Estimate of the second order electron self energy corrections in highly charged heavy ions

Abstract

An estimate for the last unknown gauge-invariant set of QED corrections of order 2 ,t he second-order self-energy correction, is presented utilizing the so-called sign approximation. This is able to reduce the present uncertainties in Lamb-shift predictions considerably. Highly charged ions provide an ideal scenario to demonstrate the validity of QED in strong fields, e.g., by measurements of the Lamb shift at utmost precision. In this respect the recent experimental progress made in measurements in hydrogen-like uranium [1] indicates that calculations of higher-order radiative corrections become relevant. The set of second-order QED diagrams includes all various combinations of the first-order self-energy (SE) and vacuum polarization (VP) effects. The present status of theoretical predictions for the Lamb shift in different one-electron ions is presented in [2]. However, the calculations of the most difficult set of diagrams (see figure 1), the second-order self-energy correction (SESE) are yet uncomplete [3‐ 6], which remains as source for major theoretical uncertainties in present Lamb-shift predictions. To close this gap is a challenge for theory. As a step towards this, we present an estimate for the two-photon self-energy contribution. The graph SESE (a) consists of the irreducible and the reducible contributions. Utilizing Feynman gauge the first part can be renormalized and calculated separately since it does not contain infrared divergencies. The irreducible part has been calculated in [3] for large nuclear charge numbers Z and in [4,6] for arbitrary values of Z .F or high-Z systems all results obtained coincide, while for low- and intermediate-Z values a discrepancy between [4] and [6] has been found. In Feynman gauge, the remaining reducible part of the SESE (a) graph and the two diagrams SESE (b) and (c) have to be calculated simultaneously in order to cancel infrared and ultraviolet divergences arising from different diagrams [7]. The general renormalization scheme for all two-photon self-energy diagrams has