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We resume a long-standing, yet not forgotten, debate on whether aChern-Simons birefringence can be generated by a local term$b_\mu\bar\psi\gamma^\mu \gamma_5\psi$ in the Lagrangian (where $b_\mu$ areconstants). In the present paper we implement a new way of managing $\gamma_5$in dimensional regularization. Gauge invariance in the underlying theory (QED)is enforced by this choice of defining divergent amplitudes. We investigate thesingular behavior of the vector meson two-point-function around the $m^2=0$ and$p^2=0$ point. We find that the coefficient of the effective Chern-Simons canbe finite or zero. It depends on how one takes the limits: they cannot beinterchanged due to the associate change of symmetry. For $m^2=0$ we evaluatealso the self-mass of the photon at the second orderin $b_\mu$. We find zero.

On effective Chern-Simons Term induced by a Local CPT-Violating Coupling using $\gamma_5$ in Dimensional Regularization