Fermion determinant for general background gauge fields

TL;DR

This paper derives an exact expression for the fermion determinant in two-dimensional Euclidean space with specific background fields, revealing its dependence on the chiral anomaly and extending duality concepts to relate to four-dimensional pair production.

Contribution

It provides a novel exact representation of the fermion determinant for general background gauge fields and extends duality concepts from one to two-variable fields.

Findings

Determinant depends only on the chiral anomaly in certain limits.

Extended duality relates 2D Euclidean determinants to 4D pair production.

Bound on the determinant is connected to the sign of the effective action derivative.

Abstract

An exact representation of the Euclidean fermion determinant in two dimensions for centrally symmetric, finite-ranged Abelian background fields is derived. Input data are the wave function inside the field's range and the scattering phase shift with their momenta rotated to the positive imaginary axis and fixed at the fermion mass for each partial-wave. The determinant's asymptotic limit for strong coupling and small fermion mass for square-integrable, unidirecitonal magnetic fields is shown to depend only on the chiral anomaly. The concept of duality is extended from one to two-variable fields, thereby relating the two-dimensional Euclidean determinant for a class of background magnetic fields to the pair production probability in four dimensions for a related class of electric pulses. Additionally, the ``diamagnetic'' bound on the two-dimensional Euclidean determinant is related to the negative sign of dImS_eff/dm^2 in four dimensions in the strong coupling, small mass limit, where S_eff is the one-loop effective action.