The stress-energy tensor of a quark moving through a strongly-coupled N=4 supersymmetric Yang-Mills plasma: comparing hydrodynamics and AdS/CFT

TL;DR

This work computes the perturbation to the stress-energy tensor $\Delta T^{\mu\nu}$ caused by a heavy quark moving through a strongly coupled $\mathcal{N}=4$ SYM plasma using gauge/string duality. By formulating a heavy quark effective theory in the gravity dual and decoupling the linearized Einstein equations into gauge-invariant sectors, the authors reconstruct the boundary stress tensor and compare it to linearized hydrodynamics with an effective source set by the drag force. The main result is that hydrodynamics reproduces the full AdS/CFT wake remarkably well down to distances of order $\sim 2/T$, capturing both the Mach cone for supersonic motion and the diffusion wake in the energy flux. This supports the use of hydrodynamics as a robust framework for modeling energy-momentum transport by high-energy partons in strongly coupled plasmas, with implications for the modeling of quark-gluon plasma in heavy-ion collisions.

Abstract

The stress-energy tensor of a quark moving through a strongly coupled N=4 supersymmetric Yang-Mills plasma is evaluated using gauge/string duality. The accuracy with which the resulting wake, in position space, is reproduced by hydrodynamics is examined. Remarkable agreement is found between hydrodynamics and the complete result down to distances less than 2/T away from the quark. In performing the gravitational analysis, we use a relatively simple formulation of the bulk to boundary problem in which the linearized Einstein field equations are fully decoupled. Our analysis easily generalizes to other sources in the bulk.

Figures (10)

• Figure 1: A cartoon of the string plus D7 brane system in the large mass limit. The D7 brane covers the asymptotically AdS space down to a minimal radial position, away from the string, denoted $u_m$. The trailing string is moving to the right at constant velocity $\bm v$. An energy flux flows down the string toward the black hole horizon, which is located at radial coordinate $u_h$. This energy is supplied by a constant $U(1)$ electric field living on the D7 brane. The D7 brane is deformed in the neighborhood of the endpoint of the string over a length scale of order $1/M$.
• Figure 2: A plot of the window function $w(q)$. The window function does not modify the Fourier space data for $u_h q < 20$.
• Figure 3: Left---Position space plot of $| \bm x| \Delta\mathcal{E}(\bm x)/(T^3 \sqrt{\lambda})$ for $v = 1/4$. Right---Position space plot of $| \bm x| \Delta S(\bm x)/(T^3 \sqrt{\lambda})$ for $v = 1/4$. The flow lines on the surface are the flow lines of the energy flux $\Delta \bm S(\bm x)$. There is a surplus of energy in front of the quark and a deficit behind it. Correspondingly, trailing the quark there is a stream of energy flux which moves in the same direction as the quark. Note the absence of structure in $\Delta\mathcal{E}(\bm x)$ for distances $| \bm x| \gg 1/ (\pi T)$.
• Figure 4: Left---Plot of $| \bm x| \Delta \mathcal{E}(\bm x)/(T^3 \sqrt{\lambda})$ for $v =3/4$. Right---Plot of $| \bm x| \Delta S(\bm x)/(T^3 \sqrt{\lambda})$ for $v = 3/4$. The flow lines on the surface are the flow lines of $\Delta\bm S(\bm x)$. There is a surplus of energy in front of the quark and a deficit behind it. Correspondingly, trailing the quark there is a narrow stream of energy flux which moves in the same direction as the quark. A Mach cone, with opening half angle $\theta_M \approx 50^{\circ}$ is clearly visible in both the energy density and the energy flux. Near the Mach cone, the bulk of the energy flux flow is orthogonal to the wavefront.
• Figure 5: Left---Plot of $| \bm x| \Delta\mathcal{E}(\bm x)/(T^3 \sqrt{\lambda})$ for $v =c_{\rm s}$. Right---Plot of $| \bm x| \Delta S(\bm x)/(T^3 \sqrt{\lambda})$ for $v = c_{\rm s}$. The flow lines on the surface are the flow lines of the energy flux $\Delta\bm S(\bm x)$. A planar Mach cone is visible in both the energy density and the energy flux. Near the Mach cone, the bulk of the energy flux flow is orthogonal to the wavefront.