On the consistency of Lorentz invariance violation in QED induced by fermions in constant axial-vector background

We show for the first time that the induced parity-even Lorentz invariance violation can be unambiguously calculated in the physically justified and minimally broken, dimensional regularization scheme, suitably tailored for a spontaneous Lorentz symmetry breaking in a field theory model. The quantization of the Lorentz invariance violating quantum electrodynamics is critically examined and shown to be consistent either for a light-like cosmic anisotropy axial-vector or for a time-like one, when in the presence of a bare photon mass.