  We resume a long-standing, yet not forgotten, debate on whether a
Chern-Simons birefringence can be generated by a local term
$b_\mu\bar\psi\gamma^\mu \gamma_5\psi$ in the Lagrangian (where $b_\mu$ are
constants). 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 the
singular 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 can
be finite or zero. It depends on how one takes the limits: they cannot be
interchanged due to the associate change of symmetry. For $m^2=0$ we evaluate
also the self-mass of the photon at the second orderin $b_\mu$. We find zero.
