Scale ambiguities in perturbative calculations and the value for the radiatively induced Chern-Simons term in extended QED

We perform a very detailed investigation into the possibility of Lorentz and $CPT$ violations in the photon sector, of the Chern-Simons form, being induced by radiative corrections arising from the Lorentz and $CPT$ noninvariant fermionic sector of an extended version of QED. In the discussions it is considered the role played by the scale ambiguities in the perturbative calculations. The origin of the possible contributions are clearly identified, due to the very general character of the calculational method applied, concerning the manipulations and calculations of divergent amplitudes. We show that constraints dictated by the consistency in the perturbative calculations, imposed by physical reasons, eliminate all the sources of violating contributions, leaving no room for the existence of a radiatively induced Lorentz and $CPT$ violations in the extended QED, putting the perturbative evaluations in accordance with the predictions made, by using very general arguments, stated by other authors.