  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.
