limit.

In this paper, we recapitulate known results such as the sum rules in the transverse channel at zero momentum, Eqs.(\[eq:sumrule-1-p0\]) and (\[eq:sumrule-3-T\]). Their derivation can be found in our previous work [@Gubler:2016hnf], but we rederive them to make our paper self-contained. The recapitulation of the evaluation of the transport peak, the continuum, and the UV tails in the three appendixes is provided for the same reason.

Preliminaries
=============

In this section, we explain the method for deriving the sum rules developed in Ref. [@Romatschke:2009ng]. Only the asymptotic behaviors of the EM current correlator in the UV and IR energy regions are necessary for this purpose. We also discuss these behaviors and give their analytic form obtained from OPE and hydrodynamics in this section.

Formalism
---------

We begin by introducing quantities that will be used in the derivation of the sum rules. The retarded Green function of the EM current ($j^\mu\equiv e\sum_f q_f \overline{\psi}_f \gamma^\mu \psi_f $) is defined as $G^R_{\mu\nu}(\omega,{{\mathbf{p}}})\equiv i \int dt \int d^3{{\mathbf{x}}}e^{i\omega t-i{{\mathbf{p}}}\cdot {{\mathbf{x}}}}
\theta(t) \langle [j_\mu(t,{{\mathbf{x}}}),j_\nu(0,{{\mathbf{0}}})] \rangle$, where the average is taken over the thermal ensemble, $e$ is the coupling constant of quantum electrodynamics, $q_f$ is the charge of each quark flavor (in units of 