Universality (beyond leading log) of soft radiative corrections to $$ \hat{q} $$ in p? broadening and energy loss

It has been known for many years that soft radiation can give potentially large double-logarithm corrections to $p_\perp$ broadening of a high-energy particle traveling through QCD matter, but this correction be absorbed into an effective value $\hat q_{\rm eff}$ the medium $p_\perp$-broadening parameter q$. Here "soft" means high energy compared scales original medium. A similar situation arises in case hard splitting particle, such as $g{\to}gg$, where double logarithms also using same eff}$. In paper, I study whether holds true large, subleading, *single*-logarthim corrections. The correspondence is more indirect single logarithms, show (in large-$N_{\rm c}$ limit) from determine radiative $g{\to}gg$ splitting. Along way, there interesting variation BDMPS-Z calculation rates q$ approximation. discuss how, soft-radiative processes, are two different types "$\hat eff}$" come play, which differ by "$i\pi$" terms multiply logarithms.