peak, continuum, and UV tail. The contribution from the transport peak is found by using Eq. (\[eq:transport-T\]) as $$\begin{aligned}
\label{eq:T-sumrule2-transport}
\begin{split}
\frac{2}{\pi}\int^\infty_{0} d\omega \omega
\delta \rho_{T}(\omega,{{\mathbf{p}}})
&=  {C_{\text{em}}}{N_c}\chi \frac{2}{3} \frac{2}{\pi}\int^\infty_{0} d\omega 
  \frac{\tau^{-1} \omega^2}{\omega^2+\tau^{-2}} \\
&~~~\times \left[1 +\frac{{{\mathbf{p}}}^2}{5} \frac{(3\omega^2-\tau^{-2})}{(\omega^2+\tau^{-2})^2}\right]\\
 &= {C_{\text{em}}}{N_c}\chi \frac{2}{3}
 \left[ \frac{2}{\pi}\tau^{-1}\varLambda +\frac{{{\mathbf{p}}}^2}{5} \right] ,
 \end{split}\end{aligned}$$ where we have introduced the UV cutoff $\varLambda$ for the transport peak because of the linear UV divergence, and used $\int^\infty_{0} d\omega \omega^2(3\omega^2-\tau^{-2})/(\omega^2+\tau^{-2})^3=\pi \tau/2$. As the Boltzmann equation cannot be used when $\omega\gtrsim gT$ since the instantaneous scattering description becomes invalid [@Moore:2006qn], we set $\varLambda\sim gT$.

The contribution from the continuum (\[eq:continuum-T\]) reads $$\begin{aligned}
\label{eq:T-sumrule2-cont}
\begin{split}
& \frac{2}{\pi}\int^\infty_{0} d\omega \omega
\delta \rho_{T}(\omega,{{\mathbf{p}}}) \\
&=  -  C_{\text{em}}{N_c}\frac{1}{3\pi^2} 
\int^\infty_{0} d\omega \omega^3
 \Biggl[{n_F}\left(\frac{\omega}{2}\right) \\
&~~~~ +{{\mathbf{p}}}^2\left\{\frac{1}{20}{n_F}''\left(\frac{\omega}{2}\right)
 -\frac{1}{\omega^2}{n_F}\left(\frac{\omega}{2}\right)\right\}\Biggr] \\
 &= -  C_{\text{em}}{N_c}\frac{1}{45} T^2
 \Biggl[ 14\pi^2T^2  +{{\mathbf{p}}}^2 \Biggr] ,
\end{split}\end{aligned}$$ where we have used $\int^\infty_{0} d\omega \omega^3 {n_F}(\omega/2)=14\pi^4T^4/15$ and $\int^\infty_{0} d\omega \omega^3 {n_F}'' (\omega/2)=8\pi^2T^2$.

The UV tail contribution is, by using Eq. (\[eq:UVtail-T\]), found to be $$\begin{aligned}
\label{eq:T-sumrule2-UVtail}
\begin{split}
 \frac{2}{\pi}\int^\infty_{0} d\omega \omega
\delta \rho_{T}(\omega,{{\mathbf{p}}})
&=  \frac{2}{\pi} {C_{\text{em}}}{N_c}{C_F}\frac{4\pi^2T^4}{27}   \alpha_s(\kappa_0)  \\
&~~~\times \int^\infty_{X_0} dX
 \left[\frac{X_0}{X}\right]^{\tilde{a}+1} \\
&~~~+{\cal O}\left({C_{\text{em}}}g^2T^2{{\mathbf{p}}}^2\right)\\
&=  {C_{\text{em}}}{N_c}{C_F}\frac{4\pi^2T^4}{9}   \frac{2}{4{C_F}+{N_f}}   \\
&~~~+{\cal O}\left({C_{\text{em}}}g^2T^2{{\mathbf{p}}}^2\right),
\end{split}\end{aligned}$$ where we have introduced $X\equiv \ln(\omega/\Lambda_{\text{QCD}})$ and $X_0\equiv \ln(\kappa_0/\Lambda_{\text{QCD}})$, used Eq. (\[eq:RG-alphas\]) and introduced the 