Transport coefficients from in-medium quarkonium dynamics

The in-medium dynamics of heavy particles are governed by transport coefficients. The heavy quark momentum diffusion coefficient, $\kappa$, is an object of special interest in the literature, but one which has proven notoriously difficult to estimate, despite the fact that it has been computed by weak-coupling methods at next-to-leading order accuracy, and by lattice simulations of the pure SU(3) gauge theory. Another coefficient, $\gamma$, has been recently identified. It can be understood as the dispersive counterpart of $\kappa$. Little is known about $\gamma$. Both $\kappa$ and $\gamma$ are, however, of foremost importance in heavy quarkonium physics as they entirely determine the in and out of equilibrium dynamics of quarkonium in a medium, if the evolution of the density matrix is Markovian, and the motion, quantum Brownian; the medium could be a strongly or weakly coupled plasma. In this paper, using the relation between $\kappa$, $\gamma$ and the quarkonium in-medium width and mass shift respectively, we evaluate the two coefficients from existing $2+1$ flavor lattice QCD data. The resulting range for $\kappa$ is consistent with earlier determinations, the one for $\gamma$ is the first nonperturbative determination of this quantity.

Figure 1

A diagrammatic representation of the leading order color-singlet self-energy diagram, ${\mathrm{\Sigma }}_s$, in pNRQCD. Single lines represent quark-antiquark color-singlet propagators, double lines quark-antiquark color-octet propagators, curly lines gluons, and crossed circles chromoelectric dipole vertices.

Figure 2

The first three entries (black bars) show $\gamma /T^3$ as obtained from Eq. (32) using lattice data of Ref. [32] for the thermal mass shift of the $J/\psi$ and of the $\mathrm{\Upsilon }(1S)$ at two different temperatures. The error bars account for the lattice uncertainties only. The last two entries (blue bars) provide $\gamma /T^3$ from the perturbative, leading order, expression of the thermal mass shift with the strong coupling computed at $\pi$ times the two different temperatures 251 and 407 MeV. We assign a 50% uncertainty to these results. The gray band gives our final range for $\gamma /T^3$, see text.

Figure 3

The first entry (black bar) shows $\kappa /T^3$ as obtained from Eq. (31) using lattice data of Refs. [31, 32] for the upper and lower bounds of the thermal decay width of the $\mathrm{\Upsilon }(1S)$. The second entry (brown bar) reports the (quenched) lattice estimate of Ref. [20]. The third and fourth entries (green bars) are the determinations based on the ALICE [24] and STAR [25] measurements of the $D$-meson azimuthal anisotropy coefficient $v_2$, respectively. The fifth entry (red bar) is the determination based on a range of values of the heavy-quark spatial diffusion coefficient obtained in [27] from an analysis of phenomenological models. The sixth entry (blue bar) is a perturbative result (see text) with the strong coupling computed at the scale $\pi \times (407\mathrm{MeV})=1.28\mathrm{GeV}$. We assign a 50% uncertainty to it.