Induced Chern-Simons Like Action by Lorentz Symmetry Breaking in (3+1)D QED

In this paper, the induced Chern-Simons like action for a system of fermions interacting with a gauge field in (3 +1) dimensions is calculated in the presence of a background field that breaks the Lorentz and CPT symmetries. The main result of this work corresponds to the obtation of a usual Chern-Simons-type action in four-dimensional spacetime resulting from the addition to the conventional QED Lagrangian of a term with a background field that breaks the Lorentz symmetry of the Lagrangian gauge theory. It is pointed out that the proportionality constant in that term depends heavily on the regularization method used in the computations. As the articles on this subject in the literature are very difficult to read, here calculations are performed clearly and objectively.