Ref.[@Ayala:2013vra]). In contrast to the procedure of this paper, one in the FESR does not subtract the zero temperature contribution as we have done in Eq.(\[eq:residue-theorem\]) and below. Instead, to avoid an ultraviolet divergence, one does not take the radius of the contour in Fig.\[fig:contour\] to infinity, but sets it to some threshold value, which however should be large enough such that the OPE is still approximately valid. Doing this, one can derive sum rules that are not exact, but practically useful, as they can constrain the spectral function below the threshold value that usually contains the most interesting physical content.

![The contour $C$, used in the integral of Eq. (\[eq:residue-theorem\]).[]{data-label="fig:contour"}](contour.eps){width="25.00000%"}

UV behavior
-----------

The behavior in the UV region can be described with the help of the OPE. At leading order in the coupling constant, the Wilson coefficients of the operators with dimension 4 read [@CaronHuot:2009ns; @Shifman:1978by] $$\begin{aligned}
\label{eq:OPE}
\begin{split}
& \delta G^{R}_{\mu\nu}(\omega,{{\mathbf{p}}}) \\
&=e^2\sum q^2_f \frac{1}{p^2} \\
&~~~\times \Bigl[\left\{2m_f \delta\left\langle\overline{\psi}_f \psi_f \right\rangle
+\frac{1}{12}\delta\left\langle \frac{\alpha_s}{\pi}G^2\right\rangle \right\}P^0_{\mu\nu}(p) \\
&~~~-2\delta\left\langle T^{\alpha\beta}_f \right\rangle A_{\mu\nu \alpha\beta}(p)
\Bigr]
+{\cal O}(\omega^{-4}), 
\end{split}\end{aligned}$$ where $A_{\mu\nu \alpha\beta}(p)\equiv g_{\mu\alpha}g_{\nu\beta}+g_{\mu\beta}g_{\nu\alpha}-2(g_{\mu\alpha}p_{\nu}p_{\beta}+
g_{\nu\alpha}p_{\mu}p_{\beta}-g_{\mu\nu}p_\alpha p_\beta)/p^2$, $G^{\mu\nu}_a\equiv \partial^\mu A^\nu_a-\partial^\nu A^\mu_a -gf_{abc} A^\mu_b A^\nu_c$ is the field strength, $G^2\equiv G^a_{\mu\nu}G^{a\mu\nu}$, $T^{\alpha\beta}_{f}\equiv i{\cal {ST}} \overline{\psi}_f\gamma^\alpha 