  We compare three available models of graphene conductivity: a non-local Kubo
model, a local model derived by Falkovsky, and finally a non-local Quantum
Field Theory based (QFT-b) model. The first two models are extensively used in
the nanophotonic community. All these models are not ab-initio since they
contain phenomenological parameters (like Fermi velocity, chemical potential
and/or mass gap parameters that depend on the chosen material and possibly on
external perturbations), and are supposed to provide coherent results since
they are derived from the same starting Hamiltonian. While we confirm that the
local model is a proper limit of the non-local Kubo model, we find some
inconsistencies in the QFT-b model as derived and used in the literature. In
particular, differently from the Kubo model, the QFT-b model shows a
plasma-like behavior for the interband transversal conductivity at low
frequencies instead of the expected behavior (an almost constant conductivity
as a function of frequency $\omega$ with a gap for frequencies $\hbar\omega <
\sqrt{(\hbar v_{F}q)^{2} + 4m^{2}}$). We show how to correctly regularize the
QFT-b model in order to satisfy the gauge invariance and, once also losses are
correctly included, we show that the Kubo and QFT-b model exactly coincide. Our
finding can be of relevant interest for both theory, predictions and
experimental tests in both the nanophotonic and Casimir effect communities.
