$e$), and $\psi_f$ the quark field with flavor $f$, respectively.

At finite temperature, the medium effect breaks Lorentz symmetry so that the tensor structure of the Green function has two independent components, $$\begin{aligned}
\label{eq:tensor-decomposition}
G^R_{\mu\nu}(p)&=G^T(p)P^T_{\mu\nu}(p)
+G^L(p)P^L_{\mu\nu}(p),\end{aligned}$$ where $p^\mu\equiv(\omega,{{\mathbf{p}}})$ is a shorthand notation for the energy and the spatial momentum, and $P^T_{\mu\nu}(p)\equiv g^{i}_\mu g^{j}_\nu (\delta^{ij}-{p^i p^j}/{{{\mathbf{p}}}^2})$ and $P^L_{\mu\nu}(p)\equiv P^0_{\mu\nu}(p) -P^T_{\mu\nu}(p)$ are the projection operators to the transverse and longitudinal parts with $P^0_{\mu\nu}(p)\equiv -(g_{\mu\nu}-{p_\mu p_\nu}/{p^2})$. The first (second) term in Eq. (\[eq:tensor-decomposition\]) corresponds to the transverse (longitudinal) component in three dimensions. When ${{\mathbf{p}}}$ is along the $z$-direction, they are related to the components of the Green function as $G_{T}=G^R_{11}=G^R_{22}$ and $G_L=p^2G^R_{00}/{{\mathbf{p}}}^2$.

Here we recapitulate the method of deriving sum rules at finite temperature that was developed in Ref.[@Romatschke:2009ng]. Using the residual theorem for the contour[^1] drawn in Fig. \[fig:contour\], we get $$\begin{aligned}
\label{eq:residue-theorem}
\begin{split}
&\delta G^{R}_{\mu\nu}(i\omega,{{\mathbf{p}}})-\delta G^{R}_{\mu\nu}(\infty,{{\mathbf{p}}}) \\
&~~~= \frac{1}{2\pi i}\oint_C d\omega' 
\frac{\delta G^{R}_{\mu\nu}(\omega',{{\mathbf{p}}})-\delta G^{R}_{\mu\nu}(\infty,{{\mathbf{p}}})}{\omega'-i\omega},
\end{split}\end{aligned}$$ where $\delta$ stands for the subtraction of the $T=0$ part: $\delta G^R_{\mu\nu}\equiv G^R_{\mu\nu}-G^R_{\mu\nu}|_{T=0}$. Because of this subtraction of the zero temperature part and another subtraction of $\delta G^{R}_{\mu\nu}(\infty,{{\mathbf{p}}})$ done in the expression above, all the UV divergences are regularized in all the cases we consider. Thus, 