On effective Chern-Simons Term induced by a Local CPT-Violating Coupling using ${\gamma }_5$ in Dimensional Regularization

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.