  The parton branching process is known to be modified in the presence of a
medium. Colour decoherence processes are known to determine the process of
energy loss when the density of the medium is large enough to break the
correlations between partons emitted from the same parent. In order to improve
existing calculations that consider eikonal trajectories for both the emitter
and the hardest emitted parton, we provide in this work, the calculation of all
finite energy corrections for the gluon radiation off a quark in a QCD medium
that exist in the small angle approximation and for static scattering centres.
Using the path integral formalism, all particles are allowed to undergo
Brownian motion in the transverse plane and the offspring allowed to carry an
arbitrary fraction of the initial energy. The result is a general expression
that contains both coherence and decoherence regimes that are controlled by the
density of the medium and by the amount of broadening that each parton acquires
independently.
