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

TL;DR

This paper explores whether a local CPT-violating term in QED can generate a Chern-Simons term, using a new method for handling $oldsymbol{ ext{γ}_5}$ in dimensional regularization, and finds the result depends on the order of limits taken.

Contribution

It introduces a novel approach to manage $oldsymbol{ ext{γ}_5}$ in dimensional regularization, affecting the gauge invariance and the generation of the Chern-Simons term.

Findings

The Chern-Simons coefficient can be finite or zero depending on limit procedures.

The photon self-mass at second order in $b_$ is zero at $m^2=0$.

Gauge invariance is maintained with the new $oldsymbol{ ext{γ}_5}$ regularization method.

Abstract

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.