Large-mass expansions or one-loop effective actions and fermion currents

Abstract

The large-mass expansion of the functional determinants for second-order elliptic operators and general Dirac operators is calculated for four-dimensional flat euclidean space using zeta function regularisation and heat kernel methods. The results are applicable to one-loop boson and fermion effective actions. In addition the expansions of covariantly regularised fermion currents are derived. It is also possible for the corresponding Pauli-Villars regularised forms to be then simply obtained and the modified currents then reproduce the usual Bardeen anomaly. Although covariant methods are used it is shown how to derive the expansion for the phase of the fermion determinant, which is non-covariant and produces the anomaly, in terms of a representation as a five-dimensional integral which is related to the spectral asymmetry for a suitable spinor hamiltonian. This relation is essentially exact and is demonstrated by considering the variation of the phase with respect to the Dirac operator.