Calculation of the One- and Two-Loop Lamb Shift for Arbitrary Excited Hydrogenic States

General expressions for quantum electrodynamic corrections to the one-loop self-energy [of order $\alpha (Z\alpha {)}^6$] and for the two-loop Lamb shift [of order ${\alpha }^2(Z\alpha {)}^6$] are derived. The latter includes all diagrams with closed fermion loops. The general results are valid for arbitrary excited non-$S$ hydrogenic states and for the normalized Lamb shift difference of $S$ states, defined as ${\Delta }_n=n^3\Delta E(nS)-\Delta E(1S)$. We present numerical results for one-loop and two-loop corrections for excited $S$, $P$, and $D$ states. In particular, the normalized Lamb shift difference of $S$ states is calculated with an uncertainty of order 0.1 kHz.

Figure 1

Two-loop Feynman diagrams for the Lamb shift. The bound-electron propagator is denoted by a double line.