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The issue intensively claimed in the literature on the generation of aCPT-odd and Lorentz violating Chern-Simons-like term by radiative correctionsowing to a CPT violating interaction -- the axial coupling of fermions with aconstant vector field $b_\m$ -- is mistaken. The presence of massless gaugefield triggers IR divergences that might show up from the UV subtractions,therefore, so as to deal with the (actual physical) IR divergences, theLowenstein-Zimmermann subtraction scheme, in the framework of BPHZLrenormalization method, has to be adopted. The proof on the non generation ofsuch a Chern-Simons-like term is done, independent of any kind ofregularization scheme, at all orders in perturbation theory.

No radiative generation of Chern-Simons-like term in Lorentz-violating QED: dealing with IR divergences