  In this work, we study the radiative generation of the CPT-odd
Lorentz-violating Chern-Simons term, arising from massless fermions. For this,
we calculate the vacuum polarization tensor using the 't Hooft-Veltman
regularization scheme, in which the result obtained for the coefficient of the
Chern-Simons term is $(k_{AF})_\mu=-\frac{e^2}{4\pi^2}\,b_\mu$. This result
leads us precisely to the same conductivity found in Weyl semimetals, i.e., the
't Hooft-Veltman regularization scheme is the correct one to be used in this
context. We also discuss the temperature dependence of $(k_{AF})_\mu$, in which
at high temperature, $(k_{AF})_0\to0$ and
$(k_{AF})_i\to-\frac{e^2}{4\pi^2}b_i$. In the context of Weyl semimetals, these
results are in accordance with the fact that the chiral magnetic current
$j^\alpha=(k_{AF})_0\epsilon^{0\alpha\lambda\rho}\partial_\lambda A_\rho$
vanishes at high temperature, whereas the anomalous Hall current
$j^\alpha=(k_{AF})_i\epsilon^{i\alpha\lambda\rho}\partial_\lambda A_\rho$
remains unaffected by the finite temperature.
