $r_{G}=0.9$ and
$n_{lat}=$ 6 and WP-3 with $r_{G}=1$ and $n_{lat}=$ 10 respectively for
various LLP scenarios. In Figures 13 and 14 (bottom), the anomaly scores for
each of these LLP benchmark points, along with the minbias background, are
presented explicitly.

Figure 13:  ROC curves (top), along with anomaly scores (bottom), for minbias
and specific LLP benchmark points with $m_{X}=$ 10 GeV (left), 30 GeV
(middle), and 50 GeV (right) for WP-2. The decay length of LLP ranges from 1
cm to 100 cm.

Figure 14:  ROC curves (top), along with anomaly scores (bottom), for minbias
and specific LLP benchmark points with $m_{X}=$ 10 GeV (left), 30 GeV
(middle), and 50 GeV (right) for WP-3. The decay length of LLP ranges from 1
cm to 100 cm.

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