Lorentz and CPT violation in QED revisited: A missing analysis

We investigate the breakdown of Lorentz symmetry in QED by a CPT violating interaction term consisting of the coupling of an axial fermion current with a constant vector field b, in the framework of algebraic renormalization - a regularization-independent method. We show, to all orders in perturbation theory, that a CPT-odd and Lorentz violating Chern-Simons-like term, definitively, is not radiatively induced by the axial coupling of the fermions with the constant vector b.