Heavy quark diffusion with relativistic Langevin dynamics in the quark-gluon fluid

The relativistic diffusion process of heavy quarks is formulated on the basis of the relativistic Langevin equation in Itô discretization scheme. The drag force inside the quark-gluon plasma (QGP) is parametrized according to the formula for the strongly coupled plasma obtained by the anti-de-Sitter space/conformal field theory (AdS/CFT) correspondence. The diffusion dynamics of charm and bottom quarks in QGP is described by combining the Langevin simulation under the background matter described by the relativistic hydrodynamics. Theoretical calculations of the nuclear modification factor $R_{\mathrm{AA}}$ and the elliptic flow $v_2$ for the single electrons from the charm and bottom decays are compared with the experimental data from the relativistic heavy-ion collisions. The $R_{\mathrm{AA}}$ for electrons with large transverse momentum ($p_T>3$ GeV) indicates that the drag force from the QGP is as strong as the AdS/CFT prediction.

Figure 1

(a) A sample of 500 heavy quarks at the initial in the transverse plane, for the $\mathrm{Au}+\mathrm{Au}$ collision with impact parameter 5.5 fm. (b) Invariant cross sections of charm and bottom production in $p+p$ collisions in midrapidity ($\left|y_p\right|⩽1.0$), which is proportional to the initial momentum distribution.

Figure 2

(a) Experimental cross section for electron production in $p+p$ collision at midrapidity [26] and the leading-order pQCD result by PYTHIA. (b) The ratio of the experimental data and the LO result. Theoretical calculations are performed at $\left|y_p\right|⩽0.35$ and then properly normalized to obtain the cross section.

Figure 3

The averaged stay time $\langle t_S\rangle$ of (a) charm quarks and (b) bottom quarks with the drag coefficient $\gamma =0,0.3,1.0,3.0$, and 30.0 at midrapidity ($\left|y_p\right|⩽1.0$). The impact parameter is chosen to be 5.5 fm in $\mathrm{Au}+\mathrm{Au}$ collisions. For freeze-out condition, $f_0=0.5$ is adopted.

Figure 4

The averaged temperature $\langle \overline{T}\rangle$ of (a) charm quarks and (b) bottom quarks with the drag coefficient $\gamma =0,0.3,1.0,3.0$, and 30.0 in midrapidity ($\left|y_p\right|⩽1.0$). The collision geometry and the freeze-out condition are the same with those in Fig. 3. The fluctuation in high $p_T$ in (a) is due to statistical errors of our simulation.

Figure 5

The averaged momentum loss $\langle \Delta p_T\rangle$ of (a) charm quarks and (b) bottom quarks with drag coefficients $\gamma =0.3,1.0,3.0$, and 30.0 in midrapidity ($\left|y_p\right|⩽1.0$). The collision geometry and the freeze-out condition are the same with those in Fig. 3.

Figure 6

$R_{\mathrm{AA}}^Q$ of (a) charm quarks and (b) bottom quarks with drag coefficients $\gamma =0.3,1.0,3.0$, and 30.0 at midrapidity ($\left|y_p\right|⩽1.0$). For collision geometry, we choose the impact parameter 5.5 fm in $\mathrm{Au}+\mathrm{Au}$ collisions. For freeze-out condition, the $f_0=0.5$ is adopted.

Figure 7

$v_2^Q$ of (a) charm quarks and (b) bottom quarks with drag coefficients $\gamma =0.3,1.0,3.0$, and 30.0 in midrapidity ($\left|y_p\right|⩽1.0$). The collision geometry and the freeze-out condition are the same with those in Fig. 6. In (a), the statistical errors for $\gamma =3.0$ and 30.0 are so large at $p_T^{\mathrm{out}}>6$ GeV that we omit them.

Figure 8

(a) $R_{\mathrm{AA}}$ of electrons from charm, (b) $R_{\mathrm{AA}}$ of electrons from bottom, (c) $R_{\mathrm{AA}}$ of electrons from both charm and bottom, and (d) the ratio of electrons from the bottom and the net electrons. All results are in midrapidity ($\left|y_p\right|⩽0.35$). The drag coefficient is taken to be $\gamma =0.3,1.0$, and 3.0. The impact parameter is taken to be 5.5 fm in $\mathrm{Au}+\mathrm{Au}$ collisions. For freeze-out condition, the $f_0=0.5$ is adopted. In (d), the result of $p+p$ collision calculated in the leading-order pQCD by PYTHIA is also plotted.

Figure 9

Comparison of $R_{\mathrm{AA}}$ in our hydro $+$ heavy-quark model with the experimental data [4, 5]. The $\mathrm{Au}+\mathrm{Au}$ collision with the impact parameter (a) 3.1 fm and (b) 5.5 fm, both in midrapidity, $\left|y_p\right|⩽0.35$. The drag coefficient is chosen to be $\gamma =0.3,1.0$, and 3.0 indicated by different colors. The freeze-out condition is taken to be $f_0=1.0,0.5$, and 0.0 which correspond to upper, middle, and lower points, respectively, within the same color. As for error bars in experimental data, we only plot the statistical errors [4, 5].

Figure 10

Comparison of $v_2$ in our hydro + heavy-quark model with experimental data [4] in midrapidity ($\left|y_p\right|⩽0.35$). Experimental data of $v_2$ is obtained in minimum bias analysis, whereas our theoretical values of $v_2$ are evaluated at impact parameter 5.5 fm as a representative. The drag coefficient is chosen to be $\gamma =0.3,1.0$, and 3.0 and the freeze-out condition is $f_0=0.5$. As for error bars in experimental data, we only plot the statistical errors [4].