Linearized Boltzmann-Langevin model for heavy quark transport in hot and dense QCD matter

In relativistic heavy-ion collisions, the production of heavy quarks at large transverse momenta is strongly suppressed compared to proton-proton collisions. In addition, an unexpectedly large azimuthal anisotropy was observed for the emission of charmed hadrons in noncentral collisions. Both observations pose challenges to the theoretical understanding of the coupling between heavy quarks and the quark-gluon plasma produced in these collisions. Transport models for the evolution of heavy quarks in a QCD medium offer the opportunity to study these effects; two of the most successful approaches are based on the linearized Boltzmann transport equation and the Langevin equation. In this work, we develop a hybrid transport model that combines the strengths of both of these approaches: Heavy quarks scatter with medium partons using matrix-elements calculated in perturbative QCD, while between these discrete hard scatterings they evolve using a Langevin equation with empirical transport coefficients to capture the nonperturbative soft part of the interaction. With the hybrid transport model coupled to a state-of-the-art event-by-event bulk evolution model based on 2+1D relativistic viscous fluid dynamics, we study the azimuthal anisotropy and nuclear modification factor of heavy quarks in Pb+Pb collisions at $\sqrt{s}=5.02$ TeV. The parameters related to heavy-flavor transport are calibrated using a Bayesian analysis comparing them to available $D$-meson and $B$-meson data at the Large Hadron Collider. Using the calibrated model, we study the implications on heavy-flavor transport properties and predict observables.

Figure 1

Elastic processes: The first diagram corresponds to heavy quark ($Q$)–light quark ($q$, $\overline{q}$) scattering. The last three diagrams contribute to heavy quark ($Q$)–gluon ($g$) scattering.

Figure 2

Inelastic processes: A heavy quark collides with a medium light (anti)quark or gluon and radiates an additional gluon.

Figure 3

The approach to thermalization of the linear Boltzmann equation with elastic processes only (red dotted line), elastic and radiation processes (green dashed line), and elastic with both radiation and absorption processes (blue solid line). The static medium has a temperature $T=0.4$ GeV, and ${10}^4$ heavy quarks are initialized with $E=10$ GeV at $t=0$.

Figure 4

Top row: Energy loss fraction as function of energy for elastic processes (left) and inelastic processes (right). Bottom row: Energy loss fraction as function of path length for elastic processes (left) and inelastic processes (right).

Figure 5

$R_{AA}$ in a static medium for charm and bottom quarks. The medium size is 3 fm and has a temperature $T_0=0.3$ GeV. The initial-state spectrum of charm and bottom quarks for this calculation is a simple power law form with fit parameters from Ref. [44].

Figure 6

Left: the prior, i.e., the full range of calculations in parameter space. Right: the posterior, i.e., observables sampled from model emulators after calibration. In both figures, blue (green) lines are calculations with EPPS (nCTEQ15np) nuclear PDF.

Figure 7

Marginalized posterior probability distribution of model parameters. Diagonal plots show the marginalization on a single parameter. Off-diagonal plots show the pair correlation between parameters. Blue (green) lines and lower (upper) off-diagonal plots correspond to the extraction using EPPS (nCTEQ15np) nuclear PDF.

Figure 8

Coupling constant with three values of the medium scale parameter $\mu$ taken from the high-likelihood region of the posterior. Left: ${\alpha }_s$ evaluated at a process scale $Q=0$. Right: ${\alpha }_s$ evaluated at a process scale $Q=m_D$.

Figure 9

Posterior range of the heavy quark transverse momentum broadening parameter $\widehat{q}$ from Eq. (19). The results include the uncertainty from using different nuclear PDFs. Blue boxed region is for bottom quarks and red slashed region for charm quarks. The shaded region indicates an extraction described in Ref. [21] but with an updated medium evolution model.

Figure 10

Posterior range of the heavy quark spatial diffusion coefficient. The blue boxed region is for bottom quarks and the red slashed region is for charm quarks. The shaded region indicates an extraction described in Ref. [21] but with an updated medium evolution model. Square and diamond symbols are lattice calculations in the static heavy quark limit [19, 20]; triangular symbols are lattice calculations with physical charm quark mass [18].

Figure 11

Calculation of heavy-flavor $R_{AA}$ using a high-likelihood parameter set. Results are compared to ALICE $D$-meson measurements in Pb+Pb and $p$+Pb [63] and STAR $D$-meson measurements in Au+Au [64]. $B$ meson $R_{AA}$ are predicted.

Figure 12

Calculation of heavy-flavor flows with a high-likelihood parameter set. Results are compared to CMS $D$-meson measurements in Pb+Pb [60] and STAR $D$-meson measurements in Au+Au [65]. $D$ meson $v_3$ and $B$ meson $v_2,v_3$ are predictions.

Figure 13

Top row: Centrality dependence of $D$-meson direct flow with respect to the $n=3$ event plane. Left: The prior range. Right: Predictions using a high-likelihood parameter set. Bottom left: The transverse momentum dependence of $D$-meson direct flow with respect to the $n=3$ event plane in the 30–50% centrality bin. Bottom right: A sample TRENToevent. The third-order eccentricity would drive a $v_3$. A measurement along the $n=3$ direction would introduce reflection asymmetry in the heavy quark energy loss.

Figure 14

Comparison of the heavy quark transport coefficient (left, $\widehat{q}$; right, $D_s$) with (solid lines) and without (dashed lines) the inclusion of the inelastic contribution. Thin blue lines denote the bottom quark and thick red lines denote the charm quark region. Blue symbols are lattice calculations in static heavy quark limit [19] and red symbols are lattice calculations with physical charm quark mass [18].