method to analytically continue imaginary time data to real time, such as the maximum entropy method (MEM) [@Jarrell:1996rrw; @Asakawa:2000tr; @Gubler:2010cf], the Backus-Gilbert method [@Backus:1968; @Backus:1970; @Brandt:2015sxa] or the Schlessinger point method [@Schlessinger:1967; @Tripolt:2016cya] (see Ref.[@Tripolt:2017] for a comparison of these three methods).

In our previous work of Ref.[@Gubler:2016hnf], we derived three sum rules at zero momentum, which constrain the spectral function, and used them to improve the ansatz employed in previous lattice QCD analysis. One aim of this paper is to derive similar sum rules for the small but finite spatial momentum case. At finite momentum, the EM spectral function no longer has a single independent component, but two, corresponding to the transverse and the longitudinal channels. In the longitudinal channel, a novel and robust structure, a sharp peak corresponding to the diffusion mode of the EM charge, appears.

As shown in Ref.[@Gubler:2016hnf], the sum rules can be used to improve the analysis of lattice QCD data by constraining the shape of the spectral functions and derive transport coefficients that do not appear directly in the spectral functions. In Ref.[@Gubler:2016hnf], this was demonstrated by using two sum rules (1 and 3 in this and the previous work), but the other 