Transverse Momentum Broadening of a Fast Quark in a $\N=4$ Yang Mills Plasma

TL;DR

This work addresses how a fast heavy quark diffuses transversely in a strongly coupled $\mathcal{N}=4$ SYM plasma by relating medium-induced momentum broadening to a Wilson loop on the Schwinger-Keldysh contour. Using the AdS/CFT correspondence, the authors map this to a semi-classical string in $AdS_5\times S^5$ with a worldsheet black hole and extract the diffusion coefficient from transverse endpoint fluctuations, obtaining $\kappa_T = \sqrt{\gamma\lambda}\,\pi\,T^3$. The result, which diverges as $v\to 1$, is contrasted with previous holographic estimates of $\hat{q}$, raising questions about the precise strong-coupling definition and limits of $\hat{q}$ in this context. A velocity bound consistent with the probe's finite mass is noted, and the authors discuss potential sources of discrepancy with lightlike Wilson-loop calculations and the need for further clarification of the relation between $\kappa_T$ and $\hat{q}$ at strong coupling.

Abstract

We compute the momentum broadening of a heavy fundamental charge propagating through a $\mathcal{N}=4$ Yang Mills plasma at large t' Hooft coupling. We do this by expressing the medium modification of the probe's density matrix in terms of a Wilson loop averaged over the plasma. We then use the AdS/CFT correspondence to evaluate this loop, by identifying the dual semi-classical string solution. The calculation introduces the type ``1'' and type ``2'' fields of the thermal field theory and associates the corresponding sources with the two boundaries of the AdS space containing a black hole. The transverse fluctuations of the endpoints of the string determine $κ_T = \sqrt{γλ} T^3 π$ -- the mean squared momentum transfer per unit time. ($γ$ is the Lorentz gamma factor of the quark.) The result reproduces previous results for the diffusion coefficient of a heavy quark. We compare our results with previous AdS/CFT calculations of $\hat{q}$.

Figures (4)

• Figure 1: The contour world line of the heavy quark with velocity $v$ in the $z$ direction. The world line of the quark passing through $X_o$ may be parametrized as $X^{\mu}_{ X_o} (\tt_{{ C}}) = X_o^{\mu} + v^{\mu}(\tt_{ C} - \tt_{o\,{ C}})$, with $v^{\mu}=(1,{\bf v})$. The Wilson line $W_{{ C}}[0]$ follows the world line of the quark. The circles at $\tt_{ C}$ and $\tt'_{{ C}}$ indicate the insertions of the field strengths $F^{y\mu}v_{\mu}(\tt_{ C})$ and $F^{y\nu}v_{\nu}(\tt'_{ C})$ into the Wilson line as in Eq. (\ref{['fstrengths']}) .
• Figure 2: Graphical representation of Eq. (\ref{['frperp']}). The Wilson line indicated by the black line is denoted $W_{{ C}}[{\bf r}_\perp/2, -{\bf r}_\perp/2]$. This Wilson line is traced with the initial density matrix, $\rho^{o}_{a_1 a_2}$.
• Figure 3: Kruskal diagram for the AdS black hole. The coordinates $(t,r)$ span the right quadrant. The thick hyperbolas on the sides of the two quadrants are the boundaries at $r=\infty$. The boundary in the right and left quadrants correspond to the "1" and "2" axes of the thermal field theory respectively. The static quark corresponds to a string which spans the full Kruskal plane. At finite velocity, the asymptotic solution Eq. (\ref{['Yaffeetal']}) is discontinuous at $t=-\infty$ or $V=0$.
• Figure 4: (a) The world sheet black hole. (b) The Schwarzschild black hole in space time. The future event horizon of the world sheet and its corresponding image $z = 1/\sqrt{\gamma}$ are indicated by the solid red line. Similarly the solid blue line maps to the future event horizon of the space time black hole. The shaded region of the world sheet black hole maps into the shaded region of space-time region of the Schwarzschild geometry. Similarly there is a separate string solution in the $V<0$ half of the $(U,V)$ plane which maps to the $\hat{V}<0$ half of the $(\hat{U}, \hat{V})$ plane. The two solutions are joined at past infinity ($V=0$) by demanding analyticity in the lower complex $V$ plane, i.e. that only positive energy solutions emerge from past infinity.