Improved opacity expansion for medium-induced parton splitting

We present a new expansion scheme to compute the rate for parton splittings in dense and finite QCD media. In contrast to the standard opacity expansion, our expansion is performed around the harmonic oscillator whose characteristic frequency depends on the typical transverse momentum scale generated in the splitting. The first two orders account for the high frequency regime that is dominated by single hard scatterings together with the regime of multiple soft scatterings at low frequency. This work generalizes the findings of Ref. \cite{Mehtar-Tani:2019tvy} beyond the leading logarithmic approximation allowing to account also for the Bethe-Heitler regime and compare to the full numerical results from Ref. \cite{CaronHuot:2010bp}. We investigate the sensitivity of our results to varying the separation scale that defines the leading order. Finally, the application to Monte Carlo event generators is discussed.