CALORIMETER BASED VERTEXING FOR THE ATLAS NEXT GENERATION TRIGGER

CALORIMETER BASED VERTEXING FOR THE ATLAS NEXT GENERATION TRIGGER AUGUST 2025 AUTHOR(S): Rafail Athanasios Giannoulakis Aristotle Univeristy of Thessaloniki SUPERVISOR(S): Ioannis Xiotidis Thorsten Wengler CERN openlab Report /2025 PROJECT SPECIFICATION All major CERN experiments along with the accelerator will enter soon in the phase of a big upgrade cycle, called the High-Luminosity LHC, in order to further broaden the physics reach. ATLAS has a plethora of upgrades concerning the HL-LHC era which will equip the detector with many exciting new opportunities. One of the core upgrades in ATLAS concerns the way of reading out the calorimeter sub-detector. In contrast to previous runs ATLAS will be able to read the full granularity of the calorimeter at the level of the hardware trigger system. Having this information available in such a challenging environment provides a unique opportunity to explore Machine Learning ideas on the edge within the context of the Next Generation Trigger project. With the current project we would like to explore the concept of enhancing the vertexing capabilities of ATLAS by using only the calorimeter information for inference. The bulk of the work will be focused on designing and implementing a Symbolic Regression based model which during training will include both the tracking detector information and the calorimeter cells and eventually aim to run inference only with the calorimeter cells. CALORIMETER BASED VERTEXING FOR THE ATLAS NEXT GENERATION TRIGGER 1 CERN openlab Report /2025 ABSTRACT This report focuses on the creation and implementation of a particle tracking and vertexing technique proposed for the ATLAS Next Generation Global Trigger. Most of the project’s time was spent on developing and testing a linear and an extended Kalman filter algorithm with data from an open dataset. As a next step to that, a modified Kalman filter was developed, combining the estimator capabilities of the extended Kalman filter (EKF) and the usefulness of a novel evolutionary driven ML symbolic regression (SR) method, in order to make a possibly more accurate inference about a particle’s interaction vertex than other contemporary methods. This technique can be later applied to the the ATLAS Global Trigger by utilizing energy depositions from the sampling layers of the ATLAS electromagnetic-liquid argon calorimeter as input (seeds) for the Kalman filter. CALORIMETER BASED VERTEXING FOR THE ATLAS NEXT GENERATION TRIGGER 2 CERN openlab Report /2025 TABLE OF CONTENTS 1 INTRODUCTION 4 1.1 Status quo of particle tracking and vertexing . . . . . . . . . . . . . . . . . . . . 4 1.2 TheATLASdetector................................. 4 1.3 Trigger and DAQ systems upgrade for the HL-LHC . . . . . . . . . . . . . . . . 5 2 KALMAN FILTER 6 2.1 Logic and algorithm of the linear Kalman filter (LKF) . . . . . . . . . . . . . . 6 2.2 The extended Kalman filter (EKF) . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Application in particle tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 SYMBOLIC REGRESSION 10 3.1 GeneralInformation ................................. 10 3.2 ThePySRlibrary................................... 10 4 PROJECT PROGRESS AND RESULTS 12 4.1 TrackMLdataset ................................... 12 4.2 LKFimplementation ................................. 12 4.3 EKFimplementation................................. 16 4.4 Symbolic regression and the modified Kalman filter (MKF) . . . . . . . . . . . . 17 5 CONCLUSIONS 20 6 REFERENCES 20 CALORIMETER BASED VERTEXING FOR THE ATLAS NEXT GENERATION TRIGGER 3 CERN openlab Report /2025 1 INTRODUCTION 1.1 Status quo of particle tracking and vertexing Every 25ns, accelerated protons from the the Large Hadron Collider (LHC) collide inside the ATLAS experiment, resulting in approximately 120 interactions. For the High Luminosity LHC upgrade, this number is expected to grow to an average of 200 interactions. In these type of experiments, scientific breakthrough is done by identifying and researching those interactions and the particles that are produced in the process. To this end, scientists use the different types of detectors inside the experiment to find the trajectory paths of particles (a.k.a tracks, tracking) and their interaction origin (a.k.a vertex, vertexing). However, most interactions don’t hold much physical interest, which is why an event filter is needed to "trigger" only the interesting ones. The calorimeter-based vertexing technique proposed in this report is a new way of enhancing the HL-LHC Global Trigger of ATLAS with tracking information. 