The effective gravitational action of a massless chiral fermion and the absence of parity-odd contributions

We consider the field theory of a quantum massless left handed fermion coupled to a background graviton field, $h_{\mu\nu}$, on four-dimensional Minkowski spacetime. By using the BPHZL renormalization scheme, we prove that, up to order four in the number of graviton fields, there are no parity-odd contributions to the renormalized gravitational effective action. As a side result, we show that, modulo arbitrary UV finite diffeomorphism invariant parity-even counterterms, the gravitational effective action in question is equal, up to order four in the number of graviton fields, to half the corresponding gravitational action for a Dirac fermion non-chirally coupled to gravity. Also as a side result, we conclude the the Weyl anomaly is purely parity-even and that its value is half the value of the Weyl anomaly for a Dirac fermion non-chirally coupled to gravity.