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 nonhomogeneous 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.