A long-lived particle (LLP) is a beyond-Standard-Model state with a macroscopic lifetime: instead of decaying at the collision point like ordinary resonances, it travels a measurable distance before decaying, leaving a displaced vertex — the smoking-gun signature searched for at ATLAS, CMS and LHCb.
Decay is a Poisson process in the particle's own rest frame, so the probability of surviving past proper time τ is exponential. Boosting to the lab frame with Lorentz factor γ and velocity β gives a mean lab-frame decay length:
λ = γβ·c·τ
P(decay beyond radius r) = exp(−r/λ)
Sampled decay length: L = −λ·ln(U), U ~ Uniform(0,1)
A vertex is only reconstructible if it decays outside the innermost pixel layers (too close and the secondary vertex is indistinguishable from the primary one — the "veto radius" here) but inside the outer tracker (too far and the decay products leave no hits at all). The detection efficiency is the fraction of the exponential distribution caught between those two radii:
ε = exp(−r_veto/λ) − exp(−r_tracker/λ)
This 2D build renders the detector as a plan view (top-down cross-section of the cylindrically-symmetric tracker) so the same isotropic 3D physics — a random flight direction and an exponential proper-time draw — projects cleanly onto a flat radial picture. The lower panel plots the sampled decay-length histogram against the analytic exponential PDF, and the analytic efficiency curve ε(cτ) with a live marker, so the Monte Carlo run and the closed-form theory can be checked against each other in real time. (Verified numerically against a 2M-sample Monte Carlo run before shipping — analytic and simulated efficiency agree to within 0.1 percentage points.)
- cτ slider — the particle's intrinsic proper lifetime, in millimetres of c·τ (typical LLP benchmarks range from sub-mm to metres).
- γβ slider — the Lorentz boost from the production energy; higher-energy colliders push LLPs further before they decay.
- Veto radius — the minimum displacement a vertex-finding algorithm can resolve from the primary interaction point.
- Tracker outer radius — how far the fiducial tracking volume extends; a decay beyond it leaves no hits at all.
- Every launched particle is a genuine exponential-random-variable draw — the live efficiency number is a running Monte Carlo estimate, not a lookup.
- Drag the main view to pan, scroll/pinch to zoom — the detector geometry redraws at any scale.