1.2 The ATLAS detector The ATLAS detector at CERN’s LHC is one of the largest and most complex scientific instruments ever built measuring 46 meters in length, 25 meters in diameter, and weighing about 7,000 tonnes. Designed to record up to a billion proton-proton collisions every second, ATLAS uses a sophisticated two-level trigger approach to reduce this data to around 1000-2000 events per second to be further analyzed. It is comprised of 3 sub-detector sections: 1) the Inner Detector (ID), 2) the calorimeters and 3) the muon detectors. At its core, the ID, consisting of a pixel tracker with 80 million channels, a silicon micro strip tracker with 6 million channels, and a transition radiation tracker with 350,000 straws, measures charged particles with micrometer precision and distinguishes between particle types. Surrounding the ID are the calorimeters that measure particle energies: the Liquid Argon calorimeter, operating at -183 °C, records electromagnetic showers, while the Tile Calorimeter, built from 500,000 scintillator tiles, measures hadronic interactions. Lastly, the muon system, made of drift tubes and specialized chambers, tracks muons with exceptional accuracy [3]. Figure 1: The ATLAS detector CALORIMETER BASED VERTEXING FOR THE ATLAS NEXT GENERATION TRIGGER 4 CERN openlab Report /2025 ATLAS relies on a powerful 2T magnet system, including large solenoids and toroids, to bend particle trajectories and measure their momentum. Overall, the detector has nearly 100 million readout channels and more than 3,000 kilometers of cables, producing over 3,200 terabytes of data per year, which are analyzed worldwide through the LHC Computing Grid. 1.3 Trigger and DAQ systems upgrade for the HL-LHC All major CERN experiments along with the accelerator will soon enter in the phase of a big upgrade cycle (called the High-Luminosity LHC) in order to further broaden the physics reach. The instantaneous luminosity (quantity proportional to the number of collisions) is expected to rise, consequently making an upgrade to the Trigger and Data Acquisition systems a necessity. The aim is to retain low physics thresholds, maintain high efficiency, and deliver data of sufficient quality under extreme pile-up conditions. The upgraded architecture is based on a single hardware Level-0 (L0) trigger followed by a software-based Event Filter. The L0 trigger accepts the full 40 MHz bunch crossing rate from the LHC and reduces it to about 1 MHz. These events are then processed by the Event Filter, running offline-grade reconstruction software on large computing farms, which reduces the output to about 10 kHz for permanent storage. The hardware Level-0 trigger combines several subsystems. The calorimeter trigger uses FPGA-based processors - eFEX, jFEX, gFEX, and a new fFEX for forward coverage - to build trigger objects. In parallel, the muon trigger uses precision detectors such as the Monitored Drift Tubes into the hardware chain for the first time, significantly improving muon candidate quality. A Global Trigger then merges information from the calorimeter and muon systems at a data throughput of about 50 Tbps. Using powerful modern FPGAs, this system can run algorithms similar to offline reconstruction, such as jet clustering, pileup suppression, and topological selections. Finally, the Central Trigger Processor integrates inputs from all subsystems and issues the L0 Accept (L0A) signal with fixed latency, interfacing directly with the LHC beam timing system. Regarding the Data Acquisition (DAQ) system, a Front-End Link Exchange (FELIX) system, based on FPGA PCIe cards in commodity PCs, distributes timing signals and handles data flow. At the design rate of 1 MHz, ATLAS will read out about 4.6 TB/s from the detector. From here, the rate is reduced to around 10 kHz, suitable for long-term storage and physics analysis [1][2]. CALORIMETER BASED VERTEXING FOR THE ATLAS NEXT GENERATION TRIGGER 5 CERN openlab Report /2025 Figure 2: Pipeline of the ATLAS Phase-II trigger systems[5] 2 KALMAN FILTER 2.1 Logic and algorithm of the linear Kalman filter (LKF) Modeling and predicting the evolution of a dynamical system requires data acquired from a variety of sensors. However, like all sensors in everyday life, detectors in particle physics experiments are not ideal measuring tools. They have innate or external noise that blends with the actual signal we want to measure [9]. Furthermore, the measurements required to make accurate predictions are not always available to us (in our case we only have data from the calorimeters). The linear Kalman filter (LKF), also known as linear quadratic estimator (LQE), is a Bayesian algorithm that estimates the state of a dynamical system based on apriori informaCALORIMETER BASED VERTEXING FOR THE ATLAS NEXT GENERATION TRIGGER 6 CERN openlab Report /2025 tion about the systems behavior [6]. It is an iterative process that utilizes a set of measurements with their respective noise and uncertainties to predict the state of a phenomena with greater accuracy than by just having individual measurements alone. Moreover, the LKF is recursive, meaning that each state prediction is used as information to make the next prediction. Lastly, the LKF is an optimal state estimator for linearly evolving systems if a) there is a sufficient amount of available measurements and b) the covariances of the "white" noise in those measurements are known exactly. The algorithm consists of two phases for each iteration: •Prediction of the next state of the system based on