Momentum broadening in unstable quark-gluon plasma

Quark-gluon plasma produced at the early stage of ultrarelativistic heavy-ion collisions is unstable, if weakly coupled, due to the anisotropy of its momentum distribution. Chromomagnetic fields are spontaneously generated and can reach magnitudes much exceeding typical values of the fields in equilibrated plasma. We consider a high-energy test parton traversing an unstable plasma that is populated with strong fields. We study the momentum broadening parameter $\widehat{q}$ which determines the radiative energy loss of the test parton. We develop a formalism which gives $\widehat{q}$ as the solution of an initial value problem, and we focus on extremely oblate plasmas which are physically relevant for relativistic heavy-ion collisions. The parameter $\widehat{q}$ is found to be strongly dependent on time. For short times it is of the order of the equilibrium value, but at later times $\widehat{q}$ grows exponentially due to the interaction of the test parton with unstable modes and becomes much bigger than the value in equilibrium. The momentum broadening is also strongly directionally dependent and is largest when the test parton velocity is transverse to the beam axis. Consequences of our findings for the phenomenology of jet quenching in relativistic heavy-ion collisions are briefly discussed.

Figure 1

Momentum broadening $\widehat{q}$ as a function of time for five angles $\mathrm{\Theta }$ between the test parton velocity and the direction of the anisotropy. The values of $\mathrm{\Theta }$ for each line are $\pi /2$ (red, solid); $\pi /3$ (orange, dot-dashed); $\pi /4$ (green, dashed); $\pi /6$ (blue, dotted); and $\pi /12$ (purple, spaced dots). For comparison, the equilibrium result is $\widehat{q}=0.11g^2{C}_R{d}_{\mathrm{scale}}$.

Figure 2

The orientation of the wave vector $\vec{k}$, current $\vec{j}$, electric $\vec{E}$, and magnetic $\vec{B}$ fields of the fastest unstable filamentation mode in the oblate plasma. The beam axis $z$ and three orientations of the test parton velocity $\vec{u}$ are also shown. Momentum broadening is smallest when the parton moves along the $z$ axis ($\mathrm{\Theta }=0$), and greatest when it moves perpendicular to $\widehat{z}\left(\mathrm{\Theta }=\pi /2\right)$.

Figure 3

The dependence of $\widehat{q}$ on the scale parameter $k_{\max }$. The upper (green) lines are for $t=10/m$ and $\mathrm{\Theta }=\pi /6$, and the lower (blue) lines are for $t=6/m$ and $\mathrm{\Theta }=\pi /6$. For both times, the dotted line represents a fit of $\widehat{q}$ as a function of $k_{\max }$ with $ln(k_{\max }/m)$.

Figure 4

The dependence of $\widehat{q}$ on the scale $m_{\min }$, with $t=6/m$ and $\mathrm{\Theta }=\pi /6$. The solid (orange) line is a fit of $\widehat{q}$ as a function of $m_{\min }$ with $ln(m/m_{\min })$.

Figure 5

The relative contribution of the $A$ modes (blue, dotted line) and $G$ modes (red, solid line) for $\mathrm{\Theta }=\pi /4$.