the group velocity. For a metal with the usual quadratic dispersion, Eq. (\[eqphik\]) is true if we use constant relaxation time, which turns out to be very good approximation for metal. For graphene, the group velocity in $x$-direction is $v_F \cos(\theta)$ with the Fermi velocity a constant, thus we must demand a relaxation time proportional to $k$, which turns out in agreement with experiments [@antonio-RMP2009]. For a full lattice model numerical calculation such that $\Phi_{\bf k}$ respects the lattice symmetry, we take the drift term to be $\Phi_{\bf k} = v_1 {\rm Re} \left[ z^{*} \frac{\partial z}{\partial k_x} \right]/ (\hbar v_F^2)$, where $v_F = 3 a t/(2\hbar)$ is the graphene Fermi velocity, and $z = 
-t ( e^{-i k_x a} + e^{i (k_x a/2 + k_y a \sqrt{3}/2)} + e^{i (k_x a/2 - k_y a \sqrt{3}/2)} )$. In Fig. \[figurePi\], we compare Doppler shifted Wunsch et al. expression [@guinea06] (or that of Hwang and Sarma [@hwang-sarma07]) which is a nonlocal result at zero temperature or that of Falkovsky’s small $q$ (local) result[@falkovsky08] at $T=100\,$K. As we can see they differ a lot and do not agree with the correct way of evaluating the drifted polarization function. On this scale of 