Heavy Quark Diffusion in Strongly Coupled $\N=4$ Yang Mills

TL;DR

The paper derives a nonperturbative expression for heavy quark diffusion in a strongly coupled gauge theory by relating contour-ordered Wilson line fluctuations to heavy-quark force correlators. Using the AdS/CFT correspondence, the authors compute the diffusion coefficient in $\mathcal{N}=4$ SYM from fluctuations of a string in $AdS_5 \times S^5$ that spans the Kruskal plane, obtaining $\kappa = \pi \sqrt{\lambda} \; T^3$ and $D = 2/(\pi T \sqrt{\lambda})$. This result shows diffusion is suppressed by $1/\sqrt{\lambda}$ relative to momentum diffusion and provides a nonperturbative benchmark for heavy quark transport in strongly coupled plasmas, with validity requiring $M \gg T\sqrt{\lambda}$. The framework links real-time transport, Schwinger-Keldysh formalism, and holographic string dynamics, and suggests directions for finite-velocity generalizations and QCD-like extensions.

Abstract

We express the heavy quark diffusion coefficient as the temporal variation of a Wilson line along the Schwinger-Keldysh contour. This generalizes the classical formula for diffusion as a force-force correlator to a non-abelian theory. We use this formula to compute the diffusion coefficient in strongly coupled $\N=4$ Yang-Mills by studying the fluctuations of a string in $AdS_5\times S_5$. The string solution spans the full Kruskal plane and gives access to contour correlations. The diffusion coefficient is $D=2/\sqrtλ πT$ and is therefore parametrically smaller than momentum diffusion, $η/(e+p)=1/4πT$. The quark mass must be much greater than $T\sqrtλ$ in order to treat the quark as a heavy quasi-particle. The result is discussed in the context of the RHIC experiments.

Figures (2)

• Figure 1: The Schwinger-Keldysh contour (with $T\rightarrow \infty$). Fields evaluated along the real axes are labeled as type 1, while fields evaluated on the $-i\beta/2$ axis are labeled as type 2. The crosses indicate the insertion points of the color singlet force operator $\mathcal{F}$ (Eq. \ref{['operator']}). After integrating out the heavy quark, the crosses indicate insertions of the electric field, and the dotted lines indicate the path of the corresponding links. The electric field insertion may be rewritten as a variation (at times $0$ and $t$) of the Wilson line running along the whole contour.
• Figure 2: Kruskal diagram for the AdS black hole. The coordinates $(t,r)$ span the right (R) quadrant. The dotted lines and the dashed hyperbolas represent the future and past horizons and the singularities, respectively. The thick hyperbolas on the sides of the two quadrants are the boundaries at $r=\infty$. The Wilson line running along the Schwinger-Keldysh contour runs along along the "1" axis (the R boundary) and "2" axis (the L boundary). This corresponds to a string whose endpoints follow these boundaries. The minimal surface with these boundary conditions is the full Kruskal plane.