its behavior, we rewrite this operator as $$\begin{aligned}
\label{eq:Tf-decompose}
T^{00}_{f}=T'{}^{00}_{f}+\frac{1}{4{C_F}+{N_f}}\left(T^{00}+\frac{2}{{N_f}}\tilde{T}^{00}\right),\end{aligned}$$ where $$\begin{aligned}
T'{}^{00}_{f}\equiv T^{00}_{f}-\frac{1}{{N_f}}\sum_{f'} T^{00}_{f'},\\
T^{00}\equiv \sum_{f'} T^{00}_{f'}+T^{00}_g,\\
\tilde{T}^{00}\equiv 2C_F  \sum_{f'} T^{00}_{f'}-\frac{{N_f}}{2} T^{00}_g .\end{aligned}$$ Here, $T^{\mu\nu}_g\equiv -G^{\mu\alpha}_{a}G^\nu{}_{\alpha a} +g^{\mu\nu}G^2/4$ is the gluon component of the traceless part of the energy-momentum tensor, $N_f$ is the flavor number, and $C_F\equiv ({N_c}^2-1)/(2{N_c})$. We note that $T^{\mu\nu}$ is the traceless part of the full energy-momentum tensor, not the energy-momentum tensor itself. A standard renormalization group (RG) analysis yields the following scaling properties [@Peskin:1995ev]: $$\begin{aligned}
\label{eq:scaling-T'-Ttilde}
\begin{split} 
T'{}^{00}_{f}(\kappa)&=  \left[\frac{\ln\left(\kappa^2_0/\Lambda^2_{\text{QCD}} \right)}{\ln\left(\kappa^2/\Lambda^2_{\text{QCD}} \right)}\right]^{a'} 
T'{}^{00}_{f}(\kappa_0),\\
 \tilde{T}^{00}(\kappa)
&= \left[\frac{\ln\left(\kappa^2_0/\Lambda^2_{\text{QCD}} \right)}{\ln\left(\kappa^2/\Lambda^2_{\text{QCD}} \right)}\right]^{\tilde{a}}
 \tilde{T}^{00}(\kappa_0),
\end{split} \end{aligned}$$ while $T^{00}$ is independent of $\kappa$. Here $\kappa$ and $\kappa_0$ are renormalization scales, $\Lambda_{\text{QCD}}$ is the QCD scale parameter, $a'\equiv 8{C_F}/(3b_0)$, and $\tilde{a}\equiv 2(4{C_F}+{N_f})/(3b_0)$, where $b_0\equiv (11{N_c}-2{N_f})/3$, which appears in the expression $$\begin{aligned}
\label{eq:RG-alphas}
\alpha_s(\kappa)=\frac{4\pi}{b_0 \ln(\kappa^2/\Lambda^2_{\text{QCD}})}.\end{aligned}$$ In the $\omega\rightarrow \infty$ limit, it is natural to choose the RG scale[^3] as $\kappa^2=\omega^2$.

We see that, except for the $T^{00}$ term, all terms in Eq. (\[eq:Tf-decompose\]) are suppressed logarithmically at large $\omega$. Thus, Eqs. (\[eq:OPE-wo-RG-T\]) and (\[eq:OPE-wo-RG-L\]) become $$\begin{aligned}
\nonumber
\delta G_{T}(\omega,{{\mathbf{p}}}) 
&=e^2\sum q^2_f \frac{1}{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-T}
&~~~+\frac{8}{3}\frac{1}{4{C_F}+{N_f}}\frac{\omega^2+{{\mathbf{p}}}^2}{p^2}\delta\left\langle T^{00} \right\rangle 
\Bigr] \nonumber \\
&~~~+{\cal O}(\omega^{-4}),\\
\nonumber
\delta G^{R}_{00}(\omega,{{\mathbf{p}}}) 
