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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 $(k_{AF})_μ=-\frac{e^2}{4π^2}\,b_μ$. 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 $(k_{AF})_μ$, in which at high temperature, $(k_{AF})_0\to0$ and $(k_{AF})_i\to-\frac{e^2}{4π^2}b_i$. In the context of Weyl semimetals, these results are in accordance with the fact that the chiral magnetic current $j^α=(k_{AF})_0ε^{0αλρ}\partial_λA_ρ$ vanishes at high temperature, whereas the anomalous Hall current $j^α=(k_{AF})_iε^{iαλρ}\partial_λA_ρ$ remains unaffected by the finite temperature.