We denote the number of “colors” of the $\mathrm{SU}(N_c)$ gauge group by $N_c$, the number of real adjoint scalars in the theory by $N_X$, and the number of Dirac fermions in representation $F$ by $N_f$. $T_F$ is the index of the representation $\mathrm{Tr}[T_F^a{T}_F^b]=T_F{\delta }^{ab}$.

By construction, ${\mathcal{I}}_{\text{Match}}$ is the leading term in the expansion of ${\mathcal{I}}_{\text{Soft}}$ in powers of $m_D^2/|\mathbf{p}{|}^2$, since the HTL-resummed propagator depends on the coupling and temperature only through $m_D^2$. Because ${\mathcal{I}}_{\text{Match}}\propto m_D^2/|\mathbf{p}{|}^4$, the difference (8) follows as written.

It is also reassuring to note that the longitudinal part of the HTL-resummed contributions (in the first two lines) in the kernel (10) satisfies the Einstein relation. This is mandated by the fact that in the extreme weakly coupled limit equilibration should still be attained, and the fact that this part of the kernel is quadratic means that the associated stochastic process is Gaussian.

We note that our calculation is based on the QCD Wilson line $\mathrm{P}\exp (ig\int A_{\mu }d{x}^{\mu })$. A different calculation in $\mathcal{N}=4$ SYM would involve Wilson lines that also include a direct coupling of the HQ to adjoint scalars, like earlier studies of HQ diffusion in this theory have done [19, 50]. However, as discussed in [32], the distinction between the two types of Wilson lines is inconsequential for the calculation of the Wilson loop (2) at strong coupling [51, 52, 53], which is what we aim to compare against here.