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No Track left behind: Graph-based Vertexing for long-lived Particle Reconstruction

This paper introduces a graph-based algorithm for reconstructing displaced vertices from long-lived particle decays, implemented as a modular Delphes tool that demonstrates high efficiency and resolution in FCC-ee simulations to enable automated sensitivity projections for exotic Higgs branching fractions.

Original authors: Jonathan Kriewald

Published 2026-08-05

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CC BY 4.0

Original authors: Jonathan Kriewald

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

Imagine the inside of a particle collider as the world's most chaotic, high-speed dance floor. In the center, two beams of particles crash together, creating a burst of energy that spawns a shower of new, fleeting particles. Most of these new dancers spin off and vanish almost instantly, leaving no trace but a tiny flash. But sometimes, a rare, mysterious dancer is born that doesn't want to leave the party right away. These are "long-lived particles." Instead of disappearing in a blink, they travel a noticeable distance—maybe a few millimeters, maybe several meters—before they finally decay and break apart.

Finding these elusive dancers is like trying to spot a specific person in a crowded stadium who suddenly drops their hat and runs away. Physicists call the spot where they break apart a "displaced vertex." It's a crucial clue because these long-lived particles could be the messengers of "new physics," potentially explaining mysteries like why neutrinos have mass or what dark matter is made of. However, finding them is incredibly hard. The detectors are designed to catch the fast, ordinary particles, and the tools used to reconstruct where particles came from often get confused by these slow, wandering ones. If you can't find the vertex, you can't prove the particle existed.

This is where a new study by Jonathan Kriewald comes in. The author has built a clever, "plug-and-play" software tool designed to hunt down these long-lived particles in simulated data. Think of it as a smart, automated detective that doesn't just look for a single clue but connects the dots between many scattered pieces of evidence. The paper introduces a new way to group tracks (the paths left by particles) using a "graph" method—imagine a social network where the algorithm figures out which tracks are friends because they all seem to be heading toward the same secret meeting spot. Once the group is identified, a robust mathematical fitting process pinpoints exactly where they met.

The author tested this new detective tool in a simulated version of a future particle collider called FCC-ee, using a specific scenario where a Higgs boson decays into heavy, long-lived particles. The results are promising: the tool successfully found these displaced vertices with very high efficiency (over 98% in many cases) and with incredible precision, pinpointing the location within micrometers. It worked well whether the particles decayed just a few millimeters away or traveled all the way to the outer edges of the detector. By using this tool, the study projects that the FCC-ee could be sensitive enough to detect these exotic Higgs decays even if they happen very rarely—about one in a hundred thousand times. This isn't a discovery of new particles yet, but it provides a powerful, ready-to-use toolkit that could help real experiments in the future spot these hidden secrets of the universe.

Technical Summary: No Track left behind: Graph-based Vertexing for Long-lived Particle Reconstruction

Problem Statement

Reconstruction of displaced vertices (DVs) is essential for precision flavor physics and searches for Long-Lived Particles (LLPs) at colliders. While existing vertexing algorithms are highly optimized for primary vertices (PV) and short-lived secondary vertices, they face significant limitations when applied to the large displacements and heterogeneous topologies characteristic of LLP decays.

Currently, public fast-simulation frameworks (e.g., Delphes) lack automated pattern recognition for identifying which tracks belong to a common displaced vertex. Consequently, phenomenological studies of LLPs often rely on truth-level objects or simplified parametric smearing, leaving the impact of realistic vertex reconstruction unexplored. Existing robust vertexing tools (e.g., CMSSW, ACTS, RAVE) are either tightly coupled to specific experiments or too computationally heavy for fast simulation chains. There is a critical gap for a "turn-key" tool that integrates automated displaced vertex finding with robust fitting into standard phenomenological workflows.

Methodology

The paper presents a self-contained Delphes module, GraphDisplacedVertexFinder, which combines a graph-based track clustering strategy for vertex finding with a robust vertex fitting procedure.

1. Robust Vertex Fitting

The author provides a unified derivation of vertex fitting within a Gauss-Newton (GN) framework, extending it to include timing information and outlier rejection.

Mathematical Formulation: The fit is formulated as a non-linear least squares problem minimizing the precision-weighted distance between tracks and a common vertex. The author derives the standard transverse-information formulation via Schur complements, explicitly exposing the underlying block structure and geometric interpretation.
4D Extension: While full 4D (space-time) fitting is possible, the author advocates for decoupling spatial and timing components due to the orders-of-magnitude difference in resolution (ps vs. $\mu$ m). A precision-weighted average is used to determine the vertex time after the spatial fit converges.
Outlier Handling: To handle mis-assigned tracks, the algorithm employs an Iteratively Re-weighted Least Squares (IRLS) approach using a Sigmoid weight function. Tracks with high $\chi^2$ contributions are smoothly down-weighted rather than abruptly removed, ensuring stability.
Track Re-fitting: After the vertex converges and outliers are identified, track parameters are re-fitted to the vertex constraint using a Kalman-filter-like update, treating the fitted vertex as a pseudo-measurement.

