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How Much do Heavy Quarks Thermalize in a Heavy Ion Collision?

Guy D. Moore, Derek Teaney

TL;DR

The paper addresses how much heavy quarks thermalize in a QGP created in relativistic heavy-ion collisions. It computes the heavy-quark diffusion coefficient $D$ in perturbative QGP via Hard Thermal Loop methods and connects it to collisional energy loss and momentum broadening, then develops a Boltzmann-Langevin framework to evolve heavy-quark spectra. Analytic solutions for a Bjorken-expanding medium and numeric Langevin runs embedded in hydrodynamics yield predictions for the nuclear modification factor $R_{AA}$ and elliptic flow $v_2(p_T)$ of charm quarks, showing a tight correlation between suppression and flow governed by $D$. The results provide a concrete, testable link between microscopic transport coefficients and observable heavy-quark dynamics, with implications for lattice QCD comparisons and RHIC phenomenology.

Abstract

We investigate the thermalization of charm quarks in high energy heavy ion collisions. To this end, we calculate the diffusion coefficient in the perturbative Quark Gluon Plasma and relate it to collisional energy loss and momentum broadening. We then use these transport properties to formulate a Langevin model for the evolution of the heavy quark spectrum in the hot medium. The model is strictly valid in the non-relativistic limit and for all velocities $γv < \alphas^{-1/2}$ to leading logarithm in $T/m_D$. The corresponding Fokker-Planck equation can be solved analytically for a Bjorken expansion and the solution gives a simple estimate for the medium modifications of the heavy quark spectrum as a function of the diffusion coefficient. Finally we solve the Langevin equations numerically in a hydrodynamic simulation of the heavy ion reaction. The results of this simulation are the medium modifications of the charm spectrum $R_{AA}$ and the expected elliptic flow $v_2(p_T)$ as a function of the diffusion coefficient.

How Much do Heavy Quarks Thermalize in a Heavy Ion Collision?

TL;DR

The paper addresses how much heavy quarks thermalize in a QGP created in relativistic heavy-ion collisions. It computes the heavy-quark diffusion coefficient in perturbative QGP via Hard Thermal Loop methods and connects it to collisional energy loss and momentum broadening, then develops a Boltzmann-Langevin framework to evolve heavy-quark spectra. Analytic solutions for a Bjorken-expanding medium and numeric Langevin runs embedded in hydrodynamics yield predictions for the nuclear modification factor and elliptic flow of charm quarks, showing a tight correlation between suppression and flow governed by . The results provide a concrete, testable link between microscopic transport coefficients and observable heavy-quark dynamics, with implications for lattice QCD comparisons and RHIC phenomenology.

Abstract

We investigate the thermalization of charm quarks in high energy heavy ion collisions. To this end, we calculate the diffusion coefficient in the perturbative Quark Gluon Plasma and relate it to collisional energy loss and momentum broadening. We then use these transport properties to formulate a Langevin model for the evolution of the heavy quark spectrum in the hot medium. The model is strictly valid in the non-relativistic limit and for all velocities to leading logarithm in . The corresponding Fokker-Planck equation can be solved analytically for a Bjorken expansion and the solution gives a simple estimate for the medium modifications of the heavy quark spectrum as a function of the diffusion coefficient. Finally we solve the Langevin equations numerically in a hydrodynamic simulation of the heavy ion reaction. The results of this simulation are the medium modifications of the charm spectrum and the expected elliptic flow as a function of the diffusion coefficient.

Paper Structure

This paper contains 17 sections, 107 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Non-Relativistic Heavy Quarks in a Thermal Medium
  3. Energy loss
  4. Why $2\leftrightarrow 2$ dominates for $\gamma v \sim 1$
  5. Relativistic heavy quarks; momentum loss and momentum diffusion
  6. The Boltzmann Fokker-Planck Equation
  7. A Langevin Model For Heavy Ion Collisions
  8. Solution to the Fokker-Planck Equation for a Bjorken Expansion
  9. Elliptic Flow and Suppression of Charm Quarks
  10. Comparison with Boltzmann Simulations
  11. Discussion
  12. Solution of The Fokker-Plank equation fork a Bjorken expansion
  13. Details of leading-order calculations
  14. Matrix elements
  15. Momentum Diffusion -- Non-Relativistic Case
  16. ...and 2 more sections

Figures (5)

  • Figure 1: (Color online) (a) The diffusion coefficient of a heavy quark in a QGP with $N_f$ flavors of light quarks. (b) The ratio of the diffusion coefficient of a heavy quark to the hydrodynamic diffusion coefficient $\eta/(e+p)$.
  • Figure 2: (Color online) (a) The drag coefficient as a function of the heavy quark momentum, $\frac{dp}{dt}=\eta_D(p) p$. (b) The transverse ($\kappa_T(p)$) and longitudinal ($\kappa_L(p)$) momentum diffusion coefficients as a function of heavy quark momentum for $m_{D}/T=1.5$ . For comparison we also show the longitudinal momentum diffusion coefficient in a Boltzmann-Langevin approach ($\frac{2 T}{v}\, \frac{dp}{dt}$).
  • Figure 3: (Color online) The transverse momentum spectrum of charm quarks at time $t_f=6\,\hbox{fm}$ for a Bjorken expansion with $\tau_0=1\,\hbox{fm}$ and $T_0=300\,\hbox{MeV}$ and $T(t_f)=165\,\hbox{MeV}$. The initial transverse momentum spectrum is given by leading order perturbation theory (LO pQCD).
  • Figure 4: (Color online) (a) The nuclear modification factor $R_{AA}$ for charm quarks for representative values of the diffusion coefficient. (b) $v_2(p_T)$ for charm quarks for the same set of diffusion coefficients given in the legend in (a). In perturbation theory, $D \times (2\pi T) \approx 6\,(0.5/\alpha_{\rm s})^2$. The model for the drag and fluctuation coefficients is referred to as LO QCD in the text. The band estimates the light hadron elliptic flow for impact parameter $b=6.5\,\hbox{fm}$ using STAR data Adams:2004bi.
  • Figure 5: (Color online) (a) The charm quark nuclear modification factor $R_{AA}$ and (b) elliptic flow for representative values of the diffusion coefficient given in the legend. In this model, the drag is proportional to the velocity, $\frac{dp}{dt}\propto v$. For further explanation see Fig. \ref{['RAA']}.