Gravitational anomalies: Antisymmetric tensor gauge field versus bispinor field

It is shown perturbatively that the gravitational anomaly of the chiral bispinor field is twice as large as that of the self-dual antisymmetric tensor gauge field; in the previous literature they were supposed to agree with each other. This implies that a naive application of Fujikawa's path integral method leads to a wrong result for the anomaly of the self-dual antisymmetric tensor gauge field.