On the trace anomaly of a Weyl fermion

Abstract

Abstract:We calculate the trace anomaly of a Weyl fermion coupled to gravity by using Fujikawa's method supplemented by a consistent regulator. The latter is constructed out of Pauli-Villars regulating fields. The motivation for presenting such a calculation stems from recent studies that suggest that the trace anomaly of chiral fermions in four dimensions might contain an imaginary part proportional to the Pontryagin density. We find that the trace anomaly of a Weyl fermion is given by half the trace anomaly of a Dirac fermion, so that no imaginary part proportional to the Pontryagin density is seen to arise.

Pontryagin trace anomaly

“…So, one cannot a priori exclude the possibility that massless Majorana and Weyl fermions, when defined on curved space, do not have the same path integral measure and are different when quantised. Let us add here that the claim apparently opposite to our, made in [12], also rests on the EPJ Web of Conferences 182, 02100 (2018) https://doi.org/10.1051/epjconf/201818202100 ICNFP 2017 EPJ Web of Conferences assumption of massless Majorana / Weyl equivalence, and so just proves that the massless Majorana fermion does not produce the Pontryagin trace anomaly, an uncontroversial claim supported also by our calculations. For the detailed account of (and respond to) criticism, and we direct readers to Refs.…”

Axial gravity: a non-perturbative approach to split anomalies

“…This turned out to be the case, as was shown in [3]. After these two papers a negative result was obtained in [37]. Using a heat kernel method with a Pauli-Villars regularization the authors found a vanishing odd parity trace anomaly in 4d.…”

“…We do not know if it is possible to solve this problem, but we are sure the solution is not the choice made in [37], because the operator chosen by the authors there includes both chiralities. Of course, with this choice, the result for the odd trace anomaly cannot be but 0.…”

“…The literature on the odd parity trace anomaly in 4d (still in a settling phase) consists of [1][2][3][36][37][38][39][40]. Textbooks on anomalies are [41][42][43].…”

Axial gravity and anomalies of fermions

“…For our purposes it is more convenient and transparent to split the Dirac fermion ψ into its two independent and Lorentz irreducible chiral components λ and ρ of opposite chiralities, ψ = λ + ρ. We use the conventions of [4,5] for spinors and gamma matrices. In particular our chiral spinors satisfy γ 5 λ = λ and γ 5 ρ = −ρ.…”