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

TL;DR

This paper calculates the anomalous diffusion exponent for transverse momentum broadening in planar al N=4 SYM, showing its dependence on the coupling and exploring the transition from weak to strong coupling regimes.

Contribution

It provides a systematic perturbative calculation of the anomalous exponent b3(7) up to second order in coupling using BFKL equation insights.

Findings

b3(7) increases with coupling 7.

At weak coupling, b3(7) is computed up to b3 b2.

The transition to strong coupling b3 b1 is discussed.

Abstract

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 $\gamma\to 1$.