Evidence for mass zeros of the fermionic determinant in four-dimensional quantum electrodynamics

TL;DR

This paper investigates the behavior of the fermionic determinant in four-dimensional QED as a function of fermionic mass, revealing conditions under which it exhibits zeros related to the chiral anomaly and background gauge fields.

Contribution

It provides a theoretical analysis of the mass dependence of the fermionic determinant in QED, highlighting the existence of zeros influenced by the chiral anomaly and specific gauge field configurations.

Findings

The fermionic determinant's remainder may vanish at least once as mass varies.

The behavior depends on the sign of integrals involving the field strength tensor.

The analysis links the zeros to the chiral anomaly and gauge field properties.

Abstract

The Euclidean fermionic determinant in four-dimensional quantum electrodynamics is considered as a function of the fermionic mass for a class of $O(2)\times O(3)$ symmetric background gauge fields. These fields result in a determinant free of all cutoffs. Consider the one-loop effective action, the logarithm of the determinant, and subtract off the renormalization dependent second-order term. Suppose the small-mass behavior of this remainder is fully determined by the chiral anomaly. Then either the remainder vanishes at least once as the fermionic mass is varied in the interval $0 < m < \infty$ or it reduces to its fourth-order value in which case the new remainder, obtained after subtracting the fourth-order term, vanishes at least once. Which possibility is chosen depends on the sign of simple integrals involving the field strength tensor and its dual.