Retardation effects on high Rydberg states: A retardedR−5polarization potential

The finiteness of the speed of light is known to change the long-range interaction of atomic systems. Stimulated by recent advances in precision measurements of high Rydberg states, we consider these "retardation effects," not on the interaction of two separate systems, but on the energy eigenvalues of the high Rydberg states of an isolated heliumlike ion, where the outer electron has quantum numbers $n>l\gg 1$, and where the core—the nucleus and the inner electron—is in its (spherically symmetric) ground state. We analyze the time-ordered Feynman-like graphs that contain one or two instantaneous or transverse photons, and find a retardation correction, $\frac{11e^2\hslash {\alpha }_d}{\left(4\mathrm{}\pi \mathrm{mc}{R}^2\right)}$ to the leading ($-\frac{e^2{\alpha }_d}{2R^4}$) polarization potential; ${\alpha }_d$ is the static electric dipole polarizability of the ion core, and $R$ is the nuclear-outer-electron separation. The correction is also applicable to scattering problems, to high Rydberg states of atoms and ions with more than two electrons, and, with $m\to m_{\mu }$, to a muon bound to a nucleus. Very recently, Bernabéu and Tarrach used dispersion theory to obtain an identical term as the retardation correction for the interaction of a charged particle with a neutral polarizable system; their procedure is not applicable, as it stands, to the present situation.