apriori knowledge of the parameters describing the system evolution. •Update of this state based on sensor data. The state of the system at the kiteration can be described by two variables : 1) a state vector xkcontaining all the system evolution parameters 2) and a covariance matrix with the covariances of all those parameters Pxx,k. From here onwards, we deal with the mathematical representation of the two phases of the KF algorithm for a system whose state is represented by N parameters [7][11]: •x– State vector (N×1) •P– Covariance matrix (N×N) •Q– Noise matrix (N×N) •d– Measurement state vector (M×1). Represents only the M measurable parameters of the state vector. •RMeasurement noise matrix (M×M). This noise comes directly from the measurement resolution of the sensor. •A– The transition/prediction matrix (N×N) that encompasses the evolution equations of the system. •H– The observation matrix (M×N) that transforms the parameters of a system to their measurable counterparts. •K– Kalman gain factor (N×M) •. . . c – Current value •. . . p – Predicted value •. . . m – Measured value •. . . u – Updated value •. . . k – At iteration k •. . . x – For State vector k •. . . d – For Measurement vector k CALORIMETER BASED VERTEXING FOR THE ATLAS NEXT GENERATION TRIGGER 7 CERN openlab Report /2025 1. Prediction of next state xp k=Akxc k(1) Pp xx,k =AkPc xx,kAT k+Qxx,k (2) 2. Get measurement vector and covariance dp k=Hxp k(3) Pp xd,k =Pp xx,kHT(4) Pp dd,k =HPp xx,kHT+Rdd,k (5) =HPp xd,k +Qdd,k (6) 3. Calculation of Kalman gain Kk=Pp xx,kHTPp dd,k−1(7) =Pp xd,k Pp dd,k−1(8) 4. Update state and covariance xu k=xp k+Kk(dm k−dp k)(9) Pu xx,k =Pp xx,k −KkPp xd,k T(10) = (I−KkHk)Pp xx,k (11) 2.2 The extended Kalman filter (EKF) In the physical world, most phenomena are explained by complex equations where the linearity between subsequent states assumed by the LKF does not apply. The extended Kalman filter (EKF) was created to, as the name suggests, extend the applications of the KF to non-linearly evolving systems. Unlike the LKF, the EKF uses a first order approximation multivariate Taylor expansion to linearize the model and each iteration state’s errors. Higher order models are also possible but the increased computation time needed renders them undesirable in almost all situations. Even though the EKF is a de-facto standard in the theory of non-linear state estimation, it is not an optimal estimator, unlike the LKF. If the system is not modeled correctly the filter can easily diverge owning to it’s linearization. For the remainder of this subsection, we provide the mathematical equations that differentiate the EKF from the LKF. The logic of the two phase algorithm remains the same: 1. We replace the A and H matrices in the prediction of the state vector with their respective analytical transition and observation functions aand h: xp k=ak(xc k)(12) dp k=h(xp k)(13) 2. For the rest of the calculations we use the Jacobian matrices of the aand hfunctions, Ak CALORIMETER BASED VERTEXING FOR THE ATLAS NEXT GENERATION TRIGGER 8 CERN openlab Report /2025 Figure 8: Track parameter and vertex z-position residuals for the LKF Figure 9: Track parameter and vertex z-position pulls for the LKF The residual distributions provided information about the estimation of a parameter’s value (mean ≈0implies good estimation) and the actual uncertainty of the estimation (standard deviation of the distribution). The pulls were created from the residuals divided by the uncertainty of each parameter at the last step. By monitoring the pull standard deviation, we made an inference about our model’s behavior and apply the appropriate changes in the P and Q matrices (std.dev. >1and std.dev. <1imply underestimation and overestimation of the model’s actual uncertainty, respectively). Moreover, since our project focuses on vertexing, we also included the residual and pulls for the reconstructed vertex. The reconstruction was done by extrapolating the last step of the LKF to the beamline to find the vertex z-position. CALORIMETER BASED VERTEXING FOR THE ATLAS NEXT GENERATION TRIGGER 15 CERN openlab Report /2025 4.3 EKF implementation The next step in the project was to develop an extended Kalman filter algorithm. The EKF implementation used 90%of the LKF’s structure with the only difference being the transition and observations matrix, as was mentioned in Subsection 2.2. The residuals and pulls produced by the EKF on the same data and starting conditions can be seen in Figures 11 and 12 respectively. Evidently, the distributions look very similar to the ones produced from the LKF. This is caused by the choice to allow negative covariances during evolution. By allowing those values, the EKF did not diverge in tracks that it otherwise should (since it has more restrictions than the LKF due to the Jacobian matrices) and so the differences between LKF and EKF are hidden. This problem was realized very late into the project, where time constraints didn’t permit for the retuning of the P and Q matrices, in order for the models to properly differ from each other. For this reason, no conclusive results can be gained from the residuals about the expected improvement in the vertex resolution. Figure 10: Track reconstruction with EKF (Reco1) and LKF (Reco2) in the ρ−zplane. The reconstructions alignment is caused by the decision of allowing negative covariances. CALORIMETER BASED VERTEXING FOR THE ATLAS NEXT GENERATION TRIGGER 16 CERN openlab Report /2025 Figure 11: Track parameter and vertex z-position residuals for the EKF Figure 12: Track parameter and vertex z-position pulls for the EKF 4.4 Symbolic regression and the modified Kalman filter (MKF) The evolution equations we utilized so far in the creation of our models assumed that the particles propagating through our detector abide to classical electromagnetism laws only. In a real life scenario, that is not the case. For more accurate estimator models, one needs to account for perturbations in the expected behavior such as reconstruction artifacts, multiple scattering and other detector effects present during particle propagation. The simulation of these processes is very arduous, time consuming or even impossible due to stochasticity, and therefore cannot be applied easily to a real-time trigger selection. As a solution to this issue we used a symbolic regression model that corrects the reconstructed hits of our dataset to more closely resemble the actual truth hits. The model we implemented with the assistance of the aforementioned PySR library, as seen below, takes as input the reconstructed hits in ϕ, z coordinates and the truth hits provided by our dataset and tries to formulate two template functions that correct our ϕ, z coordinates, respectively. CALORIMETER BASED VERTEXING FOR THE ATLAS NEXT GENERATION TRIGGER 17 CERN openlab Report /2025 from pysr import PySRRegressor model =PySRRegressor( niterations = 100, populations = 100 , model_selection="best", unary_operators=["cos","sin"], binary_operators=["+","-","*"], ) Listing 1: Symbolic Regression model. "niterations=100" implies 20000 iterations while "modelselection =’best’ " means that model searches for a function that is both accurate and not too complex. The produced template functions can be evaluated by comparing the standard deviations of the "reco-truth" and "modreco-truth" residuals. Here we provide the first results of this model: ϕcorr =ϕreco + 2.56 ·10−5·exp (cos (0.751 ·zreco)) (18) zcorr =zreco ·cos (5.49 ·10−4·zreco)(19) Figure 13: Residual distributions for the reconstructed ϕ−zcoordinates, before and after correction Using this corrected dataset, one could run the KF on the new hits and expect more accurate results in vertex estimation. Since our project is proposed for the ATLAS Global Trigger CALORIMETER BASED VERTEXING FOR THE ATLAS NEXT GENERATION TRIGGER 18 CERN openlab Report /2025 environment, where we do not have access to all hits simultaneously, we instead implemented the correction functions inside the EKF algorithm itself. Particularly, we altered the dm kmatrix in equation (9) of the update step to include the template functions produced by the SR resulting in the creation of a modified Kalman filter (MKF). However, in Figure 13 it is noticeable that the corrected residuals have larger standard deviations than the reconstructed ones, implying that the SR model did not produce functions that adjust the reconstructed points closer to the true ones. This effect is also visible in the behavior of the MKF compared to the EKF, as seen in Figure 14 for a single track. Figure 14: Track reconstruction with EKF (Reco1) and MKF (Reco2) in the ρ−zplane Figure 15: Track parameter and vertex z-position residuals for the MKF CALORIMETER BASED VERTEXING FOR THE ATLAS NEXT GENERATION TRIGGER 19 CERN openlab Report /2025 Figure 16: Track parameter and vertex z-position pulls for the MKF 5 CONCLUSIONS In conclusion, we report that our three Kalman filter models have vertexing resolutions of 0.511mm (LKF, EKF) and 0.5005mm (MKF) respectively. Evidently, the SR, although not properly tuned, provides a different resolution on the order of 10µm. Moreover, judging by the ≃0 mean residuals, the algorithms successfully estimate the track parameters without bias, in a 200 pile-up dataset. Also, we observe a consistent overestimation of the track parameter’s uncertainty between models. Even though more work is required in some areas, this project has laid some groundwork on the study and tuning of the KF initial conditions, and therefore can act as a stepping stone for future studies. Lastly, ρdependent SR models can be defined hence including the geometry of the detector and even allowing direct vertex inference without training information. 6 REFERENCES [1] Collaboration ATLAS. Technical Design Report for the Phase-II Upgrade of the ATLAS TDAQ System. Tech. rep. Geneva: CERN, 2017. doi:10.17181/CERN.2LBB.4IAL.url: https://cds.cern.ch/record/2285584. [2] Collaboration ATLAS. Technical Design Report for the Phase-II Upgrade of the ATLAS Trigger and Data Acquisition System - Event Filter Tracking Amendment. Tech. rep. 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