Fermion determinants in static, inhomogeneous magnetic fields

The renormalized fermionic determinant of QED in 3+1 dimensions, detᵣₑₙ, in a static, unidirectional, inhomogeneous magnetic field with finite flux can be calculated from the massive Euclidean Schwinger model's determinant detSch in the same field by integrating detSch over the fermion's mass. Since detᵣₑₙ for general fields is central to QED, it is desirable to have nonperturbative information on this determinant, even for the restricted magnetic fields considered here. To this end we continue our study of the physically relevant determinant detSch. It is shown that the contribution of the massless Schwinger model to detSch is canceled by a contribution from the massive sector of QED in 1+1 dimensions and that zero modes are suppressed in detSch. We then calculate detSch analytically in the presence of a finite flux, cylindrical magnetic field. Its behavior for large flux and small fermion mass suggests that the zero-energy bound states of the two-dimensional Pauli Hamiltonian are the controlling factor in the growth of ln detSch. Evidence is presented that detSch does not converge to the determinant of the massless Schwinger model in the small mass limit for finite, nonzero flux magnetic fields.