Abstract
We consider the 1s Lamb shift in hydrogen and helium ions, a quantity, required for an accurate determination of the Rydberg constant and the proton charge radius by means of hydrogen spectroscopy, as well as for precision tests of the bound-state QED. The dominant QED contribution to the uncertainty originates from α8m external-field contributions (i.e., the contributions at the non-recoil limit). We discuss the two- and three-loop cases and in particular, we revisit calculations of the coefficients B61,B60,C50 in standard notation.We have found a missing logarithmic contribution of order α2(Zα)6m. We have also obtained leading pure self-energy logarithmic contributions of order α2(Zα)8m and α2(Zα)9m and estimated the subleading terms of order α2(Zα)7m, α2(Zα)8m, and α2(Zα)9m. The determination of those higher-order contributions enabled us to improve the overall accuracy of the evaluation of the two-loop self-energy of the electron.We investigated the asymptotic behavior of the integrand related to the next-to-leading three-loop term (order α3(Zα)5m, coefficient C50 in standard notation) and applied it to approximate integration over the loop momentum. Our result for contributions to the 1s Lamb shift for the total three loop next-to-leading term is (−3.3±10.5)(α3/π3)(Zα)5m.Altogether, we have completed the evaluation of the logarithmic contributions to the 1s Lamb shift of order α8m and reduced the overall α8m uncertainty by approximately a factor of three for H, D, and He+ as compared with the most recent CODATA compilation.
Highlights
Already for a few years, there exists a discrepancy in the determination of the proton charge radius by means of the spectroscopy of ordinary and muonic hydrogen, commonly known as the proton radius puzzle
The largest uncertainty originates from the hydrogen spectroscopy and a serious experimental activity in this direction is in progress
The second largest uncertainty comes from the Quantum electrodynamics (QED) theory of the 1s Lamb shift in hydrogen [2]
Summary
Already for a few years, there exists a discrepancy in the determination of the proton charge radius by means of the spectroscopy of ordinary and muonic hydrogen (see, e.g., [1, 2]), commonly known as the proton radius puzzle. We considered diagrams with triple photon exchange and extracted the logarithmically divergent part; both methods gave the same correction to the B61 coefficient. While considering the non-logarithmic part of the α2(Zα)6m correction, i.e., the coefficient B60 and higherorder terms, one has to distinguish three groups of diagrams and treat them differently. The quantitatively largest contribution in our consideration of the B60 term beyond the free-loop approximation comes as a ‘tail’ of the logarithmic B61 term. There exist essentially three approaches to calculation of the higher-order two-loop contributions. For the higher-order terms, it is possible to estimate the magnitude plausibly, but not the sign of a coefficient and frequently the central values of estimations are zero.
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