Quantifying properties of hot and dense QCD matter through systematic model-to-data comparison

We systematically compare an event-by-event heavy-ion collision model to data from the CERN Large Hadron Collider. Using a general Bayesian method, we probe multiple model parameters including fundamental quark-gluon plasma properties such as the specific shear viscosity $\eta /s$, calibrate the model to optimally reproduce experimental data, and extract quantitative constraints for all parameters simultaneously. The method is universal and easily extensible to other data and collision models.

Figure 1

Top: Random functions drawn from a Gaussian process using a squared-exponential covariance function with length scale $\ell =1$. Bottom: Functions drawn from a GP conditioned on the training points indicated by dots. In both plots, the dashed line represents the GP mean and the grey band is twice the GP standard deviation (roughly 95% confidence interval).

Figure 2

The Latin hypercube experiment design projected into the $(\eta /s,{\tau }_0)$ dimensions. All other parameters also vary across the design, so the points that appear very close in the projection are not necessarily close in the full-dimensional space. The edge histograms show the distributions flattened into one dimension; note that they are space-filling and approximately uniform.

Figure 3

Model calculations from Glauber (top, blue) and KLN (bottom, green) initial conditions. Each plot has 254 lines corresponding to the 254 Latin hypercube design points. From left to right: average charged-particle multiplicity $\langle N_{\mathrm{ch}}\rangle$, elliptic flow two-particle cumulant $v_2\left\{2\right\}$, and triangular flow two-particle cumulant $v_3\left\{2\right\}$. Data points are experimental measurements from ALICE [42].

Figure 4

Principal component decomposition of the observables $\sqrt{\langle N_{\mathrm{ch}}\rangle },v_2\left\{2\right\}$ for the Glauber model in 20–25% centrality. Each data point represents a model calculation and the edge histograms show the approximate normal distribution of each observable. Arrows represent the PC vectors with lengths proportional to the explained variance.

Figure 5

Fraction of the variance $V\left(q\right)$ explained by the first $q$ principal components for Glauber (blue circles) and KLN (green triangles). $q=5$ explains approximately 99% of the total variance, a significant reduction from the original 18 dimensions.

Figure 6

Visualization of the principal component transformation matrices $U_q$ for Glauber (left) and KLN (right). The numerical values of each matrix element are annotated and color-coded, where darker red indicates more positive values, darker blue indicates more negative, and grey indicates zero.

Figure 7

Validation of the Gaussian process emulator for the Glauber model. Each plot shows emulator predictions against explicit calculations for the 64 validation design points in centrality bins 0–5% (green circles), 20–25% (orange triangles), and 40–45% (purple squares). The $x$ value of each data point is the emulator prediction with $2\sigma$ (95%) horizontal error bars, the $y$ value is the explicit calculation with $2\sigma$ (95%) vertical error bars, and the diagonal grey line represents $y=x$.

Figure 8

Posterior marginal and joint distributions of the calibration parameters for the Glauber model. On the diagonal are histograms of MCMC samples for the respective parameters, on the lower triangle are two-dimensional scatter histograms of MCMC samples showing the correlation between pairs of parameters, and on the upper triangle are approximate contours for 68%, 95%, and 99% confidence regions along with a dot indicating the median.

Figure 9

Same as Fig. 8 for the KLN model.

Figure 10

Random realizations of the calibrated posterior for Glauber (top, blue) and KLN (bottom, green) initial conditions. Similar to Fig. 3 except the lines are posterior emulator predictions instead of explicit prior calculations.

Figure 11

Comparison of posterior distributions of $\eta /s$ for Glauber (blue) and KLN (green). These are the same histograms as in Figs. 8 and 9, expanded and placed on the same axis. The vertical grey lines indicate the common values 0.08 for Glauber and 0.20 for KLN [28, 47].

Figure 12

Effect of varying the hyperparameters. Each panel shows a Gaussian process conditioned on fabricated training data using the covariance function Eq. (A2), where the line is the mean and the band is a $2\sigma$ confidence interval. The covariance function hyperparameters are different in each plot as indicated by the annotations. The data points are identical in each plot and were generated from a Gaussian process with hyperparameters annotated at the bottom.