A semi-classical recipe for wobbly limp noodles in partonic soup

Abstract

We compute the average squared distance, $s^2(t)$, travelled by a light-flavour off-mass-shell coloured parton in a strongly-coupled $\mathcal{N}=4$ $SU(N_c)$ super-symmetric Yang Mills plasma using the gauge/string duality. In fact, we derive a closed integral expression for $s^2(t;a)$ in $Ad{S}_3$-Schwarzschild, which interpolates between a heavy quark when $a=0$ and a light quark when $a=1$, that we evaluate analytically for small virtualities - labelled $s_{\text{small}}^2(t;a)$. For arbitrary virtualities, we show that for asymptotically early times the motion is ballistic, ${s^2(t;a)|}_{t\ll \beta }\sim t^2$, while at asymptotically late times the motion is diffusive, ${s^2(t;a)|}_{t\gg \beta }=s_{\text{small}}^2(t;a)\sim 2D(a)t$, from which we are able to extract the diffusion coefficient $D(a)$. Motivated by the apparent universality of the late time behaviour, we compute $s_{\text{small}}^2(t;a,d)$ and $D(a,d)$ for an arbitrary $Ad{S}_d$-Schwarzschild geometry. From $D(a,d)$ we then compute for the first time the dynamic, time-dependent transverse momentum squared per unit path length picked up by a high momentum light quark in a strongly-coupled plasma, the transport coefficient $\hat{q}(t)$, which is critically important for phenomenology in heavy ion collisions.

Comments: 36 pages, 7 figures