later. At $T=0$, Lorentz invariance guarantees the tensor structure of the correlator, $$\begin{aligned}
G^R_{\mu\nu}(p)
&= p^2P^0_{\mu\nu}(p) \tilde{G}^R(p^2),\end{aligned}$$ where $\tilde{G}^R(0)=0$ due to the renormalization condition of the electric charge [@Peskin:1995ev]. The transverse component is given as $G_T(p)= p^2\tilde{G}^R(p^2)$. $\tilde{G}^R(p^2)$ is regular at $p^2=0$ due to the renormalization condition, so it is easy to see that there are no contributions to $\sigma$, $\tau_J$, and $\kappa_B$ at $T=0$, from Eq. (\[eq:hydro-GR-T\]). The $T=0$ contribution only enters in the higher order terms, which are neglected in Eq. (\[eq:hydro-GR-T\]).

On the other hand, the longitudinal component reads $G^R_{00}(p)={{\mathbf{p}}}^2 \tilde{G}^R(p^2)$, so $G^R_{00}(p)$ at $T=0$ and $\omega=0$ is of order ${{\mathbf{p}}}^2 \tilde{G}^R(-{{\mathbf{p}}}^2)$. The $T=0$ contribution does not affect the sum rules 2 (\[eq:sumrule-2-L\]) and 3 (\[eq:sumrule-3-L\]) because as is explained in Sec. \[ssc:sumrule-L\], the relevant quantities to the derivation of these sum rules are $\omega^2\delta G^R_{00}(p)|_{\omega\rightarrow 0}$ and $\omega^4\delta G^R_{00}(p)|_{\omega\rightarrow 0}$, and the $T=0$ contributions vanish. The only sum rule which $T=0$ terms may affect is sum rule 1 (\[eq:sumrule-1-L\]). Nevertheless, we only consider terms in $G^R_{00}(p)|_{\omega\rightarrow 0}$ up to ${{\mathbf{p}}}^2$ in this paper, as is shown in Sec. \[ssc:sumrule-L\]. Therefore, sum rule 1 is also unaffected.

We note that we have so far neglected effects of 