the integral on the contour $C$ can be safely replaced with the integral on the real axis. Moreover, in deriving Eq. (\[eq:residue-theorem\]) we have used the property that the retarded Green function is analytic in the upper $\omega'$ plane.

Now, by taking the $\omega\rightarrow 0$ limit, Eq. (\[eq:residue-theorem\]) reduces to $$\begin{aligned}
\begin{split}
&\delta G^{R}_{\mu\nu}(0,{{\mathbf{p}}})-\delta G^{R}_{\mu\nu}(\infty,{{\mathbf{p}}}) \\
&~~~= {\text P}\frac{1}{\pi i}\int^\infty_{-\infty} d\omega' 
\frac{\delta G^{R}_{\mu\nu}(\omega',{{\mathbf{p}}})-\delta G^{R}_{\mu\nu}(\infty,{{\mathbf{p}}})}{\omega'},
\end{split}\end{aligned}$$ where we have used $1/(\omega'-i\omega)\rightarrow {\text P}(1/\omega')+i\pi\delta(\omega')$. We consider only the case of $\mu=\nu$ in this paper. Then, the real (imaginary) part of $\delta G^{R}_{\mu\nu}(\omega,{{\mathbf{p}}})$ is even (odd) in terms of $\omega$. This property enables us to simplify the equation above as $$\begin{aligned}
\label{eq:derivation}
\begin{split}
\delta G^{R}_{\mu\nu}(0,{{\mathbf{p}}})-\delta G^{R}_{\mu\nu}(\infty,{{\mathbf{p}}}) 
= \frac{2}{\pi}\int^\infty_{0} d\omega
\frac{\delta \rho_{\mu\nu}(\omega,{{\mathbf{p}}})}{\omega},
\end{split}\end{aligned}$$ where we have introduced the spectral function[^2] of the EM current, $\rho_{\mu\nu}(p)\equiv{\text{Im}}G^{R}_{\mu\nu}(p)$, and changed the label of the integration variable as $\omega'\rightarrow \omega$ for simplicity. We hence see that the asymptotic behaviors of the retarded Green function in the UV and IR regions determine the integral of the spectral function.

Let us here briefly discuss the differences between the sum rules derived here and the so-called finite energy sum rules (FESR), which are widely used in the literature (see for instance 