Multiple Soft Scatterings in Scalar Dark Matter Freeze-In

We present an improved calculation of the freeze-in production rate for scalar dark matter (DM) from a gauge-charged parent particle via a renormalizable interaction. Building on the previously developed 1PI-resummed framework to accurately capture the relevant regime $T \sim M$, we expand the analysis to include the Landau-Pomeranchuk-Migdal (LPM) effect, which contributes at leading order $g^2 T$ to the interaction rate in the ultra-relativistic limit. To this end, we derive an equation for the LPM rate of a scalar particle for the first time and combine it with the previous 1PI results, providing a new state-of-the art calculation. In contrast to the 1PI results, the LPM treatment neglects vacuum mass scales such that a phenomenological switch-off function between the ultra-relativistic and non-relativistic regime is required. We propose a new function motivated by a thermal loop contribution and compare it to other approaches in the literature, quantifying the resulting uncertainty of this method. Depending on the gauge coupling and mass splitting between DM and mediator particles, the LPM effect contributes between 1% and 27% to the relic density, with the impact increasing for larger gauge couplings and smaller mass splittings. Additionally, we compare our results to commonly used semi-classical Boltzmann approaches. For instance, when these include decays and scatterings regulated with thermal masses, we find deviations ranging from -30% to +20% depending on the mass splitting. Finally, we compare to results based on hard-thermal-loop (HTL) approximations.