Radiatively induced CPT-odd Chern-Simons term in massless QED

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.