zero momentum, and $\sigma/D=\chi_q$. For this reason, the sum rules in the longitudinal channel provide nontrivial information only when ${{\mathbf{p}}}$ is finite. Therefore, we consider only the finite momentum case in this subsection.

At finite momentum, the diffusion peak appears in the longitudinal spectral function in addition to the three structures that were already present in the transverse channel, as was explained in Sec. \[ssc:IR\]. To get a feeling about the possible shape of the spectral function in the longitudinal channel, we plot the diffusion peak (\[eq:hydro-rho-L\]), the transport peak (\[eq:transport-00\]), the continuum (\[eq:continuum-00\]), and the UV tail (\[eq:UVtail-00\]) in Fig. \[fig:spectral-longpaper-L\]. The parameters are the same as in Fig. \[fig:spectral-longpaper\]. Again, the approximate regions for which the above analytic descriptions are expected to be valid are indicated by the vertical lines and arrows.

Here we comment on the treatment of the diffusion peak in the traditional QCD sum rule literature: In the conventional sum rule approach, the delta function structure that is similar to the hydro result of Eq.(\[eq:hydro-rho-L-p=0\]) was suggested based on the perturbative calculation in Ref.[@Bochkarev:1985ex], and has been assumed in the subsequent works. Though the two approaches give the same form at $|{{\mathbf{p}}}|=0$ as they should 