Up to order four in graviton fields the effective action for a massless chiral fermion is parity-even and equals half the corresponding action for a non-chiral Dirac fermion, with the Weyl anomaly likewise parity-even and halved.
On the trace anomaly of a Weyl fermion
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abstract
We calculate the trace anomaly of a Weyl fermion coupled to gravity by using Fujikawa's method supplemented by the choice of 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.
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The effective gravitational action of a massless chiral fermion and the absence of parity-odd contributions
Up to order four in graviton fields the effective action for a massless chiral fermion is parity-even and equals half the corresponding action for a non-chiral Dirac fermion, with the Weyl anomaly likewise parity-even and halved.