possible hydro modes, which adds a $\omega^{3/2}$ term at zero momentum to $G^T$ [@Kovtun:2003vj]. It was suggested that this approximation is justified in the large ${N_c}$ limit [@Kovtun:2003vj]. Among our sum rules, sum rule 3 (\[eq:sumrule-3-T\]) in the transverse channel may be changed at finite ${N_c}$ due to this effect. In the analysis of lattice QCD data, this effect practically can be neglected because the IR cutoff of the current lattice QCD is not small enough for such IR energy effects to be detectable.

Sum rules at finite momentum {#sec:sumrule}
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In this section, we derive the sum rules at finite momentum in the transverse and the longitudinal channels. For later convenience, we also give the expressions for the sum rules at zero momentum in the transverse channel, which were already obtained in Ref. [@Gubler:2016hnf]. We also confirm that the sum rules are satisfied by the expressions of the spectral function in the chiral and weak coupling limits.

Transverse channel
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Before deriving the sum rules, let us discuss what kind of structure can be expected to appear in the spectral function in a perturbative analysis. First, in the low-energy region, a peak with a width of order $g^4T$ due 