there is drifted electron current: $$\begin{aligned}
    &\Pi^{<}_{{\bf k}, \rm noneq}(\omega) = 2 i N_{{\bf k}, \rm noneq}(\omega) \mathrm{Im} \Pi^{r}_{{\bf k}, \rm noneq}(\omega),\\
    &N_{{\bf k}, \rm noneq}(\omega)\approx N(\omega-k_x v_1),\end{aligned}$$ where $N(\omega) = 1/(e^{\hbar \omega/(k_bT)}-1)$ is the Bose distribution.

The reflection coefficient needed for the heat transfer calculation is then obtained from the relation $r = v \Pi^r/( 1 - v \Pi^r)$ with bare Coulomb potential in two dimensions $v = 1/(2 \epsilon_0 k)$. The optical conductivity is related to the retarded self energy by $\sigma = i \frac{\omega}{k^2} \Pi^r_{\bf k}(\omega)$.

![Imaginary part of $\Pi^r({\bf k}, \omega, v_1) - \Pi^r({\bf k}, \omega, 0)$, with ${\bf k} = (k_x,0)$, $v_1 = 5 \times 10^5$m/s, at chemical potential $\mu = 0.1\,$eV. Left graph has $k_x = K/1000$ and right 10 time larger, $K = 4\pi/(3 \sqrt{3} a)$ is the magnitude of the K-point vector from the $\Gamma$ point, $a=1.42\,$Å is carbon bond length. Temperature is set at 100K. A damping parameter in $\hbar \omega \to \hbar \omega + i\eta$ is set to $\eta  = 3.3\,$meV. The legends apply to both graphs, full black solid line: tight-binding lattice model with nearest neighbor hopping parameter $t=2.8\,$eV, red dash line: Doppler shifted Wunsch et al. expression, green 