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Probabilistic, Structure-Intrinsic Clustering with an Hierarchical Embedding (PSICHE) in Time and Space Applied In Particle Physics Jet Reconstruction

This paper introduces PSICHE, a novel probabilistic and hierarchical clustering algorithm that dynamically learns variable jet sizes and substructure in both space and time while incorporating experimental uncertainties, offering a self-consistent and computationally tractable solution for jet reconstruction in particle physics.

Original authors: Margaret Lazarovits, Christopher Rogan

Published 2026-09-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Margaret Lazarovits, Christopher Rogan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the heart of particle physics, scientists smash protons together at nearly the speed of light to recreate the conditions of the early universe. When these collisions happen, they do not produce single, clean particles that fly off in neat lines. Instead, they create sprays of debris known as jets. These jets are not solid objects but rather chaotic clouds of energy and particles that spread out as they travel through detectors. To understand what happened in the collision, physicists must reconstruct these jets from the millions of tiny signals left behind in their detectors. The challenge is immense: the detectors are noisy, the collisions are messy, and often, many collisions happen at the exact same time, creating a background fog that obscures the signal scientists are trying to find. For decades, researchers have used fixed rules to sort these sprays, deciding in advance how big a jet should be or how many pieces it must have. But nature does not follow fixed rules; some jets are small and tight, while others are large and sprawling, and they often contain hidden internal patterns that reveal what kind of particle created them.

A new approach, developed by researchers at the University of Kansas, offers a different way to see these collisions. They have created a method called PSICHE, which stands for Probabilistic, Structure-Intrinsic Clustering with an Hierarchical Embedding. Instead of forcing the data into a pre-set box, this method lets the data tell the story. It treats the reconstruction of a jet as a learning problem, where the computer looks at the energy, location, and timing of every tiny signal and figures out how they naturally group together. The system does not need to be told how many jets to expect or how large they should be. Instead, it discovers these features on its own, identifying not just the main jet but also the smaller, internal clusters within it that correspond to the decay of heavy particles.

The core of this new method is a probabilistic framework, which means it deals with likelihoods rather than hard, absolute decisions. Imagine the detector signals as a collection of points in space and time. The algorithm asks: what is the most likely way these points are connected? It builds a tree-like structure, starting with small groups and merging them into larger ones, constantly checking if the combination makes sense. If two groups of signals are close in space but far apart in time, the system recognizes they might not belong together. This ability to use time as a dimension is crucial. Modern particle detectors can now measure when a particle arrives with incredible precision, down to billionths of a second. By including this timing information, the new method can distinguish between the main event and the background noise, known as pileup, which comes from other collisions happening simultaneously.

In their work, the researchers tested this system using simulated data that mimics the conditions of the Large Hadron Collider. They created scenarios where protons collide to produce heavy particles like top quarks and W bosons, which then decay into jets. They also simulated extreme conditions where dozens of extra collisions happen at the same time, creating a chaotic environment. When they applied their new method, it successfully identified the correct number of jets and their sizes without any prior instructions. For example, when a top quark decays, it produces three distinct streams of particles. The algorithm naturally found these three streams as separate internal clusters within a single jet. In contrast, older methods, which rely on fixed sizes, often miss these details or force the data into shapes that do not match the physics.

One of the most significant findings is how the system handles the background noise. In high-energy physics, the "pileup" of extra collisions can make a jet look heavier or larger than it really is, hiding the true signal. The new method uses the timing and energy of the signals to identify which parts of a jet are likely to be noise. It can effectively isolate a soft, diffuse cloud of background particles from the hard, energetic core of the jet. By removing these noisy parts, the researchers were able to recover the true mass of the particles, even in the most extreme simulated conditions where the background was overwhelming. This suggests that the method can act as a powerful filter, cleaning up the data without throwing away important information.

The researchers also found that the system learns the structure of the jets in a way that matches physical reality. It does not just count particles; it understands the relationships between them. If a jet comes from a heavy particle, the internal clusters will have specific shapes and energy distributions that the algorithm learns to recognize. This allows the system to distinguish between different types of jets, such as those from a W boson versus those from a top quark, based on their internal architecture. The method is flexible enough to adapt to different types of collisions and detector setups, making it a versatile tool for future experiments.

The work presented in this paper is a demonstration of a new way to look at particle collisions. It shows that by letting the data speak for itself through a probabilistic lens, scientists can uncover hidden structures that fixed rules might miss. The method has been tested in simulations and has shown promise in reconstructing jets and filtering out noise. While it is not yet a replacement for all existing tools, it offers a compelling alternative that embraces the complexity of the data rather than trying to simplify it away. As particle detectors become more precise and capable of measuring time with greater accuracy, methods like this one could become essential for unlocking the secrets of the subatomic world, helping physicists see the true shapes of the particles that make up our universe.

Technical Summary: Probabilistic, Structure-Intrinsic Clustering with a Hierarchical Embedding (PSICHE)

Problem Statement
Jet reconstruction in particle physics remains an active area of research facing challenges related to jet size, multiplicity, substructure, and performance in the presence of pileup (PU) and noise. Existing clustering methods, such as the anti- $k_T$ (AK) algorithm, often rely on fixed, a priori parameters (e.g., cone size, energy scales, or multiplicity) and typically treat jet reconstruction, substructure analysis, and pileup mitigation as factorized tasks. Furthermore, standard approaches often lack native mechanisms to incorporate experimental uncertainties, such as timing resolutions, or to learn the scale of clusters dynamically from the data itself.

