Model and parameter-path dependence of charged Dirac quasi-bound spectra in Born--Infeld and Euler--Heisenberg-type geometries

Abstract

We study quasi-bound states of a charged massive Dirac field in the exact electrically charged Einstein--Born--Infeld (BI) geometry and in a leading-order Euler--Heisenberg-type (EH) approximation. The two descriptions agree in the weak-field limit. We compare families in which either the mass measured at infinity or the outer-horizon radius is held fixed, and examine several field charges and levels. We integrate the radial equations outward from the horizon and inward from large radius, then match the solutions. The resulting frequencies show that the effect of the nonlinear terms depends on both the electrodynamic theory and the chosen comparison. For the fundamental like-charge mode, the small change in its real frequency has opposite signs in the two theories when the mass is fixed, whereas it is upward in both when the horizon radius is fixed. The fundamental mode decays more slowly as the nonlinear term grows along either family. The narrowest level, however, decays more rapidly when the mass is fixed and more slowly when the horizon radius is fixed. The spectra of the two theories approach one another as the nonlinear term is reduced. We also follow several higher levels; their decay widths and their response depend on the state and on the family used for comparison. The nonlinear coefficient is used as an effective parameter and is not matched to the microscopic coupling in quantum electrodynamics (QED).