Two-loop self-energy correction in hydrogen Lamb shift (Vol. 42, No. 1)

The results of the recent measurement of the Lamb shift in muonic hydrogen [R. Pohl et al., Nature 466, 213 (2010)] created what is now widely known as "the proton charge radius puzzle". The charge radius of the proton derived in this experiment turned out to be 4% smaller than that obtained from the Lamb shift in ordinary hydrogen. This discrepancy is very surprising, given that the underlying theory, QED, is one of the most precise and well-tested fundamental theories. Half a year has passed since the announcement of the unexpected results and theoreticians have checked and double-checked QED calculations in both muonic and ordinary hydrogen. However, despite all the efforts, no plausible ideas about the cause of the discrepancy have been suggested, and the puzzle remains unsolved.

This paper deals with the most problematic QED effect in ordinary hydrogen, the two-loop self-energy correction. It is this effect that induces the main theoretical error in the hydrogen proton charge radius. Previous calculations have demonstrated a disagreement between two different methods: the one based on the perturbative expansion in the binding nuclear field and the one including binding effects to all orders. Though this disagreement is too small to explain the puzzle, it still needs to be resolved to clarify the uncertainty of the hydrogen proton charge radius. In this paper, a new technique is developed for calculations of the two-loop self-energy diagrams treated in the mixed coordinate-momentum space. This technique increases the numerical accuracy of the results, reducing without eliminating the disagreement between the two approaches (Fig.).