Anomalous dimension of transverse momentum broadening in planar $\mathcal{N}=4$ SYM

The typical transverse momentum $Q_s(t)$ (or “saturation” momentum) acquired by a hard particle propagating through a $\mathcal{N}=4$ SYM plasma increases over time like $t^{\gamma }$, with an anomalous exponent $\gamma >1\mathrm{/}2$ characteristic of super-diffusion. This anomalous exponent is a function of the 't Hooft coupling $\lambda =g^2{N}_c$. Recently, a method has been proposed to systematically compute the perturbative series of $\gamma (\lambda )$ at weak coupling. This method relies on the traveling wave interpretation of the time evolution of $Q_s(t)$ and on the dominance of soft-collinear radiative corrections at large times. In this paper, we compute $\gamma (\lambda )$ up to $\mathcal{O}({\lambda }^2)$ using the double logarithmic behaviour of the BFKL equation in planar $\mathcal{N}=4$ SYM at three loops. This calculation allows us to discuss the transition towards the strong coupling regime where AdS/CFT calculations predict $\gamma \to 1$.