Model-independent results on parity violation in the trace anomaly

Anomalous parity violation in four dimensions would be significant for phenomenology (baryogenesis, gravitational waves) and mathematical physics. Over the past decade, there has been a controversy in the literature as to whether free Weyl fermions give rise to (anomalous) parity violation in the trace of the energy momentum tensor; expressed by the Pontryagin densities $R\tilde{R}$ and $F\tilde{F}$ in the gravity and the gauge sector respectively. In Ref.${}^1$, we have shown, using path integral methods, that the trace anomaly of a free Weyl fermion does not violate parity (i.e the absence of the Pontryagin density). In a subsequent work${}^2$ we came to the stronger conclusion that for any theory compatible with dimensional regularisation, the Pontryagin-terms are equally absent. It is the $\mathit{\text{finiteness}}$ of the diffeomorphism, the Lorentz and the gauge anomalies that prevents anomalous parity violation.