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

Reconstruction of displaced vertices is a cornerstone of both precision flavour physics and searches for long-lived particles (LLPs) at colliders. While existing vertexing algorithms are highly optimised for primary and short-lived secondary vertices, they face limitations when confronted with the large displacements and heterogeneous topologies characteristic of LLP decays. In this work we present a new approach to displaced vertex reconstruction combining an initialisation-free robust vertex fitter with a graph-based track clustering strategy. The algorithm is implemented as a self-contained Delphes module and can be straightforwardly integrated into existing detector cards, providing a turn-key tool for phenomenological studies. This plug-and-play functionality fills a gap left by public software frameworks, which either lack displaced-vertex capabilities or are not readily usable in fast-simulation environments. We validate our approach in an IDEA-like FCC-ee detector, using Higgs-strahlung $e^+e^- \to Zh$ with exotic $h\to NN$ decays as a benchmark process. We demonstrate excellent efficiency, resolution, and purity across a broad range of lifetimes, and derive model-independent projections for the FCC-ee sensitivity to exotic Higgs branching fractions.