The gauge-invariant Chern-Simons-type Lorentz- and $\mathrm{CPT}$-breaking term is here reassessed and a spin-projector method is adopted to account for the breaking (vector) parameter. Issues such as causality, unitarity, spontaneous gauge-symmetry breaking, and vortex formation are investigated, and consistency conditions on the external vector are identified.

This is the second in a series of two contributions in which we set out to establish a novel momentum space framework to treat field-theoretical infinities in perturbative calculations when parity-violating objects occur. Since no analytic continuation on the space-time dimension is effected, this framework can be particularly useful to treat dimension-specific theories. Moreover, arbitrary local terms stemming from the underlying infinities of the model can be properly parametrized. We (re)analyze the indeterminacy of the radiatively generated $\mathrm{CPT}$ violating Chern-Simons term within an extended version of ${\mathrm{QED}}_4$ and calculate the Adler-Bardeen-Bell-Jackiw triangle anomaly to show that our framework is consistent and general enough to handle the subtleties involved when a radiative correction is finite.

We extend an implicit regularization scheme to be applicable in the n-dimensional space-time. Within this scheme divergences involving parity violating objects can be consistently treated without recoursing to dimensional continuation. Special attention is paid to differences between integrals of the same degree of divergence, typical of one loop calculations which are, in principle, undetermined. We show how to use symmetries in order to fix these quantities consistently. We illustrate with examples in which regularization plays a delicate role in order to both corroborate and elucidate the results in the literature for the case of $\mathrm{CPT}$ violation in extended ${\mathrm{QED}}_4,$ topological mass generation in three-dimensional gauge theories, the Schwinger model, and its chiral version.