Role of Ambiguities and Gauge Invariance in the Calculation of the Radiatively Induced Chern-Simons Shift in Extended Q.E.D.

TL;DR

This paper examines how ambiguities and gauge invariance influence the calculation of the radiatively induced Chern-Simons term in extended QED, showing that gauge invariance and anomaly cancellation conditions determine whether this term vanishes or not.

Contribution

It provides a general analysis of divergent amplitude manipulations in extended QED, clarifying how gauge invariance and anomaly cancellation affect the Chern-Simons term calculation.

Findings

Gauge invariance leads to a nonvanishing, ambiguity-free Chern-Simons term.

Anomaly cancellation conditions can result in a vanishing Chern-Simons term.

The approach avoids explicit divergent integral calculations by using general properties.

Abstract

We investigate the possibility of Lorentz and CPT violations in the photon sector, of the Chern-Simons form, be induced by radiative corrections arising from the Lorentz and CPT non-invariant fermionic sector of an extended version of QED. By analyzing the modified vacuum polarization tensor, three contributions are considered: two of them can be identified with well known amplitudes; the (identical) QED vacuum polarization tensor and the (closely related) $AVV$ triangular amplitude. These amplitudes are evaluated in their most general form (to include in our discussion automatically the question of ambiguities) on the point of view of a strategy to manipulate and calculate divergent amplitudes that can avoid the explicit calculation of divergent integrals. Rather than this only general properties are used in intermediary steps. With this treatment, the results obtained by others authors can be easily recovered and we show that, if we choose to impose U(1) gauge invariance maintenance in the pure QED calculated amplitudes, to be consistent with the renormalizability, the induced Chern-Simons term assumes a nonvanishing ambiguities free value. However if, in addition, we choose to get an answer consistent with renormalizability by anomaly cancellation of the Standard Model a vanishing value can be obtained, in accordance with what was previously conjectured by other authors.