Improved opacity expansion for medium-induced parton splitting

We revisit the calculation of the medium-induced gluon radiative spectrum and propose a novel expansion scheme that encompasses the two known analytic limits: i) the high frequency regime dominated by a single hard scattering that corresponds to the leading order in the standard opacity expansion, ii) the low frequency regime that is dominated by multiple soft scatterings. Our approach is based on expanding around the harmonic oscillator instead of vacuum in the leading logarithmic approximation. We compute the first two orders in this improved opacity expansion and show that they account for the aforementioned limits.