The stress tensor of a quark moving through N=4 thermal plasma

TL;DR

The paper develops a complete framework for computing the boundary stress tensor ⟨T_{mn}⟩ in N=4 SYM at finite temperature from a trailing string in AdS5-Schwarzschild, solving linearized graviton equations in a decoupled, gauge-fixed setup. It provides analytic near-boundary and near-horizon results and obtains large- and small-K expansions that isolate near-field and hydrodynamic responses, complemented by numerical solutions that reveal directional wakes and Mach-cone features in Fourier space. The work offers a holographic perspective on energy dissipation from a moving heavy quark and discusses potential qualitative connections to jet quenching in heavy-ion collisions, while carefully addressing normalization, conservation, and limitations of the AdS/CFT framework. Overall, it delivers a detailed, technically rigorous account of how a moving quark imprints a dissipative wake in a strongly coupled, thermal gauge theory plasma.

Abstract

We develop the linear equations that describe graviton perturbations of AdS_5-Schwarzschild generated by a string trailing behind an external quark moving with constant velocity. Solving these equations allows us to evaluate the stress tensor in the boundary gauge theory. Components of the stress tensor exhibit directional structures in Fourier space at both large and small momentum. We comment on the possible relevance of our results to relativistic heavy ion collisions.

Figures (5)

• Figure 1: The $AdS_5$-Schwarzschild background is part of the near-extremal D3-brane, which encodes a thermal state of ${\cal N}=4$ supersymmetric gauge theory Gubser:1996de. The external quark trails a string into the five-dimensional bulk, representing color fields sourced by the quark's fundamental charge and interacting with the thermal medium.
• Figure 2: Contour plots of $Q^K_A$, $Q^K_D$, and $Q^K_E$ for $v=0.95$. The darker regions are more positive. All components of $\langle T^K_{mn} \rangle$ can be deduced from $Q^K_A$, $Q^K_D$, and $Q^K_E$ using (\ref{['QtoQ']}), (\ref{['GotTmn']}), and (\ref{['QXdef']}). All three $Q^K_X$ go to zero at large $K$. The momentum vector $\vec{K} = \vec{k}/\pi T$ can be read in ${\rm GeV}/c$ if one chooses $T = 318\,{\rm MeV}$: see (\ref{['SetT']}). The range of momenta in each plot was chosen to show the most distinctive structures. The region outlined in gold in (i) is plotted in more detail in figure \ref{['fig:LobeFig']}c.
• Figure 3: Contour plots of $K_\perp |Q^K_E|$ for various values of $v$. $Q^K_E$ is proportional to the $K$-th Fourier component of the energy density after a near-field subtraction: see (\ref{['QtoQ']}), (\ref{['GotTmn']}), and (\ref{['QXdef']}). The phase space factor $K_\perp$ arises in Fourier transforming back to position space. The green line shows the Mach angle. The red curve shows where $K_\perp |Q^K_E|$ is maximized for fixed $K = \sqrt{K_1^2+K_\perp^2}$. The blue curves show where $K_\perp |Q^K_E|$ takes on half its maximum value for fixed $K$.
• Figure 4: Contour plots of $K_\perp |Q^K_E|$ for various values of $v$ at low momenta. The green line shows the Mach angle. The red curve shows where $K_\perp |Q^K_E|$ is maximized for fixed $K = \sqrt{K_1^2+K_\perp^2}$. The blue curves show where $K_\perp |Q^K_E|$ takes on half its maximum value for fixed $K$.
• Figure 5: $K_\perp |Q^K_E|$ at fixed $K = \sqrt{K_1^2+K_\perp^2}$ as a function of angle, for $v=0.95$ and for various values of $K$. To facilitate comparison with di-jet hadron pair correlations, we have parameterized the angle as $\Delta\phi = \pi-\theta$, where $\theta=\tan^{-1} K_\perp/K_1$. With the usual assignment $T = 318\,{\rm MeV}$ (see (\ref{['SetT']})), $K$ can be read in units of ${\rm GeV}/c$. In each plot, the solid curve is from numerics; the dashed curve is the analytical approximation (\ref{['QEseries']}); the green line indicates the Mach angle; the red dot is at the maximum of $K_\perp |Q^K_E|$; and the blue dots indicate the points where $K_\perp |Q^K_E|$ is half of its peak value.