&=e^2\sum q^2_f \frac{1}{p^2}\frac{{{\mathbf{p}}}^2}{p^2} \\ 
\nonumber
&~~~\times \Bigl[\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\} \\
\label{eq:OPE-w-RG-L}
&~~~+\frac{8}{3}\frac{1}{4{C_F}+{N_f}} \delta\left\langle T^{00} \right\rangle  
\Bigr]
+{\cal O}(\omega^{-6}). \end{aligned}$$ We note that, in the $\omega\rightarrow \infty$ limit, which is relevant to the derivation of the sum rule, the asymptotic freedom of QCD guarantees that the above expression is exact. In other words, all higher order $\alpha_s$ corrections vanish in this limit.

IR behavior {#ssc:IR}
-----------

On the other hand, the asymptotic behavior in the IR region is described by hydrodynamics [@Kovtun:2012rj], as long as the spatial momentum is small enough. In the channel of the EM current, the basic equations consist of the conservation law and the constituent equation, $$\begin{aligned}
\label{eq:conservation-law}
\partial_0j^0&= -\nabla\cdot {{\mathbf{j}}},\\
\nonumber
{{\mathbf{j}}}&= -D\nabla j^0+\sigma{{\mathbf{E}}}-\sigma\tau_J\partial_0 {{\mathbf{E}}}+\kappa_B\nabla\times{{\mathbf{B}}}\\ 
\label{eq:constituent-eq}
&~~~+{\cal O}(\partial^2 {{\mathbf{E}}}, \partial^2 {{\mathbf{B}}}, \partial^2 j^0 ),\end{aligned}$$ where $D$ is the diffusion constant, $\sigma$ the electrical conductivity and $\tau_J$ and $\kappa_B$ second order transport coefficients corresponding to $\partial_0{{\mathbf{E}}}$ and $\nabla\times{{\mathbf{B}}}$, respectively. ${{\mathbf{E}}}\equiv -\nabla A^0-\partial_0 {{\mathbf{A}}}$ and ${{\mathbf{B}}}\equiv\nabla\times{{\mathbf{A}}}$ are the electric and magnetic fields, where $A^\mu$ is the vector potential.

After performing the Fourier transformation that is defined as $f(p)\equiv \int d^4x e^{ip\cdot x} f(x)$, Eqs. (\[eq:conservation-law\]) and (\[eq:constituent-eq\]) become $$\begin{aligned}
\label{eq:conservation-law-p}
\omega j^0&= {{\mathbf{p}}}\cdot {{\mathbf{j}}},\\
