Synchrotron radiation in strongly coupled conformal field theories

TL;DR

The paper demonstrates that the angular distribution of synchrotron-like radiation from a uniformly rotating quark in vacuum is strikingly similar at both weak and strong coupling in ${ m N}=4$ SYM: a tightly beamed, non-broadening pulse propagating along a spiral at the speed of light. At weak coupling, radiation arises from coupled vector and scalar fields with a calculable angular pattern; at strong coupling, a rotating string in ${ m AdS}_5$ perturbs the bulk geometry, and the boundary stress tensor yields the same angular structure up to a normalization set by $\sqrt{\lambda}$. The key result is that the far-field angular distribution at strong coupling is $\displaystyle \frac{dP}{d\Omega}=\frac{v^2 \omega_0^2 \sqrt{\lambda}}{16\pi^2}\frac{2+v^2\sin^2\theta}{(1-v^2\sin^2\theta)^{5/2}}$, and the total radiated power remains $P=\frac{\sqrt{\lambda}}{2\pi}a^2$ with $a= v\omega_0\gamma^2$, matching Mikhailov’s strong-coupling result. These findings imply that strongly coupled conformal theories with gravity duals can exhibit radiation patterns closely resembling classical synchrotron radiation, informing future studies of jet quenching and nonzero-temperature dynamics in holographic plasmas.

Abstract

Using gauge/gravity duality, we compute the energy density and angular distribution of the power radiated by a quark undergoing circular motion in strongly coupled ${\cal N}=4$ supersymmetric Yang-Mills (SYM) theory. We compare the strong coupling results to those at weak coupling, and find the same angular distribution of radiated power, up to an overall prefactor. In both regimes, the angular distribution is in fact similar to that of synchrotron radiation produced by an electron in circular motion in classical electrodynamics: the quark emits radiation in a narrow beam along its velocity vector with a characteristic opening angle $α\sim 1/γ$. To an observer far away from the quark, the emitted radiation appears as a short periodic burst, just like the light from a lighthouse does to a ship at sea. Our strong coupling results are valid for any strongly coupled conformal field theory with a dual classical gravity description.

Figures (8)

• Figure 1: A cartoon of the gravitational description of synchrotron radiation at strong coupling. The arena where the gravitational dynamics takes place is the $5d$ geometry of AdS$_5$. A string resides in the geometry and an endpoint of the string is attached to the $4d$ boundary of $AdS_5$, which is where the dual quantum field theory lives. The trajectory of the endpoint of the string corresponds to the trajectory of the dual quark. Demanding that the endpoint rotates results in the string rotating and coiling around on itself as it extends in the $AdS_5$ radial direction. The presence of the string in turn perturbs the geometry and the near-boundary perturbation in the geometry induces a $4d$ stress tensor on the boundary. The induced stress has the interpretation as the expectation value of the stress tensor in the dual quantum field theory.
• Figure 2: A sketch of the solutions (\ref{['classicalsols']}). As the quark moves along its circular trajectory, it emits radiation in a narrow cone of angular width $\alpha \sim 1/\gamma$ in the direction of its velocity vector. The diagram is a snapshot at the time when the quark is at the top of the circle. The red spiral shows where the radiation emitted at earlier times is located at the time of the snapshot. The width $\Delta$ of the spiral scales like $\Delta \sim 1/\gamma^3$, as explained in Fig. \ref{['widthdiagram']}.
• Figure 3: A close-up illustration of the emission of radiation at two times $t_1$ and $t_2$. The radiation is emitted in the direction of the quark's velocity vector, within a cone of angular width $\alpha$. An observer at the point $p$ is illuminated by a pulse of radiation of duration $\Delta t \sim R_0 \alpha$ and of spatial thickness $\Delta$. The leading edge of the pulse observed at $p$ is emitted at $t_1$ and the trailing edge observed at $p$ is emitted at $t_2$. At time $t_2$ the radiation emitted at $t_1$, denoted by the solid red line, has traveled a distance $R_0 \alpha/v$ towards $p$. The chordal distance between the two emission points is $R_0 \alpha$ in the $\alpha \to 0$ limit. The width $\Delta$ is therefore $\Delta = R_0 \alpha(1/v - 1).$
• Figure 4: Left: a cutaway plot of $r^2 \mathcal{E}/P$ for $v = 1/2$. Right: a cutaway plot of $r^2 \mathcal{E}/P$ for $v = 3/4$. In both plots the quark is at $x = R_0$, $y=0$ at the time shown and its trajectory lies in the plane $z = 0$. The cutaways coincide with the planes $z = 0$, $\varphi = 0$ and $\varphi = 7 \pi / 5$. At both velocities the energy radiated by the quark is concentrated along a spiral structure which propagates radially outwards at the speed of light. The spiral is localized about $\theta = \pi/2$ with a characteristic width $\delta \theta \sim 1/\gamma$. As $v \to 1$ the radial thickness $\Delta$ of the spirals rapidly decreases like $\Delta \sim 1/\gamma^3$.
• Figure 5: Plot of $r^2 \mathcal{E}/P$ at $\theta=\pi/2$ and $\varphi=5\pi/4$ at $t=0$ as a function of $r$ for $v=1/2$. The plot illustrates the fact that the pulses of radiated energy do not broaden as they propagate outward. This implies that they do not broaden in azimuthal angle, either. Strongly coupled synchrotron radiation does not isotropize.