Transverse momentum broadening and collinear radiation at NLO in the $\mathcal{N}=4$ SYM plasma

We compute O(g) NLO corrections to the transverse scattering kernel and transverse momentum broadening coefficient $\widehat{q}$ of weakly-coupled $\mathcal{N}=4$ SYM. Based on this, we also compute NLO correction to the collinear splitting rates. For $\widehat{q}$ we find that the NLO/LO ratio is similar to the QCD one, with large NLO corrections. This is contrasted by our findings for the collinear splitting rate, which show a much better convergence in SYM than in QCD, providing further support to earlier expectations that NLO corrections have signs and relative magnitudes controlled by the specifics of the theory. We also compare the ratio of $\widehat{q}$ in QCD and in $\mathcal{N}=4$ theory to strong coupling expectations.