Data-driven analysis for the temperature and momentum dependence of the heavy-quark diffusion coefficient in relativistic heavy-ion collisions

By applying a Bayesian model-to-data analysis, we estimate the temperature and momentum dependence of the heavy quark diffusion coefficient in an improved Langevin framework. The posterior range of the diffusion coefficient is obtained by performing a Markov chain Monte Carlo random walk and calibrating on the experimental data of $D$-meson $R_{\mathrm{AA}}$ and $v_2$ in three different collision systems at the Relativistic Heavy-Ion Collidaer (RHIC) and the Large Hadron Collider (LHC): Au-Au collisions at 200 GeV and Pb-Pb collisions at 2.76 and 5.02 TeV. The spatial diffusion coefficient is found to be consistent with lattice QCD calculations and comparable with other models' estimation. We demonstrate the capability of our improved Langevin model to simultaneously describe the $R_{\mathrm{AA}}$ and $v_2$ at both RHIC and the LHC energies, as well as the higher order flow coefficient such as $D$ meson $v_3$. We show that by applying a Bayesian analysis, we are able to quantitatively and systematically study the heavy flavor dynamics in heavy-ion collisions.

Figure 1

A leading order pQCD calculation of $\widehat{q}$ with respect to temperature and momentum. For the calculation of $\widehat{q}$, a fixed coupling ${\alpha }_s=0.3$ is used, and a Debye screening mass according to $m_D^2=4\pi {\alpha }_s{T}^2$. All $s,u,t$ channel contributions for $Qq\to Qq,Qg\to Qg$ are included.

Figure 2

An example of the spatial diffusion coefficient parametrization. The linear component uses ${\left(D2\pi T\right)}^{\mathrm{linear}}=\alpha [1+\beta (T/T_{\mathrm{c}}-1)]$ with $(\alpha ,\beta )=(1.9,1.6)$ and is plotted as black dots. The dashed black line is the pQCD component, while the rainbow lines represent the diffusion coefficient Eq. (9) with parameter $\gamma$ varying from 0 to 1 while the color change from violet to red.

Figure 3

The 60 design points $\tilde{X}$ generated by the Latin hypercube algorithm, projected in the $(\alpha ,\beta )$ dimensions. The flat histograms on the edge show an uniform prior distribution of the parameters.

Figure 4

Proton-proton reference spectrum calculated by FONLL+pythia, used as the reference spectra in order to calculate $D$ meson $R_{\mathrm{AA}}$. The experimental results denoted total D meson are the sum of $D^0,D^{+}$, and $D^{*+}$ data from the ALICE Collaboration.

Figure 5

Improved Langevin model calculation of $D$-meson observables, compared to experimental data spanning the full range of the explored parameter space (i.e., the “prior”). Each frame contains 60 lines, corresponding to the 60 design points of the analysis. From top to bottom: (top) Au-Au collisions at 200 GeV: $D$ meson $R_{\mathrm{AA}}\left(p_{\mathrm{T}}\right)$ in the 0–10% centrality bin and $v_2\left(p_{\mathrm{T}}\right)$ in the 0–80%, 10–40% centrality bins; (middle) Pb-Pb collisions at 2.76 TeV: $D$ meson $R_{\mathrm{AA}}$ as function of centrality at high momentum range $5<p_{\mathrm{T}}<8$ GeV/c and $8<p_{\mathrm{T}}<16$ GeV/c, $v_2$ as function of $p_{\mathrm{T}}$ in 30–50% centrality bin; and (bottom) Pb-Pb collisions at 5.02 TeV: $D$ meson $R_{\mathrm{AA}}\left(p_{\mathrm{T}}\right)$ in 30–50% centrality bin and $v_2\left(p_{\mathrm{T}}\right)$ in 30–50% and 10–30% centrality bins. Experimental data are measured by STAR [67, 68], ALICE [69, 70, 71], and CMS [72].

Figure 6

Cumulative variance explained by the first $m^{\prime }$ principal components. The first few PCs are able to explain most of the total variance.

Figure 7

Validation of the Gaussian emulators for Pb-Pb collisions at 2.76 TeV. Each point represents the emulator's predicted value with respect to the model calculated (true) value. (Left) Prediction for high-momentum $D$ meson $R_{\mathrm{AA}}$ at different centrality bins; (right) prediction for $D$ meson $v_2$ at 30–50%.

Figure 8

Emulator predictions for 200 random input parameters sampled from the posterior distributions. This figure is similar to Fig. 5 but with the input parameters chosen from the posterior distribution, and the outputs are predictions from the GP emulators.

Figure 9

Posterior distributions of the diffusion coefficient parameters $(\alpha ,\beta ,\gamma )$ for each individual collision system. The diagonal plots are the histogram of the MCMC samples with other parameters integrated out. The off-diagonal plots display the joint distributions between pairs of parameters. The three different colors refer to the three different analyses that calibrate on three different sets of data.

Figure 10

Posterior distributions of the diffusion coefficient parameters $(\alpha ,\beta ,\gamma )$ for the analysis combining all three collision systems. The red color refers to the result that combines only the LHC energies, PbPb collisions at 2.76 and 5.02 TeV; the magenta color refers to the analysis combining all three energies at RHIC and LHC.

Figure 11

Posterior results of the spatial diffusion coefficient from the Bayesian analysis calibrated on the combined data set from three different systems at RHIC and the LHC. (Top) The spatial diffusion coefficient $D_{s}2\pi T$ as a function of temperature at fixed momentum $(p=0,p=10$, and $p=50$ GeV/c); (bottom) the spatial diffusion coefficient $D_{s}2\pi T$ as a function of momentum at fixed temperature $(T=154,T=350$, and $T=550$ MeV). The gray area refers to the prior range before calibration, while the red region refers to the posterior range after calibrating on experimental data. The black dashed line refers to the diffusion coefficient from a leading-order pQCD calculation; the red lines are the parametrized diffusion coefficient using the median value of the posterior parameter distributions.

Figure 12

Comparison of the heavy quark diffusion coefficients across multiple approaches available in the literature. (Left) Spatial diffusion coefficient at zero momentum $D_{s}2\pi T(p=0)$. (Right) Momentum diffusion coefficient $\widehat{q}/T^3$ at $p=10$ GeV.

Figure 13

$D$ meson $v_3$ predicted by the improved Langevin model at Au-Au collisions at 200 GeV and Pb-Pb collisions at 5.02 TeV, compared to experimental data measured by STAR [78] and CMS [72]. The input parameters $\left(\alpha ,\beta ,\gamma \right)$ were set to the median value of the posterior distribution for the Au-Au at 200-GeV dataset (AuAu200 median), Pb-Pb at 5.02-TeV dataset (PbPb5020 median), the combined LHC energy dataset (LHC-median), and the combined data set for all three systems (all-median).