Resolving discrepancies in the estimation of heavy quark transport coefficients in relativistic heavy-ion collisions

Heavy flavor observables provide valuable information on the properties of the hot and dense quark-gluon plasma (QGP) created in ultrarelativistic nucleus-nucleus collisions. Various microscopic models have successfully described many of the observables associated with its formation. Their transport coefficients differ, however, due to different assumptions about the underlying interaction of the heavy quarks with the plasma constituents, different initial geometries and formation times, different hadronization processes, and a different time evolution of the QGP. In this study we present the transport coefficients of these models and investigate systematically how some of these assumptions influence the heavy quark properties at the end of the QGP expansion. For this purpose we impose on these models the same initial condition and the same model for the QGP expansion and show that both have considerable influence on $R_{AA}$ and $v_2$.

Figure 1

A skeleton showing each ingredient that needs to be taken into consideration during the implementation of heavy quark evolution, which could affect the estimation of the heavy quark transport coefficients in the QGP phase.

Figure 2

Leading-order charm quark transport coefficients (drag coefficients ${\eta }_D$) estimated by each group to describe the $D$ meson $R_{\mathrm{AA}}$ and $v_2$ at AuAu and/or PbPb collisions at RHIC and the LHC.

Figure 3

Same as Fig. 2 but for longitudinal momentum transport coefficient ${\kappa }_L$.

Figure 4

Same as Fig. 2, 3 but for transverse momentum transport coefficient ${\kappa }_T$.

Figure 5

Overall heavy quark transport coefficients with radiative process considered. The values are estimated by propagating charm quarks inside a static medium for 1 fm/$c$. The solid lines (with or without markers) represent the total transport coefficients, while the dashed lines represent the contribution from collisional only process. The difference between those two is the contribution from higher-order radiative process.

Figure 6

Initial conditions at ${\tau }_0=0.6$ fm/$c$ for AuAu collisions at 200 GeV with impact parameter $b=6$ fm, generated from: (Left) PHSD initial conditions; (Middle) ${\mathtt{T}}_{\mathtt{R}}\mathtt{ENTo}$ average initial condition, which is averaged by 50 ${\mathtt{T}}_{\mathtt{R}}\mathtt{ENTo}$ events who share the similar spatial eccentricity as in PHSD initial condition; (Right) one example of single ${\mathtt{T}}_{\mathtt{R}}\mathtt{ENTo}$ event.

Figure 7

Time evolution of the spatial and momentum anisotropy of the QGP medium that is simulated by a (2+1)-dimensional [(2+1)D] hydrodynamical model (VISHNU) [18, 41]. The medium starts from two different initial conditions: PHSD initial condition, and averaged ${\mathtt{T}}_{\mathtt{R}}\mathtt{ENTo}$ initial condition.

Figure 8

(Left) Development of charm quark elliptic flow inside a (2+1)D hydrodynamical medium during the QGP phase. (Right) Charm quark $v_2\left(p_T\right)$ evaluated at the end of the QGP phase. The charm quarks interact with the medium following an improved Langevin dynamics, with Duke coefficients applied.

Figure 9

Hydro initial condition (${\tau }_0=0.6$ fm) generated from PHSD, used as input for hydro evolution.

Figure 10

Charm quark $R_{\mathrm{AA}}$ as a function of $y$ (left), and $p_T$ (middle), elliptic flow $v_2$ as a function of $p_T$ (right) at the end of the QGP phase. The charm quarks are following a Langevin dynamics with two sets of transport coefficients applied: PHSD (red) and Duke (green).

Figure 11

Charm quark $R_{\mathrm{AA}}$ as a function of $y$ (left), and $p_T$ (middle), elliptic flow $v_2$ as a function of $p_T$ (right) at the end of the QGP phase. The charm quarks are propagating in a hydrodynamical medium simulation for AuAu collisions at 200 GeV with $b=6$ fm/$c$ (top), and $b=2$ fm/v (bottom).

Figure 12

Charm quarks distribution after propagating within the Langevin evolution in a static medium with constant temperature for $t=50$ fm/c. The dashed black lines are the equilibrium Boltzmann distribution.

Figure 13

Comparison between different implementation of Einstein's relationship in a static medium. Heavy quarks are initialized with same momentum as $p_z\left(0\right)=30$ GeV, and propagate in a static medium ($T=0.3$ GeV) for 200 fm/$c$ under: linearized Boltzmann dynamics (red dashed line), Langevin dynamics taking (${\eta }_D,{\kappa }_L,{\kappa }_T$) calculated from linearized Boltzmann dynamics (blue line), Langevin dynamics taking (${\kappa }_T$) while ${\kappa }_L,{\eta }_D$ are calculated from Einstein's relationship (orange line), Langevin dynamics taking (${\kappa }_L$) while ${\kappa }_T,{\eta }_D$ are calculated from Einstein's relationship (cyan line), Langevin dynamics taking (${\kappa }_T,{\kappa }_L$) while ${\eta }_D$ are calculated from Einstein's relationship (purple line), Langevin dynamics taking (${\eta }_D$) while ${\kappa }_L,{\kappa }_T$ are calculated from Einstein's relationship (green line).

Figure 14

Charm quark $R_{\mathrm{AA}}$ as a function of $p_T$ (top), elliptic flow $v_2$ as a function of $p_T$ (bottom) at the end of the QGP phase. The charm quarks are propagating in a hydrodynamical medium simulation for AuAu collisions at 200 GeV with $b=6$ fm/$c$, with different scenario of Einstein's relationship applied.

Figure 15

Charm quark $R_{\mathrm{AA}}$ as a function of $y$ (left), and $p_T$ (middle), elliptic flow $v_2$ as a function of $p_T$ (right) at the end of the QGP phase. The charm quarks are propagating in a hydrodynamical medium simulation for AuAu collisions at 200 GeV with $b=6$ fm/$c$ (top), and “tune2” assumption for each set of transport coefficients.