Operator regularization and the phase of one-loop determinants

Abstract

Operator regularization is a symmetry preserving regularization procedure to all orders of perturbation theory that avoids explicit divergences. At one-loop order it is det H which is regularized, where H is an operator appearing in the theory. For some theories it is not possible to treat det H directly; in previous implementations det H has simply been replaced by det 1 2 H 2 in such cases. However, if H has negative or imaginary eigenvalues then information about the phase of det H may be lost in this replacement. We discuss this general problem for an arbitrary operator H and we give a prescription for finding the phase of det H that reduces, if H is Hermitian, to the η-function analysis introduced by Gilkey and further exploited by Witten and others. We consider explicitly one example of a non-Hermitian operator and three examples of a Hermitian operator. To illustrate how we treat the phase of the determinant of a non-Hermitian operator, we consider the determinant associated with a spinor coupled to an external axial vector field and show how the phase of this determinant is associated with the axial anomaly, using a technique introduced by Bukhbinder, Gusynin, and Fomin. In three-dimensional quantum electrodynamics (or, more generally, its nonabelian analogue) the η-function can be computed exactly (following Birmingham, Cho, Kantowski, and Rakowski) when there is an external vector field, showing that an effective Chern-Simons action is generated, resulting in a quantization of the gauge coupling constant. In the final two examples H is a super-operator, spanning a space containing both bosons and fermions. We compute the contribution of the two-point spinor function to the η-function when the vector field to which the spinor couples has a Chern-Simons action; it simply serves to renormalize the coefficient of the kinetic term for the spinor field. Finally, we consider the self-energy in a two-dimensional model in which a Majorana spinor couples to a vector field which has a topological action. In this case the η-function vanishes, but the ζ-function indicates that chiral symmetry is broken despite the fact that all the unregulated Feynman diagrams in this model vanish.