  We revisit the picture of jets propagating in the quark-gluon plasma. In
addition to vacuum radiation, partons scatter on the medium constituents
resulting in induced emissions. Analytical approaches to including these
interactions have traditionally dealt separately with multiple, soft, or rare,
hard scatterings. A full description has so far only been available using
numerical methods. We achieve full analytical control of the relevant scales
and map out the dominant physical processes in the full phase space. To this
aim, we extend existing expansion schemes for the medium-induced spectrum to
the Bethe--Heitler regime. This covers the whole phase space from early to late
times, and from hard splittings to emissions below the thermal scale. Based on
the separation of scales, a space-time picture naturally emerges: at early
times, induced emissions start to build from rare scatterings with the medium.
At a later stage, induced emissions due to multiple soft scatterings result in
a turbulent cascade that rapidly degrades energy down to, and including, the
Bethe--Heitler regime. We quantify the impact of such an improved picture,
compared to the current state-of-the-art factorization that includes only soft
scatterings, by both analytical and numerical methods for the medium-induced
energy distribution function. Our work serves to improve our understanding of
jet quenching from small to large systems and for future upgrades of Monte
Carlo generators.
