Two-Loop Bethe Logarithms for Higher Excited S Levels

Processes mediated by two virtual low-energy photons contribute quite significantly to the energy of hydrogenic S states. The corresponding level shift is of the order of ( α / π )2 ( Z α )6 mec2 and may be ascribed to a two-loop generalization of the Bethe logarithm. For 1 S and 2 S states, the correction has recently been evaluated by Pachucki and Jentschura [Phys. Rev. Lett. 91, 113005 (2003)]. Here, we generalize the approach to higher excited S states, which in contrast to the 1 S and 2 S states can decay to P states via the electric-dipole ( E 1) channel. The more complex structure of the excited-state wave functions and the necessity to subtract P -state poles lead to additional calculational problems. In addition to the calculation of the excited-state two-loop energy shift, we investigate the ambiguity in the energy level definition due to squared decay rates