for the ${{\mathbf{p}}}$-independent part, the contributions from the transport peak and the continuum, which are of order ${C_{\text{em}}}T^2$, cancel while the contribution from the UV tail is of higher order ($\sim {C_{\text{em}}}g^2T^2$), by looking at Eqs.(\[eq:T-sumrule1-transport\]-\[eq:T-sumrule1-UVtail\]). Therefore, the ${{\mathbf{p}}}$-independent part was shown to satisfy sum rule 1 at leading order already in Ref.[@Gubler:2016hnf]. For the ${{\mathbf{p}}}^2$ term, we see that the transport peak does not contribute, and the continuum contribution is of order $\sim {C_{\text{em}}}{{\mathbf{p}}}^2\ln (1/g)$ while the UV tail contribution is suppressed by a factor of $g^2$. Because the continuum contribution is sensitive to the IR cutoff, we need to improve the evaluation by performing the HTL resummation [@Moore:2006qn; @Braaten:1990wp], in order to confirm that this contribution becomes negligible so that sum rule 1 is satisfied at order ${{\mathbf{p}}}^2$. It is furthermore understood that sum rule 1 is mainly sensitive to the transport peak as well as the continuum, while the contribution of the UV tail is small.

### Sum rule 2 {#sssec:sumrule2-T}

In the derivation of Eq.(\[eq:derivation\]), we used only the fact that the retarded Green function is analytic in the upper $\omega$ plane. Thus, we can derive a similar equation in which $\delta G^R(\omega)$ ($\delta \rho(\omega)$) is 