One should not take the curves seriously when they are outside of the adequate energy regions.

![ The transport peak, the continuum, and the UV tail of the EM current spectral function in the transverse channel $\delta\rho_T$ as a function of $\omega$. The energy unit is $T$. To draw the figure, the parameters are set as ${N_c}={N_f}=3$, $\tau^{-1}/T=0.5$, $\kappa_0/T=1$, and $\Lambda_{\text{QCD}}/T=0.67$. The solid (dashed) lines correspond to a spatial momentum of $|{{\mathbf{p}}}|/T = 0.5$ ($|{{\mathbf{p}}}| = 0$). The vertical lines with the attached arrows indicate the regions for which the respective analytic expressions can be trusted.[]{data-label="fig:spectral-longpaper"}](spectral-longpaper-T.ranges.eps){width="49.00000%"}

### Sum rule 1

By using the asymptotic expression of $\delta G^R_T$ in the UV and IR energy regions, Eqs. (\[eq:OPE-w-RG-T\]) and (\[eq:hydro-GR-T\]), Eq. (\[eq:derivation\]) for $\mu=\nu=1$ becomes $$\begin{aligned}
\label{eq:sumrule-1-T}
\begin{split}
\kappa_B {{\mathbf{p}}}^2
+{\cal O}({{\mathbf{p}}}^4)
= \frac{2}{\pi}\int^\infty_{0} d\omega
\frac{\delta \rho_{T}(\omega,{{\mathbf{p}}})}{\omega},
\end{split}\end{aligned}$$ where the contribution from the UV part has vanished. This is the first sum rule in the transverse channel (sum rule 1). We note that $|{{\mathbf{p}}}|$ should be small enough to trust this relation, because we have assumed that the IR region is well described by hydrodynamics, which is valid only at small momentum and energy. $\kappa_B$ has been evaluated in lattice QCD 