Musings

First, there’s the matter of introducing a heavy quark into the $\mathcal{N}=4$ system. This is accomplished by introducing a probe D7-brane which which fills the $\text{AdS}_5$ from $u=u_m$ to $u=\infty$ and wraps a $u$-dependent $S^3\subset S^5$.

You might worry that a space-filling D7-brane violates some charge-conservation condition on the $S^5$, and you might wonder how the D7-brane can “end” at $u=u_m$. The answer to both these questions is most easily understood at zero temperature. The $\text{AdS}_5\times S^5$ metric is

where $L = \lambda^{1/4} l_s$ and the $S^5$ metric is $${d\Omega}^2_5 = {d\psi}^2 + \cos^2\psi {\theta}^2 +\sin^2\psi{d\Omega}_3^2$$ The D7-brane is located at $\theta=0$, $\psi=\cos^{-1}(u_m/u)$. This corresponds simply to a D7-brane, parallel to the stack of D3-branes, displaced a distance $u\cos(\psi) =u_m$ in the transverse direction. The mass of the quark is

At finite temperature, $T$, we replace the $\text{AdS}_5$ geometry by AdS-Schwarzschild,

A static heavy quark in this setup is a string which stretches from $u_m$ down to the blackhole horizon, $u_h= \pi T$. The relationship between its Lagrangian mass and $u_m$ is modified from the zero temperature expression (2) to

where $$g(x) = \frac{1}{8} x^4 - \frac{5}{128} x^8 + O(x^{12})$$ Its rest-energy is $$M_{\text{rest}}(T) = E = \frac{1}{2\pi l_s^2} \int_{u_h}^{u_m} L^2 du = \frac{1}{2\pi l_s^2} L^2(u_m-u_h) = m - \Delta m(T) + m g\left(\frac{\Delta m(T)}{m}\right)$$ The thermal contribution to the mass is $$\Delta m(T)= \frac{\sqrt{\lambda} u_h}{2\pi m} = \frac{1}{2}\sqrt{\lambda} T$$

They then go on to study moving strings, using the Nambu-Goto action in static gauge ($\sigma=u$), with an ansatz of the form $$x(u,t) = x(u) + v t$$ The Nambu-Goto equations of motion reduce to $$\frac{\partial}{\partial u}\left(h(u) u^2 \frac{x'}{\sqrt{-\gamma}}\right)$$ where $$\sqrt{-\gamma} = L^2 [ 1- h^{-1} u^2 v^2 + h u^2 x^{'2}]^{1/2}$$ The solution is $$x(u,t) = x_0 \pm v F(u) +v t$$ where $$F(u) = \frac{1}{2 u_h} \left[\pi/2 - \tan^{-1}(u/u_h) -\tanh^{-1}(u/u_h)\right]$$ This solution does not satisfy the standard Neumann boundary condition at the D7-brane ($u=u_m$). There is energy- and momentum-flux down the string, supplied by a constant electric field on the D7-brane. At the horizon, the energy and momentum flux along the string are, respectively $$\mathcal{E}|_{u=u_h}= \frac{\pi}{2} \sqrt{\lambda}T^2 \frac{v^2}{\sqrt{1-v^2}}$$ and

Treating the heavy quark as having effective mass, $M_{\text{kin}}$, and momentum $p = \frac{M_{\text{kin}} v}{\sqrt{1-v^2}}$, the steady-state motion, under the influence of viscous drag and an external driving force, $f$, $$0 = \frac{d p}{d t} = f - \mu p$$ Comparing with (5), we learn

independent of the Lagrangian mass, $m$. The “flavour diffusion constant” is $$D= \frac{T}{\mu M_{\text{kin}}}=\frac{2}{\pi \sqrt{\lambda}T}$$

To extract the viscous damping coefficient, $\mu$, itself, one needs to go on and study the damping of small fluctuations about the long, straight static string. The linearized Nambu-Goto equation of motion is $$\frac{\partial}{\partial u} ( h u^2 x') = \frac{u^2}{h} \ddot{x}$$ Assuming a time-dependence of the form $e^{-\mu t}$ reduces this to an ODE. Demanding a purely outgoing solution at the horizon, $u=u_h$, and Neumann boundary conditions at the D7-brane, $u=u_m$, yields a discrete eigenvalue problem for the quasinormal mode frequencies. In the limit of large quark mass, one finds $$\mu = \frac{\pi \sqrt{\lambda} T^2}{2 m}$$ which, unsurprisingly, agrees with the previous result (6), since $M_{\text{kin}}\to m$ in this limit. In the limit of low quark mass (below $u_m \sim 1.02 u_h$, the D7-brane becomes unstable), $$\mu \to 2\pi T$$

This, apparently, is the upper limit on the viscous damping parameter, and may be universal. But it’s a little unclear that one should trust the quasinormal mode analysis in this limit, as quantum fluctuations of the short string are important, in contrast to the long-string (heavy-quark) limit.

Chris Herzog has a very nice little followup paper in which he, among other things, studies the same problem in the background of a certain non-extremal $\text{AdS}_5$ blackhole background.