Analytical Treatment of the Bethe–Salpeter Equation

Abstract

The leading contributions to the helium fine structure beyond the first-order relativistic contribution (NVPA, see, Sect. 2.6) were first derived in 1957 by Araki [1] and Sucher [6, 7], starting from the Bethe–Salpeter (BS) equation [5] and including the nonrelativistic as well as the relativistic momentum regions. Following the approach of Sucher, Douglas and Kroll [2] have derived all terms of order α4 H(artree)1, where no contributions in the relativistic region were found. The same approach was later used by Zhang [8, 12] to derive corrections of order α5logα H and of order α5 H in the nonrelativistic region and recoil corrections to order α4 m ∕ M H (see also [10]). Later some additional effects of order α5 H due to relativistic momenta were found by Zhang and Drake [11]. The radiative parts are treated more rigorously by Zhang in a separate paper [9]. Using a different approach, Pachucki and Sapirstein [4] have derived all contributions of order α5 H and reported some disagreement with the early results of Zhang [8].2