Origin of the discrepancy in FV and Dirac radiative transition rates

Determine whether the discrepancy between the radiative transition rates reported for the eight-component FV theory and the Dirac theory originates in the velocity form of the coupling, retardation effects, or the definition of the rate in the Coulomb gauge, and thereby establish the gauge-consistent FV radiative transition rate.

Background

The paper establishes that the Robson–Staudte eight-component FV theory reproduces the Dirac Stark shifts and level-to-level electric-field couplings through the orders and hydrogenic states examined. However, earlier work reported substantial differences between FV and Dirac radiative transition rates, including for the $2P_{3/2}\!\to\!1S_{1/2}$ transition in $\mathrm{U}^{91+}$. The paper argues that, in the long-wavelength length gauge and when rates are defined using biorthogonal products over complete degenerate manifolds, the rates for the transitions considered should coincide with the Dirac values.

The unresolved issue is whether the earlier discrepancy is caused by using the velocity form of the interaction, by retardation effects, or by how the transition rate is defined in the Coulomb gauge. The authors identify calculations of FV radiative transition rates in both length and velocity gauges, with and without retardation, as the next step needed to resolve this question.