Ballistic diffusion of heavy quarks in the early stage of relativistic heavy ion collisions at RHIC and the LHC

We study the diffusion of charm quarks in the early stage of high energy nuclear collisions at the RHIC and the LHC. The main novelty of the present study is the introduction of the color current carried by the heavy quarks that propagate in the evolving glasma (EG), which is responsible of the energy loss via polarization of the medium. We compute the transverse momentum broadening, ${\sigma }_p$, of charm in the prethermalization stage, and the impact of the diffusion on the nuclear modification factor in nucleus-nucleus collisions. The net effect of energy loss is marginal in the prethermalization stage. The study is completed by the calculation of coordinate spreading, ${\sigma }_x$, and by a comparison with Langevin dynamics. ${\sigma }_p$ in EG overshoots the result of standard Langevin dynamics at the end of the prehydro regime. We interpret this as a result of memory of the color force acting on the charm quarks that implies ${\sigma }_p\propto t^2$. Moreover, ${\sigma }_x\propto t^2$ in the prehydro stage shows that the charm quark in the EG is in the regime of ballistic diffusion.

Figure 1

Distribution function, at $t=0.2\mathrm{fm}/\mathrm{c}$ (blue lines), $t=0.6\mathrm{fm}/\mathrm{c}$ (orange lines), and at $t=1\mathrm{fm}/\mathrm{c}$ (green lines); the solid lines correspond to calculations with the color current while the dashed line to those without the current. Initial momentum is $p_0=0.5\mathrm{GeV}$ in the upper panel and $p_0=5\mathrm{GeV}$ in the lower panel. Calculations correspond to ${\alpha }_s=0.3$.

Figure 2

Time evolution of ${\sigma }_p$ of charm quarks with different initial momenta. Solid lines correspond to the calculations with the color current, dashed lines stand for the results without color current. In the lower panel we enlarge the early time region for the case $p_0=0.5\mathrm{GeV}$ to remark on the nonlinear behavior of ${\sigma }_p(t)$.

Figure 3

Nuclear modification factor of charm quarks versus $p_T$, computed at different times. Calculations with and without current are represented by solid and dashed lines, respectively.

Figure 4

Transverse coordinate dispersion of charm quarks computed with and without the color current. We take $g^2\mu =3.4\mathrm{GeV}$.

Figure 5

Comparison of ${\sigma }_p$ of charm quarks versus time, between evolution in EG and in a thermal medium via Langevin equations. Initialization corresponds to $p_0=0.5\mathrm{GeV}$. All Langevin calculations have the same $\mathcal{D}$ of EG, $\mathcal{D}=3.37{\mathrm{GeV}}^2/\mathrm{fm}$. Langevin 1 uses the same $\gamma$ of EG, while in Langevin 2 and 3 we have used the $\gamma$ that would be required by the fluctuation-dissipation theorem, $\gamma =\mathcal{D}/ET$, for $T=1.5\mathrm{GeV}$ and $T=1\mathrm{GeV}$, respectively: these are $\gamma =1.72$ in Langevin 2 and $\gamma =2.59$ in Langevin 3. Lower panel corresponds to an enlargement to $0.2\mathrm{fm}/\mathrm{c}$.