ζ-function regularization of chiral Jacobians for singular Dirac operators

We propose a definition of the chiral Jacobian which uses the invariance of the generating functional under chiral rotations. This definition takes into account the contributions of all terms which, after rotation, depend on the chiral parameter {alpha}. We show that when the Dirac operator has zero eigenvalues the presence of fermionic sources gives an additional dependence on {alpha}. Our definition, by considering this {alpha} dependence, reconciles the {zeta}-function method of calculating chiral Jacobians with Fujikawa's.