Heavy Quark Diffusion in Perturbative QCD at Next-to-Leading Order

We compute the momentum diffusion coefficient of a heavy quark in a hot QCD plasma, to next-to-leading order in the weak-coupling expansion. Corrections arise at $\mathcal{O}(g_s)$; physically they represent interference between overlapping scatterings, as well as soft, electric scale ($p\sim gT$) gauge field physics, which we treat using the hard thermal loop effective theory. In 3-color, 3-flavor QCD, the momentum diffusion constant of a fundamental representation heavy quark at next-to-leading order is $\kappa =\frac{16\pi }{3}{\alpha }_s^2{T}^3(\ln \frac{1}{g_s}+0.07428+1.9026g_s)$. The convergence of the perturbative expansion is poor.

Figure 1

Leading-order contribution to heavy quark diffusion and its correspondence to scattering processes. On the left the double line represents the Wilson line; on the right it is the heavy quark external states.

Figure 2

Diagrams required at NLO. The double line is the Wilson line; otherwise all propagators are soft and HTL resummed and all vertices include the HTL vertex. All lines attached to the Wilson line are longitudinal.

Figure 3

Comparison of leading and NLO results for $N_f=3$ QCD as a function of coupling.