Large quark mass Euler-Heisenberg type action for QCD with quark sources and its Abelian limit

The quark determinant in the presence of a background gluon field is calculated in a large quark mass approach within a derivative expansion by considering quark sources. The resulting low-energy QCD Effective Field Theory (EFT) is valid for non-homogeneous fields and it presents, besides the usual Euler-Heisenberg type gluon self-interactions, different types of corrections to the minimal quark-gluon interaction among which there are several imaginary terms. The Abelian limit of this effective action, i.e. an expanded Euler-Heisenberg effective action with fermion sources, is also found and it is shown to contain, therefore, corrections to the fermion-photon interactions, including imaginary terms. Different resulting properties are calculated such as: their contributions to the energy-momentum tensor and its trace, the Poynting vector, the chromo-electric permittivity and chromo-magnetic permeability. Conditions for these quantities, as well as for the corresponding Abelian limits, to be real are exhibited.