2. Graph-based Vertex Finding

The core innovation is a graph-based approach to identify sets of tracks belonging to the same displaced vertex:

Track Categorization: Tracks are first classified as "prompt" or "displaced" based on their impact parameter significance ( $S_{IP}$ ) and their compatibility with the primary vertex.
Graph Construction: A scored undirected compatibility graph is built where nodes are displaced tracks. An edge exists between two tracks if a two-track fit yields a $\chi^2_{pair} \leq 9$ and, if available, timing compatibility is satisfied.
Pruning and Validation:
- Pre-filtering: A geometric check ensures track trajectories in the $(x,y)$ plane overlap before attempting a fit.
- Mutual k-NN Pruning: Edges are retained only if they are within mutual $k$ -nearest-neighbor lists to remove spurious connections in dense environments.
- Edge Support: A triplet fit is performed for retained edges. An edge receives a "support score" based on how many additional tracks can form valid triplets with it. This helps disentangle overlapping vertices.
- Bridge Finding: Tarjan's algorithm identifies edges connecting strongly connected sub-graphs; these are retained only if they have low $\chi^2$ and high support, preventing over-fragmentation.
Component Extraction: Connected components are extracted via Depth-First Search (DFS). A heuristic "triplet score" checks for multi-modality; if a component is likely composed of multiple vertices, it is split.
Final Fit: Tracks in the final clusters are fitted independently using the robust vertex fitter.

3. LLP Kinematic Reconstruction

A companion module, LLPReconstruction, associates neutral energy (jets) with displaced vertices using a momentum-weighted overlap score. This allows for the reconstruction of the LLP four-momentum and, subsequently, the parent particle (e.g., the Higgs boson) using the "corrected mass" formalism to account for invisible energy.

Key Contributions

First Automated Displaced Vertex Finder for Delphes: The work provides the first self-contained, plug-and-play Delphes module for automated displaced vertex finding, filling a gap in public fast-simulation frameworks.
Unified Fitting Framework: It offers a streamlined, unified derivation of vertex fitting that recovers standard transverse-information results while explicitly exposing geometric interpretations and Schur-complement representations, facilitating future extensions (e.g., timing).
Graph-Based Clustering: The introduction of a graph-based strategy with specific pruning techniques (mutual k-NN, edge support, bridge finding) tailored to the heterogeneous topologies of LLP decays.
Validation in FCC-ee Context: The algorithm is validated in an IDEA-like FCC-ee detector configuration using Higgs-strahlung ( $e^+e^- \to Zh$ ) with exotic $h \to NN$ decays (where $N$ are Heavy Neutral Leptons) as a benchmark.

Results

The algorithm was tested on samples with proper lifetimes $c\tau \in [10, 100, 1000]$ mm.

Efficiency: The reconstruction efficiency is excellent, reaching $>98\%$ for transverse displacements $L_{xy} \gtrsim 50$ mm. For smaller displacements, efficiency ranges from 95% to 97%. Efficiency drops to $\sim70\%$ for very large displacements ( $L_{xy} \ge 2000$ mm) due to tracking volume limits.
Purity: Vertex purity is very high ( $\ge 99\%$ ) for vertices with more than 5 tracks. For 3-track vertices, purity is 80–85%, primarily due to heavy-flavor decays, but can be suppressed via kinematic cuts.
Resolution:
- Inner Detector: Spatial resolution is $\le 2$ $\mu$ m.
- Outer Vertex Detector: Resolution degrades to 5–12 $\mu$ m (transverse) and $\sim 5$ $\mu$ m (longitudinal).
- Drift Chamber: Resolution is 20–70 $\mu$ m (transverse) and 100–200 $\mu$ m (longitudinal).
- The "predicted" uncertainties from the fit covariance matrix agree well with "measured" residuals (within 2–10%).
Kinematic Reconstruction: The reconstructed invariant mass of the HNLs and the parent Higgs boson shows excellent agreement with truth values. The "corrected mass" effectively recovers the true mass distribution, allowing for efficient signal/background separation.
Sensitivity Projections: Using the reconstructed vertices, the author projects FCC-ee sensitivity to exotic Higgs branching fractions. They estimate a 95% CL upper limit of $BR(h \to NN) \lesssim (6-7) \times 10^{-6}$ for proper lifetimes of 10–1000 mm and masses of 20–60 GeV.

Significance and Claims

The paper claims that this work bridges the gap between phenomenological studies and experimental analysis by providing a flexible, realistic tool for LLP searches.

Accessibility: By integrating into the standard MadGraph–Pythia–Delphes chain, it enables fast and realistic LLP analyses without the computational overhead of full offline reconstruction frameworks.
Physics Reach: The demonstrated efficiency and resolution allow for model-independent projections of sensitivity to exotic Higgs decays, potentially constraining mixing in the SM Higgs sector with BSM scalars and the mass scales of mediators in models like the Left-Right Symmetric Model (LRSM).
Generalizability: While validated on FCC-ee, the author states the approach is readily applicable to other collider scenarios, including CEPC, CLIC, muon colliders, and hadron colliders.

The author emphasizes that the tool is "turn-key" and "plug-and-play," designed to be immediately usable by the phenomenology community to improve the realism of LLP sensitivity projections.