Simulating the Smallest QED Atom: Lightweight True Muonium Generator For Three-Photon and Radiative Production

True muonium (TM), the bound state of a muon and antimuon, offers a uniquely clean probe of QED and a benchmark channel for displaced e+e− signatures in fixed-target experiments such as the Heavy Photon Search experiment at Jefferson laboratory. We present a lightweight, detector-agnostic Monte Carlo generator that models TM production in two complementary modes—three-photon (3γ) fusion and radiative (bremsstrahlung-like) production—at beam energies around 3.74 GeV and heavy-Z targets. For each mode we implement an analytically motivated differential spectrum in the lab-frame energy fraction x≡ETM​/Ebeam​. Unweighted samples are drawn via inverse-CDF sampling, ensuring numerical stability and excellent agreement with the theoretical shapes. TM decays are simulated as isotropic two-body decays TM→e+e− in the rest frame; daughter four-vectors are boosted to the lab using a vectorized Lorentz transform. A simple, physics-inspired production-angle model captures the expected forward collimation at large x. The generator writes standard Les Houches Event (LHE) files, enabling immediate downstream studies with existing simulation chains in HPS software.

We validate the implementation with quality-assurance plots: sampled x spectra versus normalized theory, cumulative-distribution checks, and inclusive lepton-angle distributions. Initial generator-level studies highlight clear, experimentally relevant contrasts between production modes: the radiative mechanism favors higher x and narrower lepton opening angles, while the 3γ mode populates broader kinematics. Hit-map projections at a representative detector plane show compact, central impact patterns consistent with forward production. The framework cleanly separates physics modeling from detector effects and is designed for fast iteration.

Planned extensions include material-aware absolute normalization (parameters B and Λ ∝ A^-1/3), explicit TM state handling with lifetimes and branching fractions, and optional ROOT/HDF5 outputs for rapid analysis. By providing an open, reproducible tool that bridges analytic cross-sections to LHE, this work equips upcoming fixed-target searches with realistic signal kinematics for optimization, sensitivity estimates, and cross-checks.