Methodology
The paper introduces PSICHE (Probabilistic, Structure-Intrinsic Clustering with a Hierarchical Embedding), a modular, fully probabilistic algorithm designed to address these limitations. PSICHE integrates two primary components within a single variational framework:

Variational Gaussian Mixture Model (vGMM): This component models local associations within a self-contained group of observations. It employs multivariate Gaussians to describe clusters, inferring posterior distributions for cluster location, shape, and multiplicity. Crucially, the vGMM is augmented to include observational precisions ( $\Lambda^*_n$ ) and weights ( $\omega_n$ ) for each data point. These terms encode experimental uncertainties (e.g., spatial and temporal resolution) and the relative importance of observations (e.g., energy-dependent weights), allowing the model to handle non-uniform resolutions across clustering dimensions.
Bayesian Hierarchical Clustering (BHC): Multiple vGMM instances are embedded into a tree-based hierarchical structure. This module recursively considers mergers of partitions (agglomerative approach) to learn high-level associations (jets) while simultaneously resolving lower-level substructure (subjets). The algorithm evaluates the probability of merging two partitions against the hypothesis that they remain distinct, utilizing a lower bound (ELBO) on the model evidence.

Key Algorithmic Innovations
To ensure the learning of meaningful structure and prevent overfitting or combinatorial explosion, PSICHE incorporates several specific inductive biases and mechanisms:

Hierarchical Inheritance for Initialization: To resolve the "chicken-and-egg" problem of initialization in variational algorithms, PSICHE uses the posterior parameters of parent vGMMs to initialize child models during the merge process. This regulates subcluster multiplicity, ensuring it does not exceed the sum of its parents' multiplicities unless probabilistically favorable.
Probabilistic Subcluster Regulation: A comparison step evaluates potential subcluster merges to remove redundancies, effectively implementing a Bayesian Occam's Razor to favor parsimonious substructures.
Spatiotemporal Clustering: The algorithm operates in a 4D space (pseudorapidity $\eta$ , azimuthal angle $\phi$ , time $t$ , and energy $E$ ). It utilizes local jet reference frames and coordinate transformations to break probabilistic degeneracies between jets, enforcing projective geometry constraints.
Nearest-Neighbor Constraints: To achieve computational tractability ( $O(N \log N)$ ), the BHC module restricts potential merges to nearest neighbors in a Voronoi tessellation of the spatial dimensions, while time is handled via the probabilistic likelihood rather than the geometric neighbor search.

Results and Demonstrations
The paper demonstrates PSICHE using simulated proton-proton collisions at $\sqrt{s} = 13$ TeV (Run II LHC conditions), comparing performance against standard AK algorithms (with cone sizes $R=0.4, 0.8, 1.5$ ).

Baseline Performance: In simple dijet events without significant substructure, PSICHE reconstructs jet multiplicity and kinematic properties (energy, mass, time) consistent with AK algorithms.
Dynamic Jet Sizing: In simulated top-quark ( $t\bar{t}$ ) events, PSICHE dynamically adjusts jet sizes based on the mass and momentum of the resonance. Unlike AK jets, which are constrained by fixed cone sizes, PSICHE successfully reconstructs both resolved and boosted top quarks, adapting the jet size to the physical scale of the decay products.
Unsupervised Substructure Learning: For boosted $W$ -boson decays (two-prong structure), PSICHE consistently identifies the correct two-subcluster multiplicity without prior specification. In contrast, standard AK jets require post-hoc subclustering which often fails to resolve the correct scale without specific tuning.
Pileup Mitigation: The paper presents a proof-of-concept strategy for pileup mitigation using PSICHE's learned observables. By classifying subclusters into categories (e.g., "Resolved," "Boosted," "PU-like," "PU-affected") based on relative energy, mass, and spatiotemporal variance, the algorithm can identify and remove "PU-like" subclusters (characterized by large spatial variance and low energy). Even under extreme pileup conditions ( $\lambda_{PU} = 140$ ), this method recovers the sharp $W$ -boson mass peak and correct subcluster multiplicity, which are otherwise obscured in standard reconstructions.

Significance and Claims
The authors claim that PSICHE represents a shift from deterministic, parameter-dependent clustering to a self-consistent, fully probabilistic framework that learns jet properties simultaneously. Key claims regarding the algorithm's significance include:

Simultaneous Learning: PSICHE learns jet multiplicity, size, and substructure in a single step, rather than factorizing these tasks.
Intrinsic Handling of Uncertainty: By natively incorporating experimental precisions and weights, the algorithm accounts for detector resolution and noise without external filtering steps.
Time-Awareness: The inclusion of time as a clustering dimension allows for the identification of time-coherent structures and offers new handles for pileup mitigation and Beyond Standard Model (BSM) searches involving long-lived particles.
Domain Adaptability: While demonstrated in particle physics, the modular design suggests applicability to other unsupervised clustering tasks involving hierarchical data with observational uncertainties.

The paper concludes that PSICHE provides a robust, unsupervised method for jet reconstruction that is particularly effective in complex environments with high pileup and diverse kinematic scales, offering a pathway to discover latent structures in particle physics data without relying on fixed, heuristic parameters.

Technical Summary: Probabilistic, Structure-Intrinsic Clustering with a Hierarchical Embedding (PSICHE)

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