  In this Letter we report the first calculation of all ${\cal O}(\alpha_s)$
medium-induced branching processes to any order in opacity. Our splitting
functions results are presented as iterative solutions to matrix equations with
initial conditions set by the leading order branchings in the vacuum. The
flavor and quark mass dependence of the in-medium $q \rightarrow qg$,
$g\rightarrow gg$, $q \rightarrow g q$, $g \rightarrow q\bar{q}$ processes is
fully captured by the light-front wavefunction formalism and the color
representation of the parent and daughter partons. We include the explicit
solutions to second order in opacity as supplementary material and present
numerical results in a realistic strongly-interacting medium produced in high
center-of-mass energy heavy ion collisions at the Large Hadron Collider. Our
numerical simulations show that the second order in opacity corrections can
change the energy dependence of the in-medium shower intensity. We further find
corrections to the longitudinal and angular distributions of the in-medium
splitting kernels that may have important implications for jet substructure
phenomenology.
