Applying Bayesian parameter estimation to relativistic heavy-ion collisions: Simultaneous characterization of the initial state and quark-gluon plasma medium

We quantitatively estimate properties of the quark-gluon plasma created in ultrarelativistic heavy-ion collisions utilizing Bayesian statistics and a multiparameter model-to-data comparison. The study is performed using a recently developed parametric initial condition model, ${\mathrm{T}}_{\mathrm{R}}\text{ENTo}$, which interpolates among a general class of particle production schemes, and a modern hybrid model which couples viscous hydrodynamics to a hadronic cascade. We calibrate the model to multiplicity, transverse momentum, and flow data and report constraints on the parametrized initial conditions and the temperature-dependent transport coefficients of the quark-gluon plasma. We show that initial entropy deposition is consistent with a saturation-based picture, extract a relation between the minimum value and slope of the temperature-dependent specific shear viscosity, and find a clear signal for a nonzero bulk viscosity.

Figure 1

Several randomly generated ${\mathrm{T}}_{\mathrm{R}}\text{ENTo}$ Pb+Pb initial condition events using generalized mean parameter $p=0$, nucleon width $w=0.5$ fm, and $\gamma$ fluctuation factor $k=1.4$.

Figure 2

Cross section of the participant nucleon density in a midcentral Pb+Pb collision at $\sqrt{s_{NN}}=2.76$ TeV as a function of the transverse coordinate $x$ parallel to impact parameter $b$. The gray band indicates the region bounded by the minimum and maximum values of the local participant thickness functions ${\tilde{T}}_A$ and ${\tilde{T}}_B$, while the blue band indicates the region spanned by the generalized mean of ${\tilde{T}}_A$ and ${\tilde{T}}_B$ with parameter $-1<p<1$. The solid blue line shows an example of a discrete mapping specified by a generalized mean with $p=0$.

Figure 3

Average charged particle density per participant pair $(d{N}_{\text{ch}}/d\eta )/(N_{\text{part}}/2)$ at midrapidity as a function of participant number for Pb+Pb, $\mathrm{p}+\mathrm{Pb}$, and Au+Au systems at various collision energies. Lines are ${\mathrm{T}}_{\mathrm{R}}\text{ENTo}$ calculations with generalized mean parameter $p=0$, and symbols are experimental data from PHENIX [76] and ALICE [80, 81]. The average minimum bias participant number for $\mathrm{p}+\mathrm{Pb}$ is shifted for clarity.

Figure 4

Profiles of the initial thermal distribution predicted by the KLN (left), EKRT (middle), and wounded nucleon (right) models (dashed black lines) compared to a generalized mean with different values of the parameter $p$ (solid blue lines). Staggered lines show different slices of the initial entropy density $dS/\left(d^2{r}_{\perp }dy\right)$ as a function of the participant nucleon density ${\tilde{T}}_A$ for several values of ${\tilde{T}}_B=1,2,3 {\mathrm{[fm}}^{-2}$]. The EKRT mapping is shown with model parameters $K=0.64$ and $\beta =0.8$ [19]. Entropy normalization is arbitrary.

Figure 5

Eccentricity harmonics ${\varepsilon }_2$ and ${\varepsilon }_3$ as a function of impact parameter $b$ for Pb+Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV calculated from IP-Glasma and ${\mathrm{T}}_{\mathrm{R}}\text{ENTo}$ initial conditions. IP-Glasma events are evaluated after $\tau =0.4$ fm/$c$ classical Yang-Mills evolution [18]; ${\mathrm{T}}_{\mathrm{R}}\text{ENTo}$ events after $\tau =0.4$ fm/$c$ free streaming [72, 98] and using parameters $p=0\pm 0.1, k=1.6$, and nucleon width $w=0.4$ fm to match IP-Glasma [99].

Figure 6

Validation of Gaussian process emulator predictions. Each panel shows predictions compared to explicit model calculations at the 50 validation design points. The horizontal location and error bar of each point indicates the predicted value and uncertainty, vertical indicates the explicitly calculated value and statistical uncertainty, and the diagonal gray line represents perfect agreement. Left: charged pion yields $d{N}_{{\pi }^{\pm }}/dy$, middle: mean pion transverse momenta ${\langle p_T\rangle }_{{\pi }^{\pm }}$, right: flow cumulant $v_2\left\{2\right\}$; each in centrality bins 0–5% (blue circles) and 30–40% (orange squares).

Figure 7

Posterior distributions for the model parameters from calibrating to identified particles yields (blue, solid lines, lower triangle) and charged particles yields (red, dashed lines, upper triangle). The diagonal has marginal distributions for each parameter, while the off-diagonal contains joint distributions showing correlations among pairs of parameters. ${}^{†}\mathrm{th}\mathrm{e}$ units for $\eta /s$ slope are ${\mathrm{[GeV}}^{-1}$].

Figure 8

Simulated observables compared to experimental data from the ALICE experiment [107, 108]. Top row: explicit model calculations for each of the 300 design points; bottom: emulator predictions of 100 random samples drawn from the posterior distribution. Left column: identified particle yields $dN/dy$; middle: mean transverse momenta $\langle p_T\rangle$; right: flow cumulants $v_n\left\{2\right\}$.

Figure 9

Posterior distribution of the ${\mathrm{T}}_{\mathrm{R}}\text{ENTo}$ entropy deposition parameter $p$ introduced in Eq. (14). Approximate $p$ values are annotated for the KLN ($p\approx 0.67\pm 0.01$), EKRT ($p\approx 0.0\pm 0.1$), and wounded nucleon ($p=1$) models.

Figure 10

Estimated temperature dependence of the shear viscosity $(\eta /s)\left(T\right)$ for $T>T_c=0.154$ GeV. The gray shaded region indicates the prior range for the linear $(\eta /s)\left(T\right)$ parametrization Eq. (31), the blue line is the median from the posterior distribution, and the blue band is a 90% credible region. The horizontal gray line indicates the KSS bound $\eta /s\ge 1/4\pi$ [12, 13, 14].

Figure 11

Model calculations using the high-probability parameters listed in Table 4. Solid lines are calculations using parameters based on the identified particle posterior, dashed lines are based on the charged particle posterior, and points are data from the ALICE experiment [107, 108]. Top row: calculations of identified or charged particle yields $dN/dy$ or $d{N}_{\text{ch}}/d\eta$ (left), mean transverse momenta $\langle p_T\rangle$ (middle), and flow cumulants $v_n\left\{2\right\}$ (right) compared to data. Bottom: ratio of model calculations to data, where the gray band indicates $\pm 10\%$.