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 $\gamma>1/2$ characteristic of super-diffusion. This anomalous exponent is a function of the 't Hooft coupling $\lambda=g^2N_c$. 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 $\gamma\to 1$.