Gauge Dependence of the Grabvitational Conformal Anomaly for the Electromagnetic Field

The conformal anomaly for the electromagnetic field in the background gravitational field is obtained in the general covariant gauge with a gauge parameter. We have used the path integral method of Fujikawa with the zeta-regularization. The gauge dependence is observed in the coeffisient of the □R term in the anomaly; various contradicting values have been assigned to the coefficient of this term in the previous calculations on the basis of different regularization schemes in the Feynman gauge. We emphasize that at the cost of the classical conformal invariance those terms can be eliminated altogether by adding a finite counterterm proportional to √gR2 to the Lagrangian. A calculational scheme with regularized Green functions was also examined, and it gives rise to the same results for the anomaly in the conventional Faddeev-Popov formalism as well as in the B-field formalism. The regularization of Green functions introduces an auxiliary mass parameter for each field and the Ward-Takahashi identities impose a condition among those mass parameters. In the framework of the zeta-function regularization, the gauge dependence of the □R term in the anomaly thus appears to be inevitable.