  We study the DC conductivity of a weakly disordered 2D electron gas with two
bands and spectral nodes, employing the field theoretical version of the
Kubo--Greenwood conductivity formula. Disorder scattering is treated within the
standard perturbation theory by summing up ladder and maximally crossed
diagrams. The emergent gapless (diffusion) modes determine the behavior of the
conductivity on large scales. We find a finite conductivity with an
intermediate logarithmic finite-size scaling towards smaller conductivities but
do not obtain the logarithmically divergence of the weak-localization approach.
Our results agree with the experimentally observed logarithmic scaling of the
conductivity in graphene with the formation of a plateau near $e^2/\pi h$.
