The full Lorentz-violating vacuum polarization tensor: low and high energy limits

We compute the full vacuum polarization tensor in the fermion sector of Lorentz-violating QED. Even if we assume momentum routing invariance of the Feynman diagrams, it is not possible to fix all surface terms and find an unambiguity free vacuum polarization tensor. The high and low energy limits of this tensor is presented. In the high energy limit, only $c_{\mu \nu }$ coeffcients contribute. In the low energy limit, we fnd that Lorentz-violating induced terms depend only on $b_{\mu }$, $c_{\mu \nu }$ and $g_{\mu \nu \lambda }$ coeffcients and they are suppressed by powers of $\frac{p^2}{m^2}$. This limit allows to obtain implications for condensed matter systems, explicitly, for the Hall effect in Weyl semimetals.