  When an energetic parton propagates in a hot and dense QCD medium it loses
energy by elastic scatterings or by medium-induced gluon radiation. The gluon
radiation spectrum is suppressed at high frequency due to the LPM effect and
encompasses two regimes that are known analytically: at high frequencies
$\omega >\omega_c = \hat q L^2$, where $\hat q $ is the jet quenching transport
coefficient and $L$ the length of the medium, the spectrum is dominated by a
single hard scattering, whereas the regime $\omega <\omega_c$ is dominated by
multiple low momentum transfers. In this paper, we extend a recent approach
(dubbed the Improved Opacity Expansion (IOE)), which allows an analytic (and
systematic) treatment beyond the multiple soft scattering approximation,
matching this result with the single hard emission spectrum. We calculate in
particular the NNLO correction analytically and numerically and show that it is
strongly suppressed compared to the NLO indicating a fast convergence of the
IOE scheme and thus, we conclude that it is sufficient to truncate the series
at NLO. We also propose a prescription to compare the GW and the HTL potentials
and relate their parameters for future phenomenological works.
