QCD interaction with other particles. The electric conductivity is also an important quantity since it may increase the lifetime of magnetic fields generated in the early stage of the heavy ion collision [@McLerran:2013hla; @Tuchin:2013apa; @Tuchin:2015oka; @Hattori:2016emy]. Furthermore, how the spectrum of the vector meson is modified at finite $T$ has been discussed for a long time from the point of view of the chiral symmetry restoration [@Bernard:1988db]. The spectral function of the EM current at finite $T$, which is the focus of this paper, contains information on all the above three quantities.

There are many approaches for evaluating the EM spectral function at finite $T$, such as perturbative QCD [@Baier:1988xv], holographic QCD [@CaronHuot:2006te], model calculations [@Gale:2014dfa], sum rules [@Gubler:2015yna; @Huang:1994fs; @Kapusta:1993hq; @Zschocke:2002mn; @Kim:2017nyg], low-energy effective theory [@Chanfray; @Klingl], and lattice QCD [@Ding:2010ga; @Meyer:2011gj; @Brandt:2015aqk; @Amato:2013naa; @Aarts:2007wj]. Nevertheless, none of these approaches are perfect. Especially in the lattice QCD approach, which allows fully nonperturbative first principle QCD calculations, the spectral function cannot be analyzed directly since it is a quantity defined in real, not imaginary time. Therefore, one needs to make some assumption about the functional form of the spectral function to analyze it, or otherwise has to rely on some 