Energy disturbances due to a moving quark from gauge-string duality

TL;DR

This work computes the real-space energy density ${\cal E}(\vec{X})$ around a heavy, infinitely massive quark moving through a thermal ${\cal N}=4$ SYM plasma using the AdS/CFT trailing-string setup. By Fourier transforming momentum-space data and employing a UV/IR decomposition with a Coulomb near-field subtraction, the authors obtain a comprehensive picture that spans from the Coulombic near-field to a Mach-cone–dominant far-field, including nontrivial small-scale structure. The results interpolate between analytic near-field corrections and linearized hydrodynamics at large distances, providing gauge-invariant insights into energy deposition and potential seeds for hydrodynamic simulations of the quark-gluon plasma, while recognizing limitations due to the conformal nature of ${\cal N}=4$ SYM. Overall, the paper demonstrates how AdS/CFT methods yield a coherent, multi-scale description of energy disturbances induced by a moving probe in a strongly coupled thermal medium.

Abstract

Using AdS/CFT, we calculate the energy density of a quark moving through a thermal state of N=4 super-Yang-Mills theory. Relying on previous work for momentum-space representations as well as asymptotic behaviors, we Fourier transform to position space and exhibit a sonic boom at a speed larger than the speed of sound. Nontrivial structure is found at small length scales, confirming earlier analytical work by the authors.

Figures (5)

• Figure 1: (Color online.) $E \equiv {\cal E}-{\cal E}_{\rm Coulomb}$ as a function of dimensionless position $X_1$ for various values of offset $X_p = X_\perp$ from the quark and velocity $v$ of the quark. In each plot, the black curve is $E$. The red curve is ${\cal E}_{\rm IR}$. The green curve is ${\cal E}_{\rm res}$ with $C^1$ smoothing (see the main text). The blue curve is ${\cal E}_{\rm UV}-{\cal E}_{\rm Coulomb}$. The dotted purple curve is from inviscid linearized hydrodynamics (see the discussion in section \ref{['CONCLUSIONS']}). The green dot corresponds to the Mach angle. The blue dots show a width typical of hydrodynamical broadening (see the main text). The red dots show the points where the smoothing of ${\cal E}_{\rm res}$ starts. The quark is at $X_1 = X_\perp = 0$.
• Figure 2: (Color online.) An estimate of hydrodynamical broadening of the Mach cone. The gray line, offset by a distance $x_p = x_\perp$ from the quark's trajectory, is the axis along which we plot $E \equiv {\cal E}-{\cal E}_{\rm Coulomb}$ in figure \ref{['ComparisonColumns']}. The green dot in that figure is at the intersection of the gray and green lines in this one; likewise the blue dots are at intersections of the gray line with the blue lines.
• Figure 3: (Color online.) Contour plots of $E \equiv {\cal E}-{\cal E}_{\rm Coulomb}$ as a function of dimensionless position coordinates $X_1$ and $X_p=X_\perp$, for $v=0.25$. Note that the top plot shows the smallest structures while the bottom plot shows the largest. Orange and red regions correspond to $E>0$; white regions correspond to $E\approx 0$; and blue regions correspond to $E < 0$. The energy density of the thermal bath is not included in $E$. The three-dimensional energy density profile is axially symmetric around the $X_1$ axis. The black dot is the position of the quark: $X_1=X_\perp = 0$.
• Figure 4: (Color online.) Contour plots of $E \equiv {\cal E}-{\cal E}_{\rm Coulomb}$ as a function of dimensionless position coordinates $X_1$ and $X_p=X_\perp$, for $v=0.58$. Note that the top plot shows the smallest structures while the bottom plot shows the largest. Orange and red regions correspond to $E>0$; white regions correspond to $E\approx 0$; and blue regions correspond to $E < 0$. The energy density of the thermal bath is not included in $E$. The three-dimensional energy density profile is axially symmetric around the $X_1$ axis. The black dot is the position of the quark: $X_1=X_\perp = 0$. The dashed green line shows the Mach cone.
• Figure 5: (Color online.) Contour plots of $E \equiv {\cal E}-{\cal E}_{\rm Coulomb}$ as a function of dimensionless position coordinates $X_1$ and $X_p=X_\perp$, for $v=0.75$. Note that the top plot shows the smallest structures while the bottom plot shows the largest. Orange and red regions correspond to $E>0$; white regions correspond to $E\approx 0$; and blue regions correspond to $E < 0$. The energy density of the thermal bath is not included in $E$. The three-dimensional energy density profile is axially symmetric around the $X_1$ axis. The black dot is the position of the quark: $X_1=X_\perp = 0$. The dashed green line shows the Mach cone.