Heavy-flavor in-medium momentum evolution: Langevin versus Boltzmann approach

Abstract

The propagation of heavy quarks in the quark-gluon plasma was often treated within the framework of the Langevin equation (LV), i.e., assuming the momentum transfer is small or the scatterings are sufficiently forward peaked, small screening mass ${m}_{D}$. We address a direct comparison between the Langevin dynamics and the Boltzmann collisional integral (BM) when a bulk medium is in equilibrium at fixed temperature. We show that unless the cross section is quite forward peaked $({m}_{D}\ensuremath{\cong}T)$ or the mass to temperature ratio is quite large $({M}_{\mathrm{HQ}}/T\ensuremath{\gtrsim} 8--10)$ there are significant differences in the evolution of the $p$ spectra and consequently on the nuclear modification factor ${R}_{AA}({p}_{T})$. However, for charm quark we find that very similar ${R}_{AA}({p}_{T})$ between the LV and BM can be obtained, but with a modified diffusion coefficient of about $\ensuremath{\sim}15%--50%$ depending on the angular dependence of the cross section which regulates the momentum transfer. Studying also the momentum spread suffered by the single heavy quarks we see that at temperatures $T\ensuremath{\gtrsim}\phantom{\rule{0.16em}{0ex}}250\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$ the dynamics of the scatterings is far from being of Brownian type for charm quarks. In the case of bottom quarks we essentially find no differences in the time evolution of the momentum spectra between the LV and the BM dynamics independently of the angular dependence of the cross section, at least in the range of temperature relevant for ultrarelativistic heavy-ion collisions (HICs). Finally, we have shown the possible impact of this study on ${R}_{AA}({p}_{T})$ and ${v}_{2}({p}_{T})$ for a realistic simulation of relativistic HICs. For larger ${m}_{D}$ the elliptic flow can be about $50%$ larger for the Boltzmann dynamics with respect to the Langevin. This is helpful for a simultaneous reproduction of ${R}_{AA}({p}_{T})$ and ${v}_{2}({p}_{T})$.