Light quark energy loss in strongly-coupled N = 4 supersymmetric Yang-Mills plasma

TL;DR

Using gauge/gravity duality, the paper studies the penetration depth and energy loss of energetic light quarks in a strongly coupled N=4 SYM plasma by modeling quarks as endpoints of falling open strings in AdS-Schwarzschild with D7-branes. It demonstrates, both analytically via near-null string approximations and numerically through full string evolutions, that the maximum penetration depth scales as Δx_max ∝ E^{1/3}, with a coefficient around 0.5, i.e., Δx_max(E) = (C/T) (E/(T√λ))^{1/3}. The instantaneous energy-loss rate is non-universal and sensitive to initial conditions, but the late-time dynamics exhibit an explosive transfer of energy to the plasma as the quark thermalizes. These results illuminate how light jets propagate and dissipate in strongly coupled plasmas and reveal qualitative differences from heavy-quark energy loss in the same medium.

Abstract

We compute the penetration depth of a light quark moving through a large $N_c$, strongly coupled $\mathcal N = 4$ supersymmetric Yang-Mills plasma using gauge/gravity duality and a combination of analytic and numerical techniques. We find that the maximum distance a quark with energy $E$ can travel through a plasma is given by $Δx(E) = (\mathcal C/T) (E/T \sqrtλ)^{{1}/{3}}$ with $\mathcal C \approx 0.5$.

Figures (7)

• Figure 1: A log-log plot of the quark stopping distance $\Delta x$ as a function of total quark energy $E$ for many falling strings with initial conditions of the form shown in Eq. (\ref{['IC']}). All data points fall below the red line given by $\Delta x = ({0.526}/{T})( {E}/{T\sqrt{\lambda}} )^{1/3}$.
• Figure 2: Examples of perturbative diagrams contributing to the energy loss of a quark. One may regard time as running to the right. An energetic quark can scatter, emit gluons (which themselves may radiate or split into $q \bar{q}$ pairs), or annihilate with an antiquark in the medium. However, the total baryon number of the state remains constant.
• Figure 3: A cartoon of the bulk-to-boundary problem at finite temperature. The endpoints of strings are charged under a $U(1)$ gauge field $\mathcal{A}_M$ which lives on the $D7$ brane which fills the AdS-BH geometry. The boundary of the geometry, located at radial coordinate $u = 0$, behaves like a perfect conductor. Consequently, the string endpoints induce a mirror current density $j^{\mu}$ on the boundary. Via gauge/gravity duality, the induced mirror current density has the interpretation of minus the baryon current density of a quark. Similarly, the presence of the string induces a perturbation $h_{MN}$ in the metric of the bulk geometry. The behavior of the metric perturbation near the boundary encodes the information contained in the perturbation to the SYM stress-energy tensor caused by the presence of the jet.
• Figure 4: A typical falling string studied in this paper, plotted in blue at four different instants in time. The string is created at a point and, as time passes, evolves into an increasingly extended object. Well after the creation event, but long before the plunge into the horizon, the string profile approaches a universal null string configuration which is largely insensitive to the initial conditions. Consequently, the string endpoint trajectories, shown in green and yellow, approach null geodesics.
• Figure 5: The inflation of a perturbation on a expanding string. The thin blue lines show the string at eight different instants of time. The uppermost black curve shows the endpoint trajectory. The perturbation to the stationary profile is the bump initially located close to the string endpoint. For clarity, we greatly exaggerate the size of the perturbation. The two red curves are lightlike geodesics which enclose the perturbation at all times. Even though the perturbation is initially highly localized, the two geodesics which bound the perturbation rapidly separate, and correspondingly the size of the perturbation rapidly inflates as it falls into the horizon.