The Lamb shift of the $1s$ state in hydrogen: two-loop and three-loop contributions

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 ${\alpha }^{8}m$ 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 $B_{61},B_{60},C_{50}$ in standard notation. We have found a missing logarithmic contribution of order ${\alpha }^2(Z\alpha {)}^{6}m$. We have also obtained leading pure self-energy logarithmic contributions of order ${\alpha }^2(Z\alpha {)}^{8}m$ and ${\alpha }^2(Z\alpha {)}^{9}m$ and estimated the subleading terms of order ${\alpha }^2(Z\alpha {)}^{7}m$, ${\alpha }^2(Z\alpha {)}^{8}m$, and ${\alpha }^2(Z\alpha {)}^{9}m$. 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 ${\alpha }^3(Z\alpha {)}^{5}m$, coefficient $C_{50}$ 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 $(-3.3\pm 10.5)({\alpha }^3/{\pi }^3)(Z\alpha {)}^{5}m$. Altogether, we have completed the evaluation of the logarithmic contributions to the $1s$ Lamb shift of order ${\alpha }^{8}m$ and reduced the overall ${\alpha }^{8}m$ uncertainty by approximately a factor of three for H, D, and He${}^{+}$ as compared with the most recent CODATA compilation.

arXiv comment7 pages, 3 figures

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• High-precision calculation of the 4-loop QED contribution to the slope of the Dirac form factor
• Muonium Lamb shift: theory update and experimental prospects
• Quantum electrodynamics of two-body systems with arbitrary masses
• Sub-part-per-trillion test of the Standard Model with atomic hydrogen
• Two-photon exchange in (muonic) deuterium at N3LO in pionless effective field theory
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• Flat Bundles on Function Manifolds and Evolution Equations in Quantum Field Theories