Medium-induced coherent gluon radiation for $2\to 2$ processes with general kinematics

High-energy proton-nucleus (pA) collisions offer valuable insight into the role of cold nuclear matter effects in hadron production. In particular, multiple rescatterings of an incoming parton by the nuclear target are known to induce the radiation of many soft gluons. Hadron production is affected by fully coherent energy loss (FCEL), owing to those radiated gluons with a long formation time. Here we present a recently derived formula for the induced single soft gluon radiation spectrum beyond leading logarithmic accuracy, whose main features are demonstrated with the example of $qg\to qg$ scattering.