replaced with $\omega^2\delta G^R(\omega)$ ($\omega^2\delta \rho(\omega)$), $$\begin{aligned}
\begin{split}
&\omega^2\delta G_{T}(\omega,{{\mathbf{p}}})|_{\omega\rightarrow 0}-\omega^2\delta G_{T}(\omega,{{\mathbf{p}}})|_{\omega\rightarrow \infty} \\
&= \frac{2}{\pi}\int^\infty_{0} d\omega \omega
\delta \rho_{T}(\omega,{{\mathbf{p}}}),
\end{split}\end{aligned}$$ for the transverse component. By using Eqs. (\[eq:OPE-w-RG-T\]) and (\[eq:hydro-GR-T\]), this equation becomes $$\begin{aligned}
\label{eq:sumrule-2-T}
\begin{split}
&-e^2\sum q^2_f 
\Biggl[\left\{2m_f \delta\left\langle\overline{\psi}_f \psi_f \right\rangle
+\frac{1}{12}\delta\left\langle \frac{\alpha_s}{\pi}G^2\right\rangle \right\}\\
&~~~+\frac{8}{3}\frac{1}{4{C_F}+{N_f}}\delta\left\langle T^{00} \right\rangle 
\Biggr] 
= \frac{2}{\pi}\int^\infty_{0} d\omega \omega
\delta \rho_{T}(\omega,{{\mathbf{p}}}). 
\end{split}\end{aligned}$$ We call this equation sum rule 2 in what follows. It should be noted that there is no explicit $|{{\mathbf{p}}}|$ dependence on the left-hand side, but again we are implicitly assuming that $|{{\mathbf{p}}}|$ is small enough so that the hydrodynamics well describes the behavior of $\omega^2\delta G_{T}(\omega,{{\mathbf{p}}})|_{\omega\rightarrow 0}$. Also, we emphasize that the expectation values of the local operators on the left-hand side can be evaluated nonperturbatively by lattice QCD without suffering from the problem of analytic continuation. Therefore, sum rule 2 can be used to constrain the shape of the spectral function. As it was already discussed in the previous paper [@Gubler:2016hnf], lattice QCD results show that the left-hand side of Eq.(\[eq:sumrule-2-T\]) is found to be dominated by the $\left\langle T^{00} \right\rangle$ for almost all temperatures around and above $T_c$.

Let us evaluate the contributions to sum rule 2 from the transport 