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The CREMA collaboration is pursuing a measurement of the ground-statehyperfine splitting (HFS) in muonic hydrogen ($\mu$p) with 1 ppm accuracy bymeans of pulsed laser spectroscopy. In the proposed experiment, the $\mu$p atomis excited by a laser pulse from the singlet to the triplet hyperfinesub-levels, and is quenched back to the singlet state by an inelastic collisionwith a H$_2$ molecule. The resulting increase of kinetic energy after thiscycle modifies the $\mu$p atom diffusion in the hydrogen gas and the arrivaltime of the $\mu$p atoms at the target walls. This laser-induced modificationof the arrival times is used to expose the atomic transition. In this paper wepresent the simulation of the $\mu$p diffusion in the H$_2$ gas which is at thecore of the experimental scheme. These simulations have been implemented withthe Geant4 framework by introducing various low-energy processes including themotion of the H$_2$ molecules, i.e. the effects related with the hydrogentarget temperature. The simulations have been used to optimize the hydrogentarget parameters (pressure, temperatures and thickness) and to estimate signaland background rates. These rates allow to estimate the maximum time needed tofind the resonance and the statistical accuracy of the spectroscopy experiment.

Diffusion of muonic hydrogen in hydrogen gas and the measurement of the 1$s$ hyperfine splitting of muonic hydrogen