Dynamics of Heavy Quarks in Strongly Coupled $\mathcal{N}=4$ SYM Plasma

TL;DR

This paper calculates the full probability distribution for heavy quark momentum changes in strongly coupled $ ext{N}=4$ SYM plasma, revealing non-Gaussian features and deriving a comprehensive evolution equation that describes quark thermalization.

Contribution

It provides the first calculation of all higher order moments of heavy quark momentum distribution in strongly coupled plasma, including non-Gaussian correlations, and derives a complete evolution equation beyond Fokker-Planck.

Findings

Non-Gaussian features can dominate in realistic scenarios.

Derived a Kolmogorov equation capturing all higher order correlations.

Heavy quarks reach kinetic equilibrium with the full evolution equation.

Abstract

We calculate the probability distribution $P({\bf k})$ for a heavy quark with velocity $v$ propagating through strongly coupled $\mathcal{N}=4$ SYM plasma in the 't Hooft limit at a temperature $T$ to acquire a momentum ${\bf k}$ due to interactions with the plasma. This distribution encodes the well-known drag coefficient $\eta_D$ and the transverse and longitudinal momentum diffusion coefficients $\kappa_T$ and $\kappa_L$. Furthermore, our calculation determines all of the higher order and mixed moments to leading order in $1/\sqrt{\lambda}$ for the first time. These non-Gaussian features of $P({\bf k})$ include qualitatively novel correlations between longitudinal energy loss and transverse momentum broadening at nonzero $v$. We demonstrate that these non-Gaussian characteristics can be sizable in magnitude and even dominant in physically relevant situations. We use these results to derive a Kolmogorov equation for the evolution of the probability distribution for the total momentum of a heavy quark that propagates through strongly coupled plasma. This evolution equation accounts for all higher order correlations between transverse momentum broadening and longitudinal energy loss, which we have calculated from first principles. It reduces to a Fokker-Planck (FP) equation when truncated to only include the effects of $\eta_D$, $\kappa_T$ and $\kappa_L$. Remarkably, while heavy quarks do not reach kinetic equilibrium with the plasma if evolved with this FP equation, we demonstrate that heavy quarks do reach kinetic equilibrium if evolved with the all-order Kolmogorov equation we have derived. Our results thus provide a dynamically complete framework for understanding the thermalization of a heavy quark that may be initially far from equilibrium in the strongly coupled $\mathcal{N}=4$ SYM plasma -- as well as new insight into heavy quark transport and equilibration in quark-gluon plasma.