  We address a recent puzzling result from the LHC: the jet fragmentation
functions measured in $PbPb$ and $pp$ collisions appear very similar in spite
of a large medium-induced energy loss (we will call this "jet fragmentation
scaling", JFS). To model the real-time non-perturbative effects in the
propagation of a high energy jet through the strongly coupled QCD matter, we
adopt an effective dimensionally reduced description in terms of the $(1+1)$
quasi-Abelian Schwinger theory. This theory is exactly soluble at any value of
the coupling and shares with QCD the properties of dynamical generation of
"mesons" with a finite mass and the screening of "quark" charge that are
crucial for describing the transition of the jet into hadrons. We find that
this approach describes quite well the vacuum jet fragmentation in $e^+e^-$
annihilation at $z\geq0.2$ at jet energies in the range of the LHC heavy ion
measurements ($z$ is the ratio of hadron and jet momenta). In QCD medium, we
find that the JFS is reproduced if the mean free path $\lambda$ of the jet is
short, $\lambda \leq 0.3$ fm, which is in accord with the small shear viscosity
inferred from the measurements of the collective flow. The JFS holds since at
short mean free path the quantum interference (analogous to the
Landau-Pomeranchuk-Migdal effect in QED) causes the produced mesons to have low
momenta $p \sim m$, where $m \simeq 0.6$ GeV is the typical meson mass.
Meanwhile the induced jet energy loss at short mean free path is much larger
than naively expected in string models.
