The high momentum behavior of a quark wake

TL;DR

The paper addresses the high-momentum behavior of the color wake generated by a heavy quark moving through a thermal $\mathcal{N}=4$ SYM plasma. It uses the AdS/CFT correspondence, modeling the quark as a string in AdS-Schwarzschild, and employs a WKB analysis to solve the dilaton fluctuation in the large-momentum regime, obtaining an analytic expression for the dissipative part of $\langle \widehat{\mathcal{O}_{\Phi}}(k_-,k_{\perp}) \rangle$. The result reveals a fireball-like near-field region and a directional ridge in momentum space, with a ridge angle given by $\tan^2\theta = v^2 - 1 + v^2/\alpha^2$ (and $\alpha = v k_- / \tilde{k}$, $\tilde{k}^2 = k^2 - v^2 k_-^2$), plus a leading zero-temperature-like term $-\frac{\sqrt{\lambda_{YM}(1-v^2)}}{16}\tilde{k}$ in the dissipative structure. The analysis connects to jet-quenching phenomenology in heavy-ion collisions, while noting cautions about direct extrapolation to QCD and outlining future work on the energy-momentum tensor and broader setups.

Abstract

Using the AdS/CFT duality, we analytically evaluate the high momentum (or short distance) behavior of the color field strength due to a moving quark in an N=4 SYM plasma. We find a fireball-like behavior in the near quark vicinity as predicted in the literature on general grounds, in the context of heavy ion collisions. Our approach to analytically solving the problem is based on a WKB approximation which may be extended to other setups as well.

Figures (3)

• Figure 1: Contour plots of the momentum space distribution of the leading dissipative behavior of the dimensionless quantity $\frac{1}{T\sqrt{\lambda_{YM}}}\langle \widehat{\mathcal{O}_{\Phi}}(k_-,k_{\bot}) \rangle = -\frac{1}{T\sqrt{\lambda_{YM}}}\frac{1}{2 g_{YM}^2} \text{Tr}\left(\widehat{F^2}+ \ldots \right)$ in response to a quark moving at a constant velocity of $v=0.8$. The horizontal axis corresponds to the momentum conjugate to the comoving coordinate of the quark, $k_-$, while the vertical one corresponds to momentum conjugate to the direction transverse to the quark motion, $k_{\bot}$. The momentum is measured in units of temperature (denoted [T] in the plots). Positive imaginary values are shaded dark while imaginary values closer to zero correspond to lighter shading. The origin of momentum space has been colored signifying that this region can not be accessed by our high momentum approximation. Figure (a) depicts the leading dissipative contribution to $\langle \widehat{\mathcal{O}_{\Phi}} \rangle$. The red line follows the extremum of the gradient of $\langle \widehat{\mathcal{O}_{\Phi}} \rangle$ in the $k_-$ direction. This directional behavior may be graphically enhanced (figure (b)) by multiplying $\langle \widehat{\mathcal{O}_{\Phi}} \rangle$ by $k_{\bot}$.
• Figure 2: Contour plots of the leading dissipative behavior of the dimensionless quantity $\frac{1}{T^4\sqrt{\lambda_{YM}}}\langle {\mathcal{O}}_{\Phi}(x-vt,r) \rangle = -\frac{1}{T^4\sqrt{\lambda_{YM}}}\frac{1}{2 g_{YM}^2} \left(\text{Tr}F^2+ \ldots\right)$ in response to a quark moving at a constant velocity of $v=0.8$. The horizontal axis corresponds to the comoving coordinate of the quark $x-vt$, while the vertical one corresponds to the direction transverse to the quark motion, $r$. Distances are measured in units of inverse temperature (denoted [1/T] in the plot). Large negative values are shaded in dark while values closer to zero correspond to lighter shading. The red line marks the extremum of the gradient in the $x_-$ direction.
• Figure 3: A plot of the Schrödinger potential (equation (\ref{['E:VPhi']})) corresponding to the homogeneous term in the equation of motion for the dilaton (equation (\ref{['E:EOMdilatonmy']})) for $Z_0 = 40$. As $\alpha$ increases from 0 to 1, the minimum of the potential increases such that both extrema of the potential vanish at $\alpha=1$. For $\alpha>1$, the zero of the potential gets closer to the origin, decreasing the parametric length of the